Articles

12.E: Exercises


1. In each of the following, find matrices (A, x,) and (b) such that the given system of linear equations can be expressed as the single matrix equation (Ax = b.)

[ (a)~~ left. egin{array}{ccccccc} 2x_1 &-& 3x_2& + &5x_3 &= &7 9x_1& - &x_2& +& x_3& =& -1 x_1& + &5x_2& +& 4x_3 &= &0 end{array} ight} ~~~ (b)~~ left. egin{array}{ccccccccc} 4x_1&&& -& 3x_3& +& x_4& =& 1 5x_1& +& x_2&&& -& 8x_4& =& 3 2x_1& - &5x_2& + &9x_3& -& x_4& =& 0 &&3x_2& - &x_3& +& 7x_4& =& 2end{array} ight} ]

2. In each of the following, express the matrix equation as a system of linear equations.

[ (a) left[ egin{array}{ccc} 3 & -1 & 2 4 & 3 & 7 -2& 1 & 5 end{array} ight] left[ egin{array}{c} x_1 x_2 x_3 end{array} ight] = left[ egin{array}{c} 2 -1 4 end{array} ight] ~~~ (b)left[ egin{array}{cccc} 3 & -2 & 0&1 5 & 0 & 2 & -2 3& 1 & 4&7 -2&5&1&6 end{array} ight] left[ egin{array}{c} w x y z end{array} ight] = left[ egin{array}{c} 0 0 0 0 end{array} ight] ]

3. Suppose that (A, B, C, D,) and (E) are matrices over (mathbb{F}) having the following sizes:

[A {it{~is~}} 4 imes 5,~~ B {it{~is~}} 4 imes 5,~~ C {it{~is~}} 5 imes 2,~~ D {it{~is~}} 4 imes 2,]

Determine whether the following matrix expressions are defined, and, for those that are defined, determine the size of the resulting matrix.

[(a)~ BA ~~~(b)~ AC + D ~~~(c)~ AE + B~~~ (d)~ AB + B~~~ (e)~E(A + B)~~~ (f) E(AC)]

4. Suppose that (A, B, C, D,) and (E) are the following matrices:

[ A=left[ egin{array}{cc} 3 & 0 -1 & 2 1&1 end{array} ight],~ B= left[ egin{array}{cc} 4 & -1 0 & 2 end{array} ight], ~ C= left[ egin{array}{ccc} 1 & 4 &2 3 & 1&5 end{array} ight], D= left[ egin{array}{ccc} 1 & 5 &2 -1 & 0 & 1 3& 2 & 4end{array} ight], {it{~and ~}}E= left[egin{array}{ccc} 6 & 1 &3 -1 & 1 & 2 4& 1 & 3end{array} ight]. ]

Determine whether the following matrix expressions are defined, and, for those that are defined, compute the resulting matrix.

((a)~ D + E~~ (b)~ D - E~~ (c)~ 5A~~ (d)~ -7C~~ (e)~ 2B - C
(f)~ 2E - 2D~~ (g)~ -3(D + 2E)~~ (h)~A - A~~ (i)~ AB~~ (j)~ BA
(k)~ (3E)D~~ (l)~ (AB)C ~~(m)~ A(BC)~~ (n)~(4B)C + 2B ~~(o)~ D - 3E
(p)~ CA + 2E ~~(q)~ 4E - D ~~(r)~ DD)

5. Suppose that (A, B,) and (C) are the following matrices and that (a = 4) and (b = 7.)

[ A= left[ egin{array}{ccc} 1 & 5 & 2 -1 & 0 & 1 3 & 2 & 4 end{array} ight],B = left[ egin{array}{ccc} 6 & 1 & 3 -1 & 1 & 2 4 & 1 & 3 end{array} ight], {it{~and~}} C = left[ egin{array}{ccc} 1 & 5 & 2 -1 & 0 & 1 3 & 2 & 4 end{array} ight]. ]

Verify computationally that
( (a)~ A + (B + C) = (A + B) + C ~~~(b) ~(AB)C = A(BC)
(c)~ (a + b)C = aC + bC ~~~(d)~ a(B - C) = aB - aC
(e)~ a(BC) = (aB)C = B(aC) ~~~(f)A(B - C) = AB - AC
(g)~ (B + C)A = BA + CA ~~~(h) a(bC) = (ab)C
(i)~ B - C = -C + B)

6. Suppose that (A) is the matrix
[A=left[ egin{array}{cc} 3 & 1 2 & 1 end{array} ight] ]
Compute (p(A)), where (p(z)) is given by
((a)~ p(z) = z - 2 ~~~(b)~ p(z) = 2z^2 - z + 1
(c)~ p(z) = z^3 - 2z + 4~~~ (d)~ p(z) = z^2 - 4z + 1)

7. Define matrices (A, B, C, D,) and (E) by

[ A=left[ egin{array}{cc} 3 & 1 2 & 1 end{array} ight],~ B= left[ egin{array}{cc} 4 & -1 0 & 2 end{array} ight], ~ C= left[ egin{array}{ccc} 2 & -3 &5 9 & -1&1 1&5&4end{array} ight], D= left[ egin{array}{ccc} 1 & 5 &2 -1 & 0 & 1 3& 2 & 4end{array} ight], {it{~and ~}}E= left[egin{array}{ccc} 6 & 1 &3 -1 & 1 & 2 4& 1 & 3end{array} ight]. ]

(a) Factor each matrix into a product of elementary matrices and an RREF matrix.
(b) Find, if possible, the LU-factorization of each matrix.
(c) Determine whether or not each of these matrices is invertible, and, if possible, compute the inverse.

8. Suppose that (A, B, C, D,) and (E) are the following matrices:

[ A=left[ egin{array}{cc} 3 & 0 -1 & 2 1&1 end{array} ight],~ B= left[ egin{array}{cc} 4 & -1 0 & 2 end{array} ight], ~ C= left[ egin{array}{ccc} 1 & 4 &2 3 & 1&5 end{array} ight], D= left[ egin{array}{ccc} 1 & 5 &2 -1 & 0 & 1 3& 2 & 4end{array} ight], {it{~and ~}}E= left[egin{array}{ccc} 6 & 1 &3 -1 & 1 & 2 4& 1 & 3end{array} ight]. ]

Determine whether the following matrix expressions are defined, and, for those that are defined, compute the resulting matrix.

( (a)~ 2A^T + C~~~ (b)~ D^T - E^T~~~ (c)~ (D - E)^T
(d)~ B^T + 5C^T~~~ (e) ~frac{1}{2}C^T - frac{1}{4}A~~~ (f)~ B B^T
(g) ~3E^T - 3D^T~~~ (h)~ (2E^T - 3D^T )^T~~~ (i)~ CC^T
(j)~ (DA)^T~~~ (k)~ (C^TB)A^T~~~ (l)~ (2D^T - E)A
(m)~ (BA^T - 2C)^T~~~ (n)~ B^T (CC^T - A^TA)~~~ (o)~D^TE^T - (ED)^T
(p)~ trace(DD^T)~~~ (q)~trace(4E^T - D)~~~ (r)~trace(C^TA^T + 2E^T ) )

1. Let (n in mathbb{Z}_+) be a positive integer and (a_{i,j} in mathbb{F}) be scalars for (i, j = 1, ldots , n.) Prove that
the following two statements are equivalent:
(a) The trivial solution (x_1 = cdots = x_n = 0) is the only solution to the homogeneous system of equations
[ left. egin{array}{ccc} sum_{k=1}^{n} a_{1,k}x_k & = & 0 & vdots & sum_{k=1}^{n} a_{n,k}x_k & = & 0 end{array} ight}. ]

(b) For every choice of scalars (c_1 , ldots , c_n in mathbb{F},) there is a solution to the system of equations [ left. egin{array}{ccc} sum_{k=1}^{n} a_{1,k}x_k & = & c_1 & vdots & sum_{k=1}^{n} a_{n,k}x_k & = & c_n end{array} ight}. ]
2. Let (A) and (B) be any matrices.
(a) Prove that if both (AB) and (BA) are defined, then (AB) and (BA) are both square matrices.
(b) Prove that if (A) has size (m imes n) and (ABA) is defined, then (B) has size (n imes m.)
3. Suppose that (A) is a matrix satisfying (A^T A = A.) Prove that (A) is then a symmetric matrix and that (A = A^2 .)
4. Suppose (A) is an upper triangular matrix and that (p(z)) is any polynomial. Prove or give a counterexample: (p(A)) is a upper triangular matrix.


12.E: Correlations (Exercises)

  • Contributed by Foster et al.
  • University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus
  • Sourced from University of Missouri&rsquos Affordable and Open Access Educational Resources Initiative

Correlations assess the linear relation between two continuous variables

  1. What are the three characteristics of a correlation coefficient?
  2. What is the difference between covariance and correlation?

Covariance is an unstandardized measure of how related two continuous variables are. Correlations are standardized versions of covariance that fall between negative 1 and positive 1.

  1. Why is it important to visualize correlational data in a scatterplot before performing analyses?
  2. What sort of relation is displayed in the scatterplot below?

Strong, positive, linear relation

  1. What is the direction and magnitude of the following correlation coefficients
    1. -0.81
    2. 0.40
    3. 0.15
    4. -0.08
    5. 0.29

    Your scatterplot should look similar to this:

    1. In the following correlation matrix, what is the relation (number, direction, and magnitude) between&hellip
      1. Pay and Satisfaction
      2. Stress and Health
      1. Using the data from problem 7, test for a statistically significant relation between the variables.

      Step 1: (H_0: &rho = 0), &ldquoThere is no relation between time spent studying and overall performance in class&rdquo, (H_A: &rho > 0), &ldquoThere is a positive relation between time spent studying and overall performance in class.&rdquo

      Step 2: (df = 15 &ndash 2 = 13, &alpha = 0.05), 1-tailed test, (r^* = 0.441).

      Step 3: Using the Sum of Products table, you should find: (overline = 1.61, SS_X = 17.44, overline= 2.95, SS_Y = 13.60, SP = 10.06, r = 0.65).

      Step 4: Obtained statistic is greater than critical value, reject (H_0). There is a statistically significant, strong, positive relation between time spent studying and performance in class, (r(13) = 0.65, p < .05).

      1. A researcher collects data f rom 100 people to assess whether there is any relation between level of education and levels of civic engagement. The researcher finds the following descriptive values: (overline= 4.02, s_x = 1.15, overline= 15.92, s_y = 5.01, SS_X = 130.93, SS_Y = 2484.91, SP = 159.39). Test for a significant relation using the four step hypothesis testing procedure.

      12.E: Electromagnetism (Exercises)

      • Contributed by Benjamin Crowell
      • Professor (Physics) at Fullerton College

      1. A particle with a charge of 1.0 C and a mass of 1.0 kg is observed moving past point P with a velocity ((1.0 ext/ ext)hat>). The electric field at point P is ((1.0 ext/ ext)hat>), and the magnetic field is ((2.0 ext)hat>). Find the force experienced by the particle.(answer check available at lightandmatter.com)

      2. For a positively charged particle moving through a magnetic field, the directions of the (mathbf), (mathbf), and (mathbf) vectors are related by a right-hand rule:

      Make a three-dimensional model of the three vectors using pencils or rolled-up pieces of paper to represent the vectors assembled with their tails together. Make all three vectors perpendicular to each other. Now write down every possible way in which the rule could be rewritten by scrambling up the three symbols (mathbf), (mathbf), and (mathbf). Referring to your model, which are correct and which are incorrect?

      3. A charged particle is released from rest. We see it start to move, and as it gets going, we notice that its path starts to curve. Can we tell whether this region of space has (mathbf eq 0), or (mathbf eq 0), or both? Assume that no other forces are present besides the possible electrical and magnetic ones, and that the fields, if they are present, are uniform.

      4. A charged particle is in a region of space in which there is a uniform magnetic field (mathbf=Bhat>). There is no electric field, and no other forces act on the particle. In each case, describe the future motion of the particle, given its initial velocity.

      5. (a) A line charge, with charge per unit length (lambda), moves at velocity (v) along its own length. How much charge passes a given point in time (dt)? What is the resulting current? hwans
      (b) Show that the units of your answer in part a work out correctly.

      6. Two parallel wires of length (L) carry currents (I_1) and (I_2). They are separated by a distance (R), and we assume (R) is much less than (L), so that our results for long, straight wires are accurate. The goal of this problem is to compute the magnetic forces acting between the wires.
      (a) Neither wire can make a force on itself. Therefore, our first step in computing wire 1's force on wire 2 is to find the magnetic field made only by wire 1, in the space occupied by wire 2. Express this field in terms of the given quantities.(answer check available at lightandmatter.com)
      (b) Let's model the current in wire 2 by pretending that there is a line charge inside it, possessing density per unit length (lambda_2) and moving at velocity (v_2). Relate (lambda_2) and (v_2) to the current (I_2), using the result of problem 5a. Now find the magnetic force wire 1 makes on wire 2, in terms of (I_1), (I_2), (L), and (R). hwans
      (c) Show that the units of the answer to part b work out to be newtons.

      7. Suppose a charged particle is moving through a region of space in which there is an electric field perpendicular to its velocity vector, and also a magnetic field perpendicular to both the particle's velocity vector and the electric field. Show that there will be one particular velocity at which the particle can be moving that results in a total force of zero on it. Relate this velocity to the magnitudes of the electric and magnetic fields. (Such an arrangement, called a velocity filter, is one way of determining the speed of an unknown particle.)

      8. The following data give the results of two experiments in which charged particles were released from the same point in space, and the forces on them were measured:

      q1 = 1 µ C (mathbf_1=(1 m/s)mathbf) (mathbf_1=(-1 mN)mathbf)
      q2 ​= - 2 µ C (mathbf_2=(-1 m/s)mathbf)​ (mathbf_2=(-2 mN)mathbf)

      The data are insufficient to determine the magnetic field vector demonstrate this by giving two different magnetic field vectors, both of which are consistent with the data.

      9. The following data give the results of two experiments in which charged particles were released from the same point in space, and the forces on them were measured:

      q1 = 1 nC (mathbf_1=(1 m/s)mathbf) (mathbf_1=(5 mN)mathbf+(2 mN)mathbf)
      q2 ​= 1 nC (mathbf_2=(3 m/s)mathbf)​ (mathbf_2=(10 mN)mathbf+(4 mN)mathbf)

      Is there a nonzero electric field at this point? A nonzero magnetic field?

      10. This problem is a continuation of problem 6. Note that the answer to problem 6b is given on page 930.

      1. Interchanging the 1's and 2's in the answer to problem 6b, what is the magnitude of the magnetic force from wire 2 acting on wire 1? Is this consistent with Newton's third law?
      2. Suppose the currents are in the same direction. Make a sketch, and use the right-hand rule to determine whether wire 1 pulls wire 2 towards it, or pushes it away.
      3. Apply the right-hand rule again to find the direction of wire 2's force on wire 1. Does this agree with Newton's third law?
      4. What would happen if wire 1's current was in the opposite direction compared to wire 2's?

      11. (a) In the photo of the vacuum tube apparatus in figure o on page 656, infer the direction of the magnetic field from the motion of the electron beam. (The answer is given in the answer to the self-check on that page.)
      (b) Based on your answer to part a, find the direction of the currents in the coils.
      (c) What direction are the electrons in the coils going?
      (d) Are the currents in the coils repelling the currents consisting of the beam inside the tube, or attracting them? Check your answer by comparing with the result of problem 10.

      12. A charged particle of mass (m) and charge (q) moves in a circle due to a uniform magnetic field of magnitude (B), which points perpendicular to the plane of the circle.

      1. Assume the particle is positively charged. Make a sketch showing the direction of motion and the direction of the field, and show that the resulting force is in the right direction to produce circular motion.
      2. Find the radius, (r), of the circle, in terms of (m), (q), (v), and (B).(answer check available at lightandmatter.com)
      3. Show that your result from part b has the right units.
      4. Discuss all four variables occurring on the right-hand side of your answer from part b. Do they make sense? For instance, what should happen to the radius when the magnetic field is made stronger? Does your equation behave this way?
      5. Restate your result so that it gives the particle's angular frequency, (omega), in terms of the other variables, and show that (v) drops out.(answer check available at lightandmatter.com)

      13. Each figure represents the motion of a positively charged particle. The dots give the particles' positions at equal time intervals. In each case, determine whether the motion was caused by an electric force, a magnetic force, or a frictional force, and explain your reasoning. If possible, determine the direction of the magnetic or electric field. All fields are uniform. In (a), the particle stops for an instant at the upper right, but then comes back down and to the left, retracing the same dots. In (b), it stops on the upper right and stays there.

      14. One model of the hydrogen atom has the electron circling around the proton at a speed of (2.2 imes10^6) m/s, in an orbit with a radius of 0.05 nm. (Although the electron and proton really orbit around their common center of mass, the center of mass is very close to the proton, since it is 2000 times more massive. For this problem, assume the proton is stationary.)

      1. Treat the circling electron as a current loop, and calculate the current.
      2. Estimate the magnetic field created at the center of the atom by the electron.(answer check available at lightandmatter.com)
      3. Does the proton experience a nonzero force from the electron's magnetic field? Explain.
      4. Does the electron experience a magnetic field from the proton? Explain.
      5. Does the electron experience a magnetic field created by its own current? Explain.
      6. Is there an electric force acting between the proton and electron? If so, calculate it.(answer check available at lightandmatter.com)
      7. Is there a gravitational force acting between the proton and electron? If so, calculate it.
      8. An inward force is required to keep the electron in its orbit -- otherwise it would obey Newton's first law and go straight, leaving the atom. Based on your answers to the previous parts, which force or forces (electric, magnetic and gravitational) contributes significantly to this inward force? (Based on a problem by Arnold Arons.)

      15. The equation (B_z=eta kIA/c^2r^3) was found on page 666 for the distant field of a dipole. Show, as asserted there, that the constant (eta) must be unitless.

      16. The following data give the results of three experiments in which charged particles were released from the same point in space, and the forces on them were measured:

      q1 = 1 C (mathbf_1=0) (mathbf_1=(1 N)mathbf)
      q2 ​=1 C (mathbf_2=(1 m/s)mathbf)​ (mathbf_2=(1 N)mathbf)
      q3 ​=1 C (mathbf_3=(1 m/s)mathbf)​ (mathbf_3=0)

      Determine the electric and magnetic fields.(answer check available at lightandmatter.com)

      17. If you put four times more current through a solenoid, how many times more energy is stored in its magnetic field?(answer check available at lightandmatter.com)

      18. A Helmholtz coil is defined as a pair of identical circular coils lying in parallel planes and separated by a distance, (h), equal to their radius, (b). (Each coil may have more than one turn of wire.) Current circulates in the same direction in each coil, so the fields tend to reinforce each other in the interior region. This configuration has the advantage of being fairly open, so that other apparatus can be easily placed inside and subjected to the field while remaining visible from the outside. The choice of (h=b) results in the most uniform possible field near the center. A photograph of a Helmholtz coil was shown in figure o on page 656.
      (a) Find the percentage drop in the field at the center of one coil, compared to the full strength at the center of the whole apparatus. (answer check available at lightandmatter.com)
      (b) What value of (h) (not equal to (b)) would make this percentage difference equal to zero?(answer check available at lightandmatter.com)

      19. The figure shows a nested pair of circular wire loops used to create magnetic fields. (The twisting of the leads is a practical trick for reducing the magnetic fields they contribute, so the fields are very nearly what we would expect for an ideal circular current loop.) The coordinate system below is to make it easier to discuss directions in space. One loop is in the (y-z) plane, the other in the (x-y) plane. Each of the loops has a radius of 1.0 cm, and carries 1.0 A in the direction indicated by the arrow.

      1. Calculate the magnetic field that would be produced by one such loop, at its center. (answer check available at lightandmatter.com)
      2. Describe the direction of the magnetic field that would be produced, at its center, by the loop in the (x-y) plane alone.
      3. Do the same for the other loop.
      4. Calculate the magnitude of the magnetic field produced by the two loops in combination, at their common center. Describe its direction.(answer check available at lightandmatter.com)

      20. Four long wires are arranged, as shown, so that their cross-section forms a square, with connections at the ends so that current flows through all four before exiting. Note that the current is to the right in the two back wires, but to the left in the front wires. If the dimensions of the cross-sectional square (height and front-to-back) are (b), find the magnetic field (magnitude and direction) along the long central axis.(answer check available at lightandmatter.com)

      21. In problem 16, the three experiments gave enough information to determine both fields. Is it possible to design a procedure so that, using only two such experiments, we can always find (mathbf) and (mathbf)? If so, design it. If not, why not?

      22. Use the Biot-Savart law to derive the magnetic field of a long, straight wire, and show that this reproduces the result of example 6 on page 658.

      23. (a) Modify the calculation on page 663 to determine the component of the magnetic field of a sheet of charge that is perpendicular to the sheet.(answer check available at lightandmatter.com)
      (b) Show that your answer has the right units.
      (c) Show that your answer approaches zero as (z) approaches infinity.
      (d) What happens to your answer in the case of (a=b)? Explain why this makes sense.

      24. Consider two solenoids, one of which is smaller so that it can be put inside the other. Assume they are long enough so that each one only contributes significantly to the field inside itself, and the interior fields are nearly uniform. Consider the configuration where the small one is inside the big one with their currents circulating in the same direction, and a second configuration in which the currents circulate in opposite directions. Compare the energies of these configurations with the energy when the solenoids are far apart. Based on this reasoning, which configuration is stable, and in which configuration will the little solenoid tend to get twisted around or spit out? hwhint

      25. (a) A solenoid can be imagined as a series of circular current loops that are spaced along their common axis. Integrate the result of example 12 on page 671 to show that the field on the axis of a solenoid can be written as (B=(2pi keta/c^2)(coseta+cosgamma)), where the angles (eta) and (gamma) are defined in the figure.
      (b) Show that in the limit where the solenoid is very long, this exact result agrees with the approximate one derived in example 13 on page 674 using Ampère's law.
      (c) Note that, unlike the calculation using Ampère's law, this one is valid at points that are near the mouths of the solenoid, or even outside it entirely. If the solenoid is long, at what point on the axis is the field equal to one half of its value at the center of the solenoid?
      (d) What happens to your result when you apply it to points that are very far away from the solenoid? Does this make sense?

      26. The first step in the proof of Ampère's law on page 675 is to show that Ampère's law holds in the case shown in figure f/1, where a circular Ampèrian loop is centered on a long, straight wire that is perpendicular to the plane of the loop. Carry out this calculation, using the result for the field of a wire that was established without using Ampère's law.

      27. A certain region of space has a magnetic field given by (mathbf=bxhat>). Find the electric current flowing through the square defined by (z=0), (0le x le a), and (0le y le a).(answer check available at lightandmatter.com)

      f / A nautilus shell is approximately a logarithmic spiral, of the type in problem 28.

      28. Perform a calculation similar to the one in problem 54, but for a logarithmic spiral, defined by (r=we^), and show that the field is (B=(kI/c^2u)(1/a-1/b)). Note that the solution to problem 54 is given in the back of the book.

      29. (a) For the geometry described in example 8 on page 661, find the field at a point the lies in the plane of the wires, but not between the wires, at a distance (b) from the center line. Use the same technique as in that example.
      (b) Now redo the calculation using the technique demonstrated on page 666. The integrals are nearly the same, but now the reasoning is reversed: you already know (eta=1), and you want to find an unknown field. The only difference in the integrals is that you are tiling a different region of the plane in order to mock up the currents in the two wires. Note that you can't tile a region that contains a point of interest, since the technique uses the field of a distant dipole.(answer check available at lightandmatter.com)

      30. (a) A long, skinny solenoid consists of (N) turns of wire wrapped uniformly around a hollow cylinder of length (ell) and cross-sectional area (A). Find its inductance.(answer check available at lightandmatter.com)
      (b) Show that your answer has the right units to be an inductance.

      31. Consider two solenoids, one of which is smaller so that it can be put inside the other. Assume they are long enough to act like ideal solenoids, so that each one only contributes significantly to the field inside itself, and the interior fields are nearly uniform. Consider the configuration where the small one is partly inside and partly hanging out of the big one, with their currents circulating in the same direction. Their axes are constrained to coincide.
      (a) Find the difference in the magnetic energy between the configuration where the solenoids are separate and the configuration where the small one is inserted into the big one. Your equation will include the length (x) of the part of the small solenoid that is inside the big one, as well as other relevant variables describing the two solenoids.(answer check available at lightandmatter.com)
      (b) Based on your answer to part a, find the force acting between the solenoids.(answer check available at lightandmatter.com)

      32. Verify Ampère's law in the case shown in the figure, assuming the known equation for the field of a wire. A wire carrying current (I) passes perpendicularly through the center of the rectangular Ampèrian surface. The length of the rectangle is infinite, so it's not necessary to compute the contributions of the ends.

      33. The purpose of this problem is to find how the gain of a transformer depends on its construction.
      (a) The number of loops of wire, (N), in a solenoid is changed, while keeping the length constant. How does the impedance depend on (N)? State your answer as a proportionality, e.g., (Zpropto N^3) or (Zpropto N^<-5>).
      (b) For a given AC voltage applied across the inductor, how does the magnetic field depend on (N)? You need to take into account both the dependence of a solenoid's field on (N) for a given current and your answer to part a, which affects the current.
      (c) Now consider a transformer consisting of two solenoids. The input side has (N_1) loops, and the output (N_2). We wish to find how the output voltage (V_2) depends on (N_1), (N_2), and the input voltage (V_1). The text has already established (V_2propto V_1N_2), so it only remains to find the dependence on (N_1). Use your result from part b to accomplish this. The ratio (V_2/V_1) is called the voltage gain.

      34. Problem 33 dealt with the dependence of a transformer's gain on the number of loops of wire in the input solenoid. Carry out a similar analysis of how the gain depends on the frequency at which the circuit is operated.

      35. A U-shaped wire makes electrical contact with a second, straight wire, which rolls along it to the right, as shown in the figure. The whole thing is immersed in a uniform magnetic field, which is perpendicular to the plane of the circuit. The resistance of the rolling wire is much greater than that of the U.
      (a) Find the direction of the force on the wire based on conservation of energy.
      (b) Verify the direction of the force using right-hand rules.
      (c) Find magnitude of the force acting on the wire. There is more than one way to do this, but please do it using Faraday's law (which works even though it's the Ampèrian surface itself that is changing, rather than the field).(answer check available at lightandmatter.com)
      (d) Consider how the answer to part a would have changed if the direction of the field had been reversed, and also do the case where the direction of the rolling wire's motion is reversed. Verify that this is in agreement with your answer to part c.

      36. A charged particle is in motion at speed (v), in a region of vacuum through which an electromagnetic wave is passing. In what direction should the particle be moving in order to minimize the total force acting on it? Consider both possibilities for the sign of the charge. (Based on a problem by David J. Raymond.)

      37. A wire loop of resistance (R) and area (A), lying in the (y-z) plane, falls through a nonuniform magnetic field (mathbf=kzhat>), where (k) is a constant. The (z) axis is vertical.
      (a) Find the direction of the force on the wire based on conservation of energy.
      (b) Verify the direction of the force using right-hand rules.
      (c) Find the magnetic force on the wire.(answer check available at lightandmatter.com)

      38. A capacitor has parallel plates of area (A), separated by a distance (h). If there is a vacuum between the plates, then Gauss's law gives (E=4pi ksigma=4pi kq/A) for the field between the plates, and combining this with (E=V/h), we find (C=q/V=(1/4pi k)A/h). (a) Generalize this derivation to the case where there is a dielectric between the plates. (b) Suppose we have a list of possible materials we could choose as dielectrics, and we wish to construct a capacitor that will have the highest possible energy density, (U_e/v), where (v) is the volume. For each dielectric, we know its permittivity (epsilon), and also the maximum electric field (E) it can sustain without breaking down and allowing sparks to cross between the plates. Write the maximum energy density in terms of these two variables, and determine a figure of merit that could be used to decide which material would be the best choice.

      39. (a) For each term appearing on the right side of Maxwell's equations, give an example of an everyday situation it describes.
      (b) Most people doing calculations in the SI system of units don't use (k) and (k/c^2). Instead, they express everything in terms of the constants

      Rewrite Maxwell's equations in terms of these constants, eliminating (k) and (c) everywhere.

      40. (a) Prove that in an electromagnetic plane wave, half the energy is in the electric field and half in the magnetic field.
      (b) Based on your result from part a, find the proportionality constant in the relation (dmathbf

      proptomathbf imesmathbfdv), where (d mathbf

      ) is the momentum of the part of a plane light wave contained in the volume (dv). The vector (mathbf imesmathbf) is known as the Poynting vector. (To do this problem, you need to know the relativistic relationship between the energy and momentum of a beam of light.)(answer check available at lightandmatter.com)

      41. (a) A beam of light has cross-sectional area (A) and power (P), i.e., (P) is the number of joules per second that enter a window through which the beam passes. Find the energy density (U/v) in terms of (P), (A), and universal constants.
      (b) Find ( ilde>) and ( ilde>), the amplitudes of the electric and magnetic fields, in terms of (P), (A), and universal constants (i.e., your answer should not include (U) or (v)). You will need the result of problem 40a. A real beam of light usually consists of many short wavetrains, not one big sine wave, but don't worry about that.(answer check available at lightandmatter.com)hwhint
      (c) A beam of sunlight has an intensity of (P/A=1.35 imes10^3 ext/ ext^2), assuming no clouds or atmospheric absorption. This is known as the solar constant. Compute ( ilde>) and ( ilde>), and compare with the strengths of static fields you experience in everyday life: (E sim 10^6 ext/ ext) in a thunderstorm, and (B sim 10^<-3>) T for the Earth's magnetic field.(answer check available at lightandmatter.com)

      42. The circular parallel-plate capacitor shown in the figure is being charged up over time, with the voltage difference across the plates varying as (V=st), where (s) is a constant. The plates have radius (b), and the distance between them is (d). We assume (d ll b), so that the electric field between the plates is uniform, and parallel to the axis. Find the induced magnetic field at a point between the plates, at a distance (R) from the axis. hwhint(answer check available at lightandmatter.com)

      43. A positively charged particle is released from rest at the origin at (t=0), in a region of vacuum through which an electromagnetic wave is passing. The particle accelerates in response to the wave. In this region of space, the wave varies as (mathbf=hat> ildesinomega t), (mathbf=hat> ildesinomega t), and we assume that the particle has a relatively large value of (m/q), so that its response to the wave is sluggish, and it never ends up moving at any speed comparable to the speed of light. Therefore we don't have to worry about the spatial variation of the wave we can just imagine that these are uniform fields imposed by some external mechanism on this region of space.
      (a) Find the particle's coordinates as functions of time.(answer check available at lightandmatter.com)
      (b) Show that the motion is confined to (-z_leq z leq z_), where (z_ = 1.101left(q^2 ilde ilde/m^2omega^3 ight)).

      44. Electromagnetic waves are supposed to have their electric and magnetic fields perpendicular to each other. (Throughout this problem, assume we're talking about waves traveling through a vacuum, and that there is only a single sine wave traveling in a single direction, not a superposition of sine waves passing through each other.) Suppose someone claims they can make an electromagnetic wave in which the electric and magnetic fields lie in the same plane. Prove that this is impossible based on Maxwell's equations.

      45. Repeat the self-check on page 710, but with one change in the procedure: after we charge the capacitor, we open the circuit, and then continue with the observations.

      46. On page 713, I proved that (mathbf_=mathbf_) at the boundary between two substances if there is no free current and the fields are static. In fact, each of Maxwell's four equations implies a constraint with a similar structure. Some are constraints on the field components parallel to the boundary, while others are constraints on the perpendicular parts. Since some of the fields referred to in Maxwell's equations are the electric and magnetic fields (mathbf) and (mathbf), while others are the auxiliary fields (mathbf) and (mathbf), some of the constraints deal with (mathbf) and (mathbf), others with (mathbf) and (mathbf). Find the other three constraints.

      47. (a) Figure j on page 714 shows a hollow sphere with (mu/mu_ ext=x), inner radius (a), and outer radius (b), which has been subjected to an external field (mathbf_ ext). Finding the fields on the exterior, in the shell, and on the interior requires finding a set of fields that satisfies five boundary conditions: (1) far from the sphere, the field must approach the constant (mathbf_ ext) (2) at the outer surface of the sphere, the field must have (mathbf_=mathbf_), as discussed on page 713 (3) the same constraint applies at the inner surface of the sphere (4) and (5) there is an additional constraint on the fields at the inner and outer surfaces, as found in problem 46. The goal of this problem is to find the solution for the fields, and from it, to prove that the interior field is uniform, and given by

      This is a very difficult problem to solve from first principles, because it's not obvious what form the fields should have, and if you hadn't been told, you probably wouldn't have guessed that the interior field would be uniform. We could, however, guess that once the sphere becomes polarized by the external field, it would become a dipole, and at (rgg b), the field would be a uniform field superimposed on the field of a dipole. It turns out that even close to the sphere, the solution has exactly this form. In order to complete the solution, we need to find the field in the shell ((alt rlt b)), but the only way this field could match up with the detailed angular variation of the interior and exterior fields would be if it was also a superposition of a uniform field with a dipole field. The final result is that we have four unknowns: the strength of the dipole component of the external field, the strength of the uniform and dipole components of the field within the shell, and the strength of the uniform interior field. These four unknowns are to be determined by imposing constraints (2) through (5) above.
      (b) Show that the expression from part a has physically reasonable behavior in its dependence on (x) and (a/b).

      48. Two long, parallel strips of thin metal foil form a configuration like a long, narrow sandwich. The air gap between them has height (h), the width of each strip is (w), and their length is (ell). Each strip carries current (I), and we assume for concreteness that the currents are in opposite directions, so that the magnetic force, (F), between the strips is repulsive.
      (a) Find the force in the limit of (wgg h).(answer check available at lightandmatter.com)
      (b) Find the force in the limit of (wll h), which is like two ordinary wires.
      (c) Discuss the relationship between the two results.

      49. Suppose we are given a permanent magnet with a complicated, asymmetric shape. Describe how a series of measurements with a magnetic compass could be used to determine the strength and direction of its magnetic field at some point of interest. Assume that you are only able to see the direction to which the compass needle settles you cannot measure the torque acting on it.

      50. On page 680, the curl of (xhat>) was computed. Now consider the fields (xhat>) and (yhat>).
      (a) Sketch these fields.
      (b) Using the same technique of explicitly constructing a small square, prove that their curls are both zero. Do not use the component form of the curl this was one step in deriving the component form of the curl.

      51. If you watch a movie played backwards, some vectors reverse their direction. For instance, people walk backwards, with their velocity vectors flipped around. Other vectors, such as forces, keep the same direction, e.g., gravity still pulls down. An electric field is another example of a vector that doesn't turn around: positive charges are still positive in the time-reversed universe, so they still make diverging electric fields, and likewise for the converging fields around negative charges.
      (a) How does the momentum of a material object behave under time-reversal?(solution in the pdf version of the book)
      (b) The laws of physics are still valid in the time-reversed universe. For example, show that if two material objects are interacting, and momentum is conserved, then momentum is still conserved in the time-reversed universe.(solution in the pdf version of the book)
      (c) Discuss how currents and magnetic fields would behave under time reversal. hwhint
      (d) Similarly, show that the equation (dmathbf

      proptomathbf imesmathbf) is still valid under time reversal.

      52. This problem is a more advanced exploration of the time-reversal ideas introduced in problem 51.
      (a) In that problem, we assumed that charge did not flip its sign under time reversal. Suppose we make the opposite assumption, that charge does change its sign. This is an idea introduced by Richard Feynman: that antimatter is really matter traveling backward in time! Determine the time-reversal properties of (mathbf) and (mathbf) under this new assumption, and show that (dmathbf

      proptomathbf imesmathbf) is still valid under time-reversal.
      (b) Show that Maxwell's equations are time-reversal symmetric, i.e., that if the fields (mathbf(x,y,z,t)) and (mathbf(x,y,z,t)) satisfy Maxwell's equations, then so do (mathbf(x,y,z,-t)) and (mathbf(x,y,z,-t)). Demonstrate this under both possible assumptions about charge, (q ightarrow q) and (q ightarrow -q).

      53. The purpose of this problem is to prove that the constant of proportionality (a) in the equation (dU_m=aB^2 dv), for the energy density of the magnetic field, is given by (a=c^2/8pi k) as asserted on page 665. The geometry we'll use consists of two sheets of current, like a sandwich with nothing in between but some vacuum in which there is a magnetic field. The currents are in opposite directions, and we can imagine them as being joined together at the ends to form a complete circuit, like a tube made of paper that has been squashed almost flat. The sheets have lengths (L) in the direction parallel to the current, and widths (w). They are separated by a distance (d), which, for convenience, we assume is small compared to (L) and (w). Thus each sheet's contribution to the field is uniform, and can be approximated by the expression (2pi keta/c^2).
      (a) Make a drawing similar to the one in figure 11.2.1 on page 664, and show that in this opposite-current configuration, the magnetic fields of the two sheets reinforce in the region between them, producing double the field, but cancel on the outside.
      (b) By analogy with the case of a single strand of wire, one sheet's force on the other is (ILB_1), were (I=eta w) is the total current in one sheet, and (B_1=B/2) is the field contributed by only one of the sheets, since the sheet can't make any net force on itself. Based on your drawing and the right-hand rule, show that this force is repulsive.
      For the rest of the problem, consider a process in which the sheets start out touching, and are then separated to a distance (d). Since the force between the sheets is repulsive, they do mechanical work on the outside world as they are separated, in much the same way that the piston in an engine does work as the gases inside the cylinder expand. At the same time, however, there is an induced emf which would tend to extinguish the current, so in order to maintain a constant current, energy will have to be drained from a battery. There are three types of energy involved: the increase in the magnetic field energy, the increase in the energy of the outside world, and the decrease in energy as the battery is drained. (We assume the sheets have very little resistance, so there is no ohmic heating involved.)(answer check available at lightandmatter.com)
      (c) Find the mechanical work done by the sheets, which equals the increase in the energy of the outside world. Show that your result can be stated in terms of (eta), the final volume (v=wLd), and nothing else but numerical and physical constants.(answer check available at lightandmatter.com)
      (d) The power supplied by the battery is (P=IGamma_E) (like (P=IDelta V), but with an emf instead of a voltage difference), and the circulation is given by (Gamma=-dPhi_B/dt). The negative sign indicates that the battery is being drained. Calculate the energy supplied by the battery, and, as in part c, show that the result can be stated in terms of (eta), (v), and universal constants.(answer check available at lightandmatter.com)
      (e) Find the increase in the magnetic-field energy, in terms of (eta), (v), and the unknown constant (a).(answer check available at lightandmatter.com)
      (f) Use conservation of energy to relate your answers from parts c, d, and e, and solve for (a).(answer check available at lightandmatter.com)

      54. Magnet coils are often wrapped in multiple layers. The figure shows the special case where the layers are all confined to a single plane, forming a spiral. Since the thickness of the wires (plus their insulation) is fixed, the spiral that results is a mathematical type known as an Archimedean spiral, in which the turns are evenly spaced. The equation of the spiral is (r=w heta), where (w) is a constant. For a spiral that starts from (r=a) and ends at (r=b), show that the field at the center is given by ((kI/c^2w)ln b/a).(solution in the pdf version of the book)


      16.4: Free Energy

      Q16.4.1

      What is the difference between &DeltaG, &DeltaG°, and (&DeltaG^circ_<298>) for a chemical change?

      Q16.4.2

      A reactions has (&DeltaH^circ_<298>) = 100 kJ/mol and (&DeltaS^circ_<298>= extrm<250 J/mol&sdotK>). Is the reaction spontaneous at room temperature? If not, under what temperature conditions will it become spontaneous?

      S16.4.2

      The reaction is nonspontaneous at room temperature. Above 400 K, &DeltaG will become negative, and the reaction will become spontaneous.

      Q16.4.3

      Explain what happens as a reaction starts with &DeltaG < 0 (negative) and reaches the point where &DeltaG = 0.

      Use the standard free energy of formation data in Appendix G to determine the free energy change for each of the following reactions, which are run under standard state conditions and 25 °C. Identify each as either spontaneous or nonspontaneous at these conditions.

      1. (ce(s)⟶ce(s)+ce(g))
      2. (ce

        (g)+ce(l)⟶ce(g))

      3. (ce(s)+ce(g)⟶ce(s))
      4. (ce<2LiOH>(s)+ce(g)⟶ce(s)+ce(g)) (ce(g)+ce(g)⟶ce(s,,ce)+ce(g)) (ce(g)+ce(g)⟶ce(g)+ce(g)) S16.4.3 465.1 kJ nonspontaneous &minus106.86 kJ spontaneous &minus53.6 kJ spontaneous &minus83.4 kJ spontaneous &minus406.7 kJ spontaneous &minus30.0 kJ spontaneous Q16.4.4 Use the standard free energy data in Appendix G to determine the free energy change for each of the following reactions, which are run under standard state conditions and 25 °C. Identify each as either spontaneous or nonspontaneous at these conditions. (ce(s,, ce)+ce(g)⟶ce(g)) (ce(g)+ce(g)⟶ce(g)) (ce(s)+ce(g)⟶ce(s)) (ce(s)+ce(l)⟶ce(s)) (ce(s)+ce(g)⟶ce(s)+ce(g)) (ce(s)⟶ce(s)+ce(g)) Q16.4.5 Determine the standard free energy of formation, (&DeltaG^circ_ce), for phosphoric acid. How does your calculated result compare to the value in Appendix G? Explain. S16.4.5 &minus1124.3 kJ/mol for the standard free energy change. The calculation agrees with the value in Appendix G because free energy is a state function (just like the enthalpy and entropy), so its change depends only on the initial and final states, not the path between them. Q16.4.6 Is the formation of ozone (O3(g)) from oxygen (O2(g)) spontaneous at room temperature under standard state conditions? Q16.4.7 Consider the decomposition of red mercury(II) oxide under standard state conditions. Is the decomposition spontaneous under standard state conditions? Above what temperature does the reaction become spontaneous? S16.4.7 The reaction is nonspontaneous Above 566 °C the process is spontaneous. Q16.4.8 Among other things, an ideal fuel for the control thrusters of a space vehicle should decompose in a spontaneous exothermic reaction when exposed to the appropriate catalyst. Evaluate the following substances under standard state conditions as suitable candidates for fuels. Ammonia: (ce(g)⟶ce(g)+ce(g)) Diborane: (ce(g)⟶ce(g)+ce(g)) Hydrazine: (ce(g)⟶ce(g)+ce(g)) Hydrogen peroxide: (ce(l)⟶ce(g)+dfracce(g)) Q16.4.9 Calculate &DeltaG° for each of the following reactions from the equilibrium constant at the temperature given. (ce(g)+ce(g)⟶ce(g) hspace mathrm hspace K_p=4.1×10^) (ce(g)+ce(g)⟶ce(g) hspace mathrm hspace K_p=50.0)
      5. (ce(g)+ce

        (g)⟶ce(g)+ce(g) hspace mathrm hspace K_p=1.67)

      6. (ce(s)⟶ce(s)+ce(g) hspace <20px>mathrm hspace <20px>K_p=1.04)
      7. (ce(aq)+ce(l)⟶ce(aq)+ce(aq) hspace mathrm hspace K_p=7.2×10^) (ce(s)⟶ce(aq)+ce(aq) hspace mathrm hspace K_p=3.3×10^) S16.4.9 1.5 × 10 2 kJ &minus21.9 kJ &minus5.34 kJ &minus0.383 kJ 18 kJ 71 kJ Q16.4.10 Calculate &DeltaG° for each of the following reactions from the equilibrium constant at the temperature given. (ce(g)+ce(g)⟶ce(g) hspace mathrm hspace K_p=4.7×10^) (ce(g)+ce(g)⇌ce(g) hspace mathrm hspace K_p=48.2) (ce(l)⇌ce(g) hspace mathrm hspace K_p=mathrm) (ce(s)+ce(g)⇌ce(s)+ce(g) hspace mathrm hspace K_p=4.90×10^2) (ce(aq)+ce(l)⟶ce(aq)+ce(aq) hspace mathrm hspace K_p=4.4×10^) (ce(s)⟶ce(aq)+ce(aq) hspace mathrm hspace K_p=8.7×10^) Q16.4.11 Calculate the equilibrium constant at 25 °C for each of the following reactions from the value of &DeltaG° given. (ce(g)+ce(g)⟶ce(g) hspace &DeltaG°=mathrm) (ce(s)+ce(l)⟶ce(g) hspace &DeltaG°=mathrm) (ce(s)+ce(g)⟶ce(s)+ce(g) hspace &DeltaG°=mathrm) (ce(g)⟶ce(g)+ce(g) hspace &DeltaG°=mathrm) (ce(l)⟶ce(l) hspace &DeltaG°=mathrm) S16.4.11 Q16.4.2 Calculate the equilibrium constant at 25 °C for each of the following reactions from the value of &DeltaG° given. (ce(s)+ce(g)⟶ce(g) hspace &DeltaG°=mathrm) (ce(g)+ce(s)⟶ce(g) hspace &DeltaG°=mathrm)
      8. (ce(g)+ce<3Cl2>(g)⟶ce(g)+ce(g) hspace <20px>&DeltaG°=mathrm<&minus39: kJ>)
      9. (ce<2SO2>(g)+ce(g)⟶ce<2SO3>(g) hspace <20px>&DeltaG°=mathrm<&minus141.82: kJ>)
      10. (ce(g)⟶ce(l) hspace <20px>&DeltaG°=mathrm<&minus1.88: kJ>)

      Q16.4.13

      Calculate the equilibrium constant at the temperature given.

      1. (a) (ce(g)+ce<2F2>(g)⟶ce<2F2O>(g) hspace <20px>mathrm<(T=100:°C)>)
      2. (ce(s)+ce(l)⟶ce<2IBr>(g) hspace <20px>mathrm<(T=0.0:°C)>)
      3. (ce<2LiOH>(s)+ce(g)⟶ce(s)+ce(g) hspace mathrm) (ce(g)⟶ce(g)+ce(g) hspace mathrm) (ce(l)⟶ce(g) hspace mathrm) S16.4.13 In each of the following, the value of &DeltaG is not given at the temperature of the reaction. Therefore, we must calculate &DeltaG° from the values &DeltaH° and &DeltaS° and then calculate &DeltaG from the relation &DeltaG° = &DeltaH° &minus T&DeltaS°. Q16.4.14 Calculate the equilibrium constant at the temperature given. (a) (ce(s)+ce(g)⟶ce(g) hspace mathrm) (ce(g)+ce(s)⟶ce(g) hspace mathrm)
      4. (ce(g)+ce<3Cl2>(g)⟶ce(g)+ce(g) hspace <20px>mathrm<(T=125:°C)>)
      5. (ce<2SO2>(g)+ce(g)⟶ce<2SO3>(g) hspace <20px>mathrm<(T=675:°C)>)
      6. (ce(g)⟶ce(l) hspace <20px>mathrm<(T=90:°C)>)

      Q16.4.15

      Consider the following reaction at 298 K:

      What is the standard free energy change at this temperature? Describe what happens to the initial system, where the reactants and products are in standard states, as it approaches equilibrium.

      S16.4.16

      The standard free energy change is (&DeltaG^circ_<298>=&minusRTln K=mathrm<4.84: kJ/mol>). When reactants and products are in their standard states (1 bar or 1 atm), Q = 1. As the reaction proceeds toward equilibrium, the reaction shifts left (the amount of products drops while the amount of reactants increases): Q < 1, and (&DeltaG_<298>) becomes less positive as it approaches zero. At equilibrium, Q = K, and &DeltaG = 0.

      Q16.4.17

      Determine the normal boiling point (in kelvin) of dichloroethane, CH2Cl2. Find the actual boiling point using the Internet or some other source, and calculate the percent error in the temperature. Explain the differences, if any, between the two values.

      Q16.4.18

      Under what conditions is (ce(g)⟶ce(g)+ce(g)) spontaneous?

      S16.4.18

      The reaction will be spontaneous at temperatures greater than 287 K.

      Q16.4.19

      At room temperature, the equilibrium constant (Kw) for the self-ionization of water is 1.00 × 10 &minus14 . Using this information, calculate the standard free energy change for the aqueous reaction of hydrogen ion with hydroxide ion to produce water. (Hint: The reaction is the reverse of the self-ionization reaction.)

      Q16.4.20

      Hydrogen sulfide is a pollutant found in natural gas. Following its removal, it is converted to sulfur by the reaction (ce<2H2S>(g)+ce(g)⇌dfrac<3><8>ce(s,,ce)+ce<2H2O>(l)). What is the equilibrium constant for this reaction? Is the reaction endothermic or exothermic?

      S16.4.20

      The process is exothermic.

      Q16.4.21

      Consider the decomposition of CaCO3(s) into CaO(s) and CO2(g). What is the equilibrium partial pressure of CO2 at room temperature?

      Q16.4.22

      In the laboratory, hydrogen chloride (HCl(g)) and ammonia (NH3(g)) often escape from bottles of their solutions and react to form the ammonium chloride (NH4Cl(s)), the white glaze often seen on glassware. Assuming that the number of moles of each gas that escapes into the room is the same, what is the maximum partial pressure of HCl and NH3 in the laboratory at room temperature? (Hint: The partial pressures will be equal and are at their maximum value when at equilibrium.)

      S16.4.22

      1.0 × 10 &minus8 atm. This is the maximum pressure of the gases under the stated conditions.

      Q16.4.23

      Benzene can be prepared from acetylene. (ce<3C2H2>(g)⇌ce(g)). Determine the equilibrium constant at 25 °C and at 850 °C. Is the reaction spontaneous at either of these temperatures? Why is all acetylene not found as benzene?

      Q16.4.24

      Carbon dioxide decomposes into CO and O2 at elevated temperatures. What is the equilibrium partial pressure of oxygen in a sample at 1000 °C for which the initial pressure of CO2 was 1.15 atm?

      Q16.4.25

      Carbon tetrachloride, an important industrial solvent, is prepared by the chlorination of methane at 850 K.

      What is the equilibrium constant for the reaction at 850 K? Would the reaction vessel need to be heated or cooled to keep the temperature of the reaction constant?

      Q16.4.25B

      Acetic acid, CH3CO2H, can form a dimer, (CH3CO2H)2, in the gas phase.

      The dimer is held together by two hydrogen bonds with a total strength of 66.5 kJ per mole of dimer.

      At 25 °C, the equilibrium constant for the dimerization is 1.3 × 10 3 (pressure in atm). What is &DeltaS° for the reaction?

      S16.4.25B

      Q16.4.26

      Nitric acid, HNO3, can be prepared by the following sequence of reactions:

      [ce<3NO2>(g)+ce(l)⟶ce(l)+ce(g)] How much heat is evolved when 1 mol of NH3(g) is converted to HNO3(l)? Assume standard states at 25 °C. Q16.4.27A Determine &DeltaG for the following reactions. (a) Antimony pentachloride decomposes at 448 °C. The reaction is: An equilibrium mixture in a 5.00 L flask at 448 °C contains 3.85 g of SbCl5, 9.14 g of SbCl3, and 2.84 g of Cl2. Chlorine molecules dissociate according to this reaction: 1.00% of Cl2 molecules dissociate at 975 K and a pressure of 1.00 atm. S16.4.27A Q16.4.27 Given that the (&DeltaG^circ_ce) for Pb 2+ (aq) and Cl &minus (aq) is &minus24.3 kJ/mole and &minus131.2 kJ/mole respectively, determine the solubility product, Ksp, for PbCl2(s). Q16.4.28 Determine the standard free energy change, (&DeltaG^circ_ce), for the formation of S 2&minus (aq) given that the (&DeltaG^circ_ce) for Ag + (aq) and Ag2S(s) are 77.1 k/mole and &minus39.5 kJ/mole respectively, and the solubility product for Ag2S(s) is 8 × 10 &minus51 . S16.4.28 Q16.4.29 Determine the standard enthalpy change, entropy change, and free energy change for the conversion of diamond to graphite. Discuss the spontaneity of the conversion with respect to the enthalpy and entropy changes. Explain why diamond spontaneously changing into graphite is not observed. Q16.4.30 The evaporation of one mole of water at 298 K has a standard free energy change of 8.58 kJ. [ce(l)⇌ce(g) hspace &DeltaG^circ_=mathrm] (a) Is the evaporation of water under standard thermodynamic conditions spontaneous? Determine the equilibrium constant, KP, for this physical process. By calculating ∆G, determine if the evaporation of water at 298 K is spontaneous when the partial pressure of water, (P_>), is 0.011 atm. If the evaporation of water were always nonspontaneous at room temperature, wet laundry would never dry when placed outside. In order for laundry to dry, what must be the value of (P_>) in the air? S16.4.30 (a) Under standard thermodynamic conditions, the evaporation is nonspontaneous Kp = 0.031 The evaporation of water is spontaneous (P_>) must always be less than Kp or less than 0.031 atm. 0.031 atm represents air saturated with water vapor at 25 °C, or 100% humidity. Q16.4.31 In glycolysis, the reaction of glucose (Glu) to form glucose-6-phosphate (G6P) requires ATP to be present as described by the following equation: In this process, ATP becomes ADP summarized by the following equation: Determine the standard free energy change for the following reaction, and explain why ATP is necessary to drive this process: Q16.4.32 One of the important reactions in the biochemical pathway glycolysis is the reaction of glucose-6-phosphate (G6P) to form fructose-6-phosphate (F6P): (a) Is the reaction spontaneous or nonspontaneous under standard thermodynamic conditions? Standard thermodynamic conditions imply the concentrations of G6P and F6P to be 1 M, however, in a typical cell, they are not even close to these values. Calculate &DeltaG when the concentrations of G6P and F6P are 120 &muM and 28 &muM respectively, and discuss the spontaneity of the forward reaction under these conditions. Assume the temperature is 37 °C. S16.4.32 (a) Nonspontaneous as (&DeltaG^circ_>0) (&DeltaG^circ_=&minusRTln K,) (&DeltaG = 1.7×10^3 + left(8.314 × 335 × lndfrac ight) = mathrm). The forward reaction to produce F6P is spontaneous under these conditions. Q16.4.33 Without doing a numerical calculation, determine which of the following will reduce the free energy change for the reaction, that is, make it less positive or more negative, when the temperature is increased. Explain. When ammonium chloride is added to water and stirred, it dissolves spontaneously and the resulting solution feels cold. Without doing any calculations, deduce the signs of &DeltaG, &DeltaH, and &DeltaS for this process, and justify your choices. S16.4.33 &DeltaG is negative as the process is spontaneous. &DeltaH is positive as with the solution becoming cold, the dissolving must be endothermic. &DeltaS must be positive as this drives the process, and it is expected for the dissolution of any soluble ionic compound. Q16.4.34 An important source of copper is from the copper ore, chalcocite, a form of copper(I) sulfide. When heated, the Cu2S decomposes to form copper and sulfur described by the following equation: (a) Determine (&DeltaG^circ_) for the decomposition of Cu2S(s). The reaction of sulfur with oxygen yields sulfur dioxide as the only product. Write an equation that describes this reaction, and determine (&DeltaG^circ_) for the process. The production of copper from chalcocite is performed by roasting the Cu2S in air to produce the Cu. By combining the equations from Parts (a) and (b), write the equation that describes the roasting of the chalcocite, and explain why coupling these reactions together makes for a more efficient process for the production of the copper. Q16.4.35 What happens to (&DeltaG^circ_) (becomes more negative or more positive) for the following chemical reactions when the partial pressure of oxygen is increased? 12.E: RNA Processing (Exercises)

      12.1 Nucleoside triphosphates labeled with [32P] at the a, b, or g position are useful for monitoring various aspects of transcription. For the specific process listed in a-c, give the position of the label that is appropriate for examining that step.

      a) Initiation by E. coliRNA polymerase.

      b) Forming the 5' end of eukaryotic mRNA.

      c) Elongation by eukaryotic RNA polymerase II.

      12.2 (POB) RNA posttranscriptional processing.

      Predict the likely effects of a mutation in the sequence (5')AAUAAA in a eukaryotic mRNA transcript.

      12.3 A phosphoester transfer mechanism (or transesterification) is observed frequently in splicing and other reactions involving RNA. Are the following statements about these mechanisms true or false?

      a) The mechanism requires the cleavage of high-energy bonds from ATP.

      b) The initiating nucleophile for splicing of Group I introns (including the intron of pre-rRNA from Tetrahymena) is the 3' hydroxyl of a guanine nucleotide.

      c) The initiating nucleophile for splicing of nuclear pre-mRNA is the 2' hydroxyl of an internal adenine nucleotide.

      d) The individual reactions in the phosphoester transfers are reversible, but the overall process is essentially irreversible because of circularization (includes lariat formation) of the excised intron.

      12.4 What properties are shared by the splicing mechanism of Tetrahymena pre-rRNA and Group II fungal mitochrondrial introns?

      12.5 Please answer these questions on splicing of precursors to mRNA.

      a) What dinucleotides are almost invariably found at the 5&rsquo and 3&rsquo splice sites of introns?

      b) Which splicing component binds at the 5' splice junction?

      c) What nucleotides are joined by the branch structure in the intron during splicing?

      d) What is ATP used for during splicing of precursors to mRNA?

      What is the minimum number of transesterification reactions needed to splice an intron from an mRNA transcript? Why?

      12.7 Match the following statements with the appropriate eukaryotic splicing process listed in parts a-c.

      1) A guanine nucleoside or nucleotide initiates a concerted phosphotransfer reaction.

      2) The consensus sequences at splice junctions are AG'GUAAGU. YYYAG'G (' is the junction, Y = any pyrimidine).

      3) Splicing occurs in two separate steps, cutting to generate a 3'-phosphate followed by an ATP dependent ligation.

      4) Splicing requires no protein factors.

      5) Splicing requires U1 small nuclear ribonucleoprotein complexes.

      b) Splicing of pre-tRNA in yeast

      c) Splicing of pre-rRNA in Tetrahymena

      12.8 The enzyme RNase H will cleave any RNA that is in a heteroduplex with DNA. Thus one can cleave a single-stranded RNA in any specific location by first annealing a short oligodeoxyribonucleotide that is complementary to that location and then treating with RNase H.

      This approach is useful in determining the structure of splicing intermediates. Let's consider a hypothetical case shown in the figure below. After incubation of radiolabeled precursor RNA (exon1-intron-exon2) with a nuclear extract that is capable of carrying out splicing, the products were analyzed on a denaturing polyacrylamide gel. The results showed that the exons were joined as linear RNA, but the excised intron moved much slower than a linear RNA of the same size, indicative of some non-linear structure. The excised intron was annealed to a short oligodeoxyribonucleotide that is complementary to the region at the 5' splice site (labeled oligo 1 in the figure), treated with RNase H and analyzed on a denaturing polyacrylamide gel. The product ran as a linear RNA with the size of the excised intron (less the length of the RNase H cleavage site). As summarized in the figure, the excised intron was analyzed by annealing (separately) with three other oligodeoxyribonucleotides, followed by RNase H treatment and gel electrophoresis. Use of oligodeoxyribonucleotide number 2 generated a Y-shaped molecule, use of oligodeoxyribonucleotide number 3 generated a V-shaped molecule with one 5' end and 2 3' ends, and use of oligodeoxyribonucleotide number 4 generated a circle and a short linear RNA.

      (a) What does the result with oligodeoxyribonucleotide 2 tell you?

      (b) What does the result with oligodeoxyribonucleotide 4 tell you?

      (c) What does the result with oligodeoxyribonucleotide 1 tell you?

      (d) What does the result with oligodeoxyribonucleotide 3 tell you?

      (e) What is the structure of the excised intron? Show the locations of the complementary oligos on your drawing.


      12.E: Exercises (Part 2)

      Classifying Cash Flows. Identify whether each of the following items would appear in the operating, investing, or financing activities section of the statement of cash flows. Briefly explain your answer for each item.

      1. Cash receipts from the sale of goods
      2. Cash payments for the purchases of merchandise
      3. Cash receipts from the issuance of bonds
      4. Cash payments to shareholders for dividends
      5. Cash payments to employees
      6. Cash receipts from the sale of equipment

      Operating Activities Section Using the Indirect Method. The following income statement and current sections of the balance sheet are for Manor Company.

      1. Using the indirect method, prepare the operating activities section of the statement of cash flows for Manor Company for the year ended December 31, 2012. Use the format presented in Figure 12.5.
      2. How much cash was provided by (used by) operating activities? Briefly describe what this amount tells us about the company.

      (Appendix) Operating Activities Section Using the Direct Method. The following income statement and current sections of the balance sheet are for Manor Company (this is the same information as the previous exercise).

      1. Using the direct method, prepare the operating activities section of the statement of cash flows for Manor Company for the year ended December 31, 2012. Use the format presented in Figure 12.12.
      2. How much cash was provided by (used by) operating activities? Briefly describe what this amount tells us about the company.

      Investing Activities Section. The following information is from the noncurrent asset portion of Gebhardt Company&rsquos balance sheet.

      The following activities occurred during 2012:

      • Sold equipment with a book value of $4,000 (= $90,000 cost &minus $86,000 accumulated depreciation) for $9,000 cash and depreciation expense for the year totaled $71,000
      • Purchased equipment for $50,000 cash
      • Loans totaling $62,000 were made to other entities during the year (Hint: Solve for the principal amount on loans collected during the year.)
      • Purchased long-term investments for $16,000 cash
      1. Prepare the investing activities section of the statement of cash flows for Gebhardt, Inc., for the year ended December 31, 2012. Use the format presented in Figure 12.6.
      2. How much cash was provided by (used by) investing activities? Briefly describe what this amount tells us about the company.

      Financing Activities Section. The following information is from the noncurrent liabilities and owners&rsquo equity portions of System, Inc.&rsquos balance sheet.

      The following activities occurred during 2012:

      • Paid principal amount of $70,000 for long-term notes payable
      • Received $40,000 for long-term notes payable
      • Paid principal amount on bonds totaling $15,000 (Hint: Solve for the proceeds received from the issuance of bonds.)
      • Issued common stock for $100,000 cash (Hint: Solve for the amount paid for the repurchase of stock.)
      • Earned net income totaling $170,000
      • Paid cash dividends totaling $20,000
      1. Prepare the financing activities section of the statement of cash flows for System, Inc., for the year ended December 31, 2012. Use the format presented in Figure 12.7.
      2. How much cash was provided by (used by) financing activities? Briefly describe what this amount tells us about the company.

      Operating Activities Section Using the Indirect Method and Cash Ratios. The following data are for Mills Company.

        Using the indirect method, prepare the operating activities section of the statement of cash flows for Mills Company for the year ended December 31, 2012. Use the format presented in Figure 12.5.

      Calculate the following cash measures:

      Classifying Cash Flows. Big Sky, Inc., had the following transactions during 2012:

      1. Issued common stock for $150,000 cash
      2. Paid $25,000 in principal on previously issued bonds
      3. Paid $300,000 in salaries and wages to employees
      4. Sold property for $45,000 cash
      5. Paid $3,000 in cash dividends
      6. Received $600,000 from customers for cash sales
      7. Paid $350,000 cash for merchandise
      8. Converted bonds into common stock
      9. Purchased a building for $850,000 cash
      10. Paid $310,000 for operating expenses
      11. Received $200,000 cash for the sale of long-term investments
      12. Issued bonds for $87,000 cash
      13. Repurchased common stock for $35,000 cash
      14. Issued common stock to purchase land valued at $450,000
      15. Paid $10,000 cash for interest on notes payable

      Classify each transaction as one of the following: operating activity, investing activity, financing activity, or noncash transaction. Briefly explain your answer for each item.

      Prepare a Statement of Cash Flows, Indirect Method. Glenbrook Company&rsquos most recent balance sheet, income statement, and other important information for 2012 are presented as follows.

      Additional data for 2012 are as follows:

      • Sold equipment with a book value of $30,000 (= $40,000 cost &minus $10,000 accumulated depreciation) for $28,000 cash
      • Purchased equipment for $96,000 cash
      • There were no sales of long-term investments (Hint: Solve for the purchase of long-term investments.)
      • Issued bonds for $16,000 cash
      • Repurchased common stock (treasury shares) for $45,000 cash
      • Declared and paid $12,000 in cash dividends
      1. Use the four steps described in the chapter to prepare a statement of cash flows for the year ended December 31, 2012, using the indirect method. Refer to the format presented in Figure 12.8.
      2. Briefly describe the major changes in cash identified in the statement of cash flows.

      (Appendix) Prepare a Statement of Cash Flows, Direct Method. Refer to the information for Glenbrook Company presented in the previous problem.

      1. Use the four steps described in the chapter, including the appendix, to prepare a statement of cash flows for the year ended December 31, 2012, using the direct method. Refer to the operating activities section format using the direct method presented in Figure 12.12 and the adjustment rules for the direct method presented in Figure 12.13.
      2. Briefly describe the major changes in cash identified in the statement of cash flows.

      Prepare and Analyze a Statement of Cash Flows, Indirect Method. Travel Supply, Inc.&rsquos most recent balance sheet, income statement, and other important information for 2012 are presented as follows.

      Additional data for 2012 are as follows:

      • Sold equipment with a book value of $3,000 (= $23,000 cost &minus $20,000 accumulated depreciation) for $8,000 cash
      • Purchased equipment for $47,000 cash
      • Sold long-term investments for $9,000 cash and these investments had an original cost of $13,000
      • Paid $16,000 cash for principal amount on notes payable
      • Issued common stock for $8,000 cash
      • Declared and paid $22,000 in cash dividends
      1. Use the four steps described in the chapter to prepare a statement of cash flows for the year ended December 31, 2012, using the indirect method. Refer to the format presented in Figure 12.8.
      2. The owner of Travel Supply, Inc., wants to know why cash only increased $51,000 even though the company had net income of $103,000, issued common stock for $8,000, and sold long-term investments for $9,000. Use the information in the statement of cash flows to briefly explain why cash only increased $51,000.

      Prepare a Statement of Cash Flows, Indirect Method Analyze Using Cash Ratios. Nolan Company&rsquos most recent balance sheet, income statement, and other important information for 2012 are presented as follows.

      Additional data for 2012 are as follows:

      • Sold equipment with a book value of $13,000 (= $27,000 cost &minus $14,000 accumulated depreciation) for $21,000 cash
      • Purchased equipment for $10,000 cash
      • Sold long-term investments for $6,000 cash and these investments had an original cost of $8,000
      • Received $19,000 cash related to notes payable
      • Issued common stock for $35,000 cash
      • Declared and paid $4,000 in cash dividends
      1. Use the four steps described in the chapter to prepare a statement of cash flows for the year ended December 31, 2012, using the indirect method. Refer to the format presented in Figure 12.8.
      2. The owner of Nolan Company wants to know how cash more than doubled, from $82,000 to $165,000, given the company&rsquos modest net income of $9,000. Use the information in the statement of cash flows to briefly explain why cash more than doubled.

      Calculate the following cash measures:

      1. Operating cash flow ratio
      2. Capital expenditure ratio (Hint: Capital expenditures can be found in the investing activities section of the statement of cash flows prepared in part a.)
      3. Free cash flow

      (Appendix) Prepare a Statement of Cash Flows (Direct Method) Analyze Using Cash Ratios. Refer to the information for Nolan Company presented in the previous problem.

      1. Use the four steps described in the chapter, including the appendix, to prepare a statement of cash flows for the year ended December 31, 2012, using the direct method. Refer to the operating activities section format using the direct method presented in Figure 12.12, and the adjustment rules for the direct method presented in Figure 12.13.
      2. Briefly describe the major changes in cash identified in the statement of cash flows.

      Calculate the following cash measures:

      1. Operating cash flow ratio
      2. Capital expenditure ratio (Hint: Capital expenditures can be found in the investing activities section of the statement of cash flows prepared in part a.)
      3. Free cash flow

      Prepare and Analyze a Statement of Cash Flows, Indirect Method and Direct Method. Ritz Company&rsquos most recent balance sheet, income statement, and other important information for 2012 are presented as follows.

      Additional data for 2012 are as follows:

      • Sold equipment with a book value of $15,000 (= $100,000 cost &minus $85,000 accumulated depreciation) for $32,000 cash
      • Purchased equipment for $140,000 cash
      • Sold long-term investments for $23,000 cash and these investments had an original cost of $24,000
      • Purchased long-term investments for $5,000 cash
      • Issued bonds for $105,000 cash
      • Issued common stock for $7,000 cash
      • Declared and paid $11,000 in cash dividends
      1. Use the four steps described in the chapter to prepare a statement of cash flows for the year ended December 31, 2012, using the indirect method. Refer to the format presented in Figure 12.8.
      2. The owner of Ritz Company wants to know why cash decreased from $350,000 to $278,000 given the company&rsquos net income of $18,000. Use the information in the statement of cash flows to briefly explain why cash decreased.
      3. Use the four steps described in the chapter, as well as in the appendix, to prepare a statement of cash flows for the year ended December 31, 2012, using the direct method. Refer to the operating activities section format using the direct method presented in Figure 12.12 and the adjustment rules for the direct method presented in Figure 12.13.

      One Step Further: Skill-Building Cases

      1. Southwest Airlines Statement of Cash Flows. Refer to the Note 12.3 "Business in Action 12.1" How could Southwest&rsquos cash balance increase by $147,000,000 even though the company generated $1,600,000,000 in cash from operating activities?
      2. Home Depot and Lowe&rsquos Statement of Cash Flows. Refer to the Note 12.10 "Business in Action 12.2" How much cash was generated from daily activities for each company? Where was the bulk of this cash spent for each company?

      Internet Project: Statement of Cash Flows. Using the Internet, find the most recent annual report for a company of your choice. Print the statement of cash flows and include it with your response to the following requirements.

      1. How much cash was provided by (used by) operating activities? Compare this amount to net income (often called net earnings) and explain why the two are different.
      2. What method did the company use to prepare the operating activities section, direct or indirect? Explain.
      3. How much cash was provided by (used by) investing activities? Which activity in this section had the biggest impact on investing cash flows?
      4. How much cash was provided by (used by) financing activities? Which activity in this section had the biggest impact on financing cash flows?
      5. Calculate free cash flow. Did the company generate enough cash from operating activities to cover capital expenditures? Explain.

      Group Activity: Analyzing General Motors Statement of Cash Flows.The following information is from the consolidated statement of cash flows for General Motors (GM) for the year ended December 31, 2005 (in millions).

      An investment advisor recently reviewed GM&rsquos statement of cash flows and balance sheet and stated: &ldquoGM is doing great! They are sitting on cash of more than $30,000,000,000. There is no cash flow problem with this company!&rdquo In groups of two to four students, decide whether you agree with this statement. Support your conclusion with an analysis of GM&rsquos cash flows.

      Ethics: Manipulating Data to Reach Target Cash Flow. Country Market, Inc., sells food and beverage products at its five retail stores. The company&rsquos fiscal year ends on December 31. The company&rsquos president and CEO, Jean Williams, just received a draft of the statement of cash flows from the controller, Stan Walker. Jean is very interested in the results since a significant part of her annual bonus depends on generating at least $400,000 in cash from operating activities. A summary of the statement is provided in the following:

      Becky Swanson, the chief financial officer (CFO) for Country Market, is approached by Jean:


      12.E: Networked Programs (Exercises)

      • Contributed by Chuck Severance
      • Clinical Associate Professor (School of Information) at University of Michigan

      Exercise 1: Change the socket program socket1.py to prompt the user for the URL so it can read any web page. You can use split('/') to break the URL into its component parts so you can extract the host name for the socket connect call. Add error checking using try and except to handle the condition where the user enters an improperly formatted or non-existent URL.

      Exercise 2: Change your socket program so that it counts the number of characters it has received and stops displaying any text after it has shown 3000 characters. The program should retrieve the entire document and count the total number of characters and display the count of the number of characters at the end of the document.

      Exercise 3: Use urllib to replicate the previous exercise of (1) retrieving the document from a URL, (2) displaying up to 3000 characters, and (3) counting the overall number of characters in the document. Don't worry about the headers for this exercise, simply show the first 3000 characters of the document contents.

      Exercise 4: Change the urllinks.py program to extract and count paragraph (p) tags from the retrieved HTML document and display the count of the paragraphs as the output of your program. Do not display the paragraph text, only count them. Test your program on several small web pages as well as some larger web pages.

      Exercise 5: (Advanced) Change the socket program so that it only shows data after the headers and a blank line have been received. Remember that recv is receiving characters (newlines and all), not lines.


      The scores of a random sample of (8) students on a physics test are as follows: (60, 62, 67, 69, 70, 72, 75, 78).

      1. Test to see if the sample mean is significantly different from (65) at the (0.05) level. Report the (t) and (p) values.
      2. The researcher realizes that she accidentally recorded the score that should have been (76) as (67). Are these corrected scores significantly different from (65) at the (0.05) level? (relevant section)

      A (hypothetical) experiment is conducted on the effect of alcohol on perceptual motor ability. Ten subjects are each tested twice, once after having two drinks and once after having two glasses of water. The two tests were on two different days to give the alcohol a chance to wear off. Half of the subjects were given alcohol first and half were given water first. The scores of the (10) subjects are shown below. The first number for each subject is their performance in the "water" condition. Higher scores reflect better performance. Test to see if alcohol had a significant effect. Report the (t) and (p) values. (relevant section)