**I will take the best 8 out of 10 assignment marks.**

ASSIGNMENTS THAT HAVE NOT BEEN PICKED UP HAVE BEEN PUT INTO A LABELED BOX DOWN THE HALL FROM MY OFFICE (ROOM 308e STIRLING)

Assignment 1: Assigned Jan 15, Due Jan 22 (for this assignment, you will need to refer to this link for information on the Martian atmosphere)

If you don't have the textbook yet, click here for Probs. 1.16 and 1.17.

1. (a) Compute the mass of a mole of Martian air. Assume that nitrogen and oxygen are in their usual diatomic forms. Compare the result to the mass of a mole of dry Earth air. Which is more massive?

(b) What is the volume of a mole of Martian air at the surface of Mars and at a typical Martian temperature of 250 K? What is the linear dimension corresponding to this volume (assume a simple cube geometry)?

(c) Compute the volume of a mole of Earth's air at Earth's surface at room temperature and compare it to the result from part (b).

(d) Suppose you took a mole of Martian air and transported it to the Earth's surface (not allowing it to mix with Earth's air) where it came to room temperature. Now compare its volume to the volume of a mole of Earth's air.

2. Do problem 1.16 in the text, parts a, b, and c only. Note that this problem assumes that the temperature in the Earth's atmosphere is constant; this is clearly not true but the variation in temperature is much less than the variation in pressure so your results should be approximately correct.

3. Identify each of the following quantities as

ASSIGNMENTS THAT HAVE NOT BEEN PICKED UP HAVE BEEN PUT INTO A LABELED BOX DOWN THE HALL FROM MY OFFICE (ROOM 308e STIRLING)

Assignment 1: Assigned Jan 15, Due Jan 22 (for this assignment, you will need to refer to this link for information on the Martian atmosphere)

If you don't have the textbook yet, click here for Probs. 1.16 and 1.17.

1. (a) Compute the mass of a mole of Martian air. Assume that nitrogen and oxygen are in their usual diatomic forms. Compare the result to the mass of a mole of dry Earth air. Which is more massive?

(b) What is the volume of a mole of Martian air at the surface of Mars and at a typical Martian temperature of 250 K? What is the linear dimension corresponding to this volume (assume a simple cube geometry)?

(c) Compute the volume of a mole of Earth's air at Earth's surface at room temperature and compare it to the result from part (b).

(d) Suppose you took a mole of Martian air and transported it to the Earth's surface (not allowing it to mix with Earth's air) where it came to room temperature. Now compare its volume to the volume of a mole of Earth's air.

2. Do problem 1.16 in the text, parts a, b, and c only. Note that this problem assumes that the temperature in the Earth's atmosphere is constant; this is clearly not true but the variation in temperature is much less than the variation in pressure so your results should be approximately correct.

3. Identify each of the following quantities as

*extensive*or*intensive*and provide the SI units for each:**pressure (P), temperature (T), molar specific volume (i.e. volume per mole) (v), internal energy (U), heat capacity (either C**

_{P}or C_{V}), specific heat (either c_{P}or c_{V}), stress on a rod (σ), electric charge (Q)**4. Do problem 1.17 in the text, parts a and b only.**

**For part (a), you will need to solve a quadratic equation for V/n. There will be two roots. Calculate the volume per molecule and the linear separation between molecules and provide a brief physical argument as to why one root can be chosen over the other. At low T, is the pressure higher or lower than the ideal gas pressure? At high T, is the pressure higher or lower than the ideal gas pressure?**

**Assignment 2: Assigned Jan 22, Due Jan 29**

1. (a) Expand the van der Waals equation of state in dP and, using the method of mixed second partial derivatives, confirm that dP represents an exact differential.

(b) A unique equation of state can be determined for a substance only if an exact differential has been found. Suppose that, in a laboratory, you measure the following coefficients for a substance:

1. (a) Expand the van der Waals equation of state in dP and, using the method of mixed second partial derivatives, confirm that dP represents an exact differential.

(b) A unique equation of state can be determined for a substance only if an exact differential has been found. Suppose that, in a laboratory, you measure the following coefficients for a substance:

**where v is the molar specific volume and a and b are constants. Using the same method as part (a), determine whether or not an equation of state can be found for this substance.**

**2. Starting with the Dieterici equation of state (see notes for the equation), assume that the constant, b = 0, and show that the isothermal compressibility, and volume coefficient of expansion are given, respectively, by:**

**where (again) v is the molar specific volume and a and b are constants. Use the cyclic relation at least once. Would you expect a > RTv or a < RTv? Explain briefly.**

**3. A substance is studied in the lab for which the following response functions are found:**

**where a and b are constants. Assume that the differentials are exact and find the equation of state of the substance. In addition, find the ratio of a/b.**

**4. Imagine a hypothetical molecule in a gas that is shaped like a cube with one atom at each corner of the cube. Molecular bonds exist along the edges of the cube but not diagonally across it (in other words, let vibrations exist along the bonds only). How many degrees of freedom does such a molecule have?**

**Assignment 3: Assigned Jan 29, Due Feb. 5**

Click here for the problems

1. Do problem 1.34 in the text.

2. Do problem 1. 37 in the text.

3. Suppose you have a very large adiabatic box which is divided into 2 parts. One part contains an ideal gas at a temperature, T_0. The other side is completely evacuated, hence for this side initially, temperature has no meaning. Now you put a hole in the divider so that the gas from one side flows into the other side. In the end, both sides have equal amounts of the gas. Ignoring any interactions with the wall, what is the final temperature of the gas? Explain

4. Do problem #4 on the sheet provided.

Click here for the problems

1. Do problem 1.34 in the text.

2. Do problem 1. 37 in the text.

3. Suppose you have a very large adiabatic box which is divided into 2 parts. One part contains an ideal gas at a temperature, T_0. The other side is completely evacuated, hence for this side initially, temperature has no meaning. Now you put a hole in the divider so that the gas from one side flows into the other side. In the end, both sides have equal amounts of the gas. Ignoring any interactions with the wall, what is the final temperature of the gas? Explain

*briefly*.4. Do problem #4 on the sheet provided.

**Assignment 4: Assigned Feb. 5, Due Feb. 12**

_{5}H

_{12}+ 8O

_{2}==> 5CO

_{2}+ 6H

_{2}O

(a) Find ΔH for this reaction, when all substances are in their gaseous forms. Δ

_{f}H for C

_{5}H

_{12}is -146.5 kJ

(b) Repeat (a) but now let H

_{2}O on the right hand side be in the liquid form.

(c) How much heat is given off in each case? Offer an explanation as to why there is a difference between the two cases.

_{0}, volume, V

_{0}, and temperature, T

_{0}, are also the same on both sides of the cylinder. By means of a heating coil in the gas on the left side of the piston, heat is supplied slowly to the gas on the left. It expands and compresses the gas on the right side until the final pressure has increased to 27P

_{0}/8. [Hint: think about which side(s) may or may not be undergoing adiabatic expansion or compression.] For each of the questions below, express the final result in terms of n, R, and T

_{0}.

(a) What is the final temperature of the gas on the right?

(b) What is the work done on the gas on the right?

(c) What is the final temperature of the gas on the left?

(d) How much heat flows into the gas on the left?

4. Do problem 2.17 in the text. Link to questions

**Assignment 5: Assigned Feb. 14, Due Feb. 26**

1. (a) Using the Sackur-Tetrode equation, compute the entropy of one mole of Neon at T = 298 K and atmospheric pressure. Compare the result to that of Helium which is given in Eqn. 2.50 of the text. How can you explain the difference?

1. (a) Using the Sackur-Tetrode equation, compute the entropy of one mole of Neon at T = 298 K and atmospheric pressure. Compare the result to that of Helium which is given in Eqn. 2.50 of the text. How can you explain the difference?

_{i}to V

_{f}. In a similar fashion, find an expression for ΔS when only the temperature varies from T

_{i}to T

_{f}.

(c) Use your result from part (a) and the equation of part (b) to find the absolute entropy of Neon when the temperature is raised to 350 K.

(d) Show that Eqn. 3.19 in the text (the macroscopic approach) results in the same equation as you found in part (b).

**2. Do problem 3.13 (part a only) in the text.**

3. Do problem 3.14 in the text.

3. Do problem 3.14 in the text.

^{-5}K

^{-1}, κ = 8 x 10

^{-12}m

^{2}/N and the density, ρ = 8.9 x 10

^{3}kg m

^{-3}are approximately constant during the compression. Do

**not**assume that only quadratic degrees of freedom are present.

(a) Calculate the work done on the copper per kg (the 'specific work') [hint: think about the definition of expansive/compressive work as well as the definition of κ].

(b) Calculate the heat per kg that is extracted during the process.

(c) What would be the rise in temperature if the compression were adiabatic rather than isothermal? [Note that, for copper, c

_{P}= 378 J/(K kg).]

**Midterm: Feb. 15**

READING WEEK: Feb. 19 - 23

READING WEEK: Feb. 19 - 23

**Assignment 6: Assigned Feb. 26, Due Mar. 5**

^{o}C is compressed isothermally from an initial pressure of 1 atm to a final pressure of 10,000 atms. For water, β=3.03 x 10

^{-4}/K, κ=50 x 10

^{-6}/atm, and 1 gram has a volume of 1.00 cm

^{3}. Express your results in SI units.

(a) Calculate the work done on the water.

(b) Calculate the heat extracted in the process.

(c) What is the increase in internal energy, ΔU, of the water?

2. Do problem 4.1 in the text.

3. Draw the Carnot cycle on a PV diagram, as shown in Fig. 4.3 of the text, labelling the points 1 (top left), 2 (top center), 3 (far right), and 4 (bottom center), with arrows displayed. Now, assume that the working substance is an ideal gas and draw this Carnot cycle on:

(a) the VT plane

(b) the PT plane

(c) the UT plane

2. Do problem 4.1 in the text.

3. Draw the Carnot cycle on a PV diagram, as shown in Fig. 4.3 of the text, labelling the points 1 (top left), 2 (top center), 3 (far right), and 4 (bottom center), with arrows displayed. Now, assume that the working substance is an ideal gas and draw this Carnot cycle on:

(a) the VT plane

(b) the PT plane

(c) the UT plane

**Ensure that points are labelled and arrows shown. In all cases let the second thermodynamic coordinate represent the x axis (e.g. VT puts V on the y axis and T on the x axis). Also ensure that any curvature is clear, where relevant.**

(a) Draw and label this cycle on the usual PV plane. Let the larger volume and higher temperature be V

_{2}and T

_{h}, respectively, and the smaller be V

_{1}and T

_{c}, respectively.

(b)Find an expression for the net work (W

_{net}) done on the gas around the entire cycle.

(c) Find an expression for the net heat (Q

_{net}) around the entire cycle, ensuring that terms for each segment of the cycle are shown as well as a final simplification.

(d) In the Carnot cycle, the expelled heat was lost and the absorbed heat was 'new'. However, for the Stirling engine, the heat expelled during one constant volume portion of the cycle is the

**same**heat as is absorbed again in the other constant volume portion of the cycle (this occurs in the regenerator). Therefore identify which term or terms in the expression of part (c) should be used for Q

_{h}in the efficiency equation.

(e) Find an expression for the efficiency of the Stirling engine, e, in terms of T

_{h}and T

_{c}. How does the result compare to that of the Carnot engine?

**Assignment 7: Assigned Mar. 5, Due Mar. 12**

1. Do problem 4.18 in the text.

2.

1. Do problem 4.18 in the text.

2.

**Air flows into a gasoline engine at 95 kPa and 300 K. The air is then compressed with a volumetric compression ratio of 8:1. The combustion process releases 1300 kJ/kg of energy as the fuel burns. Find the temperature and pressure after combustion. The working substance is well approximated as air with a specific heat of c_V = 717 J/kg/K.****3. Do problem 4.22 in the text.**

**4. An air conditioner operates on 800 W of power and has a coefficient of performance of 2.80 with a room temperature of 21.0 degrees C and an outside temperature of 35.0 degrees C**

(a) Calculate the rate of heat removal for this unit.

(b) Calculate the rate at which heat is discharged to the outside air.

(c) Calculate the total entropy change in the room if the air conditioner runs for 1 hour. Calculate the total entropy change for the outside air in the same time period.

(d) What is the net change in entropy for the system (room + outside air)?(a) Calculate the rate of heat removal for this unit.

(b) Calculate the rate at which heat is discharged to the outside air.

(c) Calculate the total entropy change in the room if the air conditioner runs for 1 hour. Calculate the total entropy change for the outside air in the same time period.

(d) What is the net change in entropy for the system (room + outside air)?

**5. Do Prob. 4.30 in the text but let the pressure operate between 1 bar and 12 bars (rather than 10 bars). For part (b), just calculate the COP and omit the remainder of that part of the question.**

**Assignment 8: Assigned Mar. 12, Due Mar. 19**

**1. In class, I went through the example of the electrolysis of water, where just enough electrical energy was supplied to make the reaction proceed. I then calculated the amount of heat, Q, that is extracted from the reservoir (the air) during this process. Recalculate Q but now use the fact that Q = TdS. The tables at the back of the text will be useful. Assume that T = 298 K and P = Patm.**

2. Do Prob. 5.5 in the text.

2. Do Prob. 5.5 in the text.

**3.**(This question is a modified version of Prob. 5.14 in the text.)

It can be shown that expressions for the heat capacity at constant volume and at constant pressure can be written as C

_{V}= T ∂S/∂T|

_{V}and C

_{P}= T ∂S/∂T|

_{P}, respectively (this is the result of Prob. 3.3 in the text which I am not asking you to do).

(a) Expand S in terms of T and V, and then expand V in terms of T and P.

(b) Insert dV from the second expansion into the first, then let P = constant and simplify the result.

(c) Write the result of part (b) in terms of C

_{V}and C

_{P}.

(d) Use a Maxwell relation, the cyclic relation and the definitions of β and κ to reproduce the result:

C

_{P}= C

_{V}+ (TVβ

^{2})/κ

(e) Check that this formula gives the correct value of C

_{P}- C

_{V}for an ideal gas.

**4.** Suppose that the molar-specific Gibbs free energy function is:

g = RT ln(P/P_{0}) - AP

where P is the pressure, P_{0} is a reference pressure, and A is a function which depends only on temperature, T. In the following, express all extensive quantities in molar specific form, express all results in terms of known quantities and constants, and express the results as simply as possible. Find expressions for:

(a) the equation of state

(b) the entropy, s.

(c) the remaining thermodynamic potentials (h, f, and u)

(d) the specific heats, c_{v} and c_{P}

(e) the volume expansion coefficient, β, and isothermal compressibility, κ

(f) the Joule-Thomson coefficient, μ

**Assignment 9: Assigned Mar. 19, Due Mar. 26**

**1.**
**(a)** The molar-specific energy equation for a monatomic van der Waals gas is given by u = (3/2)RT - a/v. Show that the molar-specific enthalpy is given by

h(T,v) = (5/2)RT + RTb/v - 2a/v

where a and b are the usual van der Waals constants, and v is the molar volume. HINT: For small b/v, consider what expansions might be of help.

**(b)** It can be shown that the Joule-Thomson coefficient for such a gas is given by

μ = ∂T/∂P|_{h} = (-1/c_{P})[RTv^{3}b - 2av(v-b)^{2}]/[RTv^{3} - 2a(v-b)^{2}]

where c_{P} is the molar specific heat at constant pressure.
Find an expression for the inversion temperature, T_{inv}, for this gas. The right hand side will include only v and constants.

**(c)** For N_{2}, a = 0.1408 J m^{3}/mol^{2}, and b = 3.913 x 10^{-5} m^{3}/mol and let v = 1.27 x 10^{-4} m^{3} per mole. What is the inversion temperature?

**(d)** Suppose that the volume were much larger so that b << v. Find a new expression for T_{inv} for such a case.

**(e)** Compare your results from part (c) and part (d) and comment on the importance of including the very slight departures from an ideal gas when considering throttling.

**2.** In class, I provided the function for the molar-specific Gibbs free energy for an ideal gas which included the function, A. I indicated that A is a function only of T and some constants. I then showed that the molar specific internal energy can be given by,

u = A - A' T - RT

I would now like you to reconcile this equation with the known equation for an ideal gas, i.e.

u = f/2 RT

To do this, you need the complete form for A which is

A = c_{P}(T - T_{0}) - c_{P}T ln(T/T_{0}) - s_{0}(T - T_{0}) + g_{0}

The values that are subscripted with 0 are reference values (constants) in the event that you wish to measure u with respect to those reference values. Evaluate the first form for u and see if you can reproduce the well-known second form for u. Your end result will still include the reference constants. Comment on what they mean.

**3.** A container with total volume, V, is divided into 3 partitions containing, respectively, 1 kmole of He, 2 kmole of Ne, and 3 kmole of Ar (all gaseous). Each compartment is at T = 300 K and P = 2 atm. The partitions are then removed and the gases diffuse together with no change in temperature. Find

**(a)** the final fraction, by number, of each gas.

**(b)** the final partial pressures of each gas.

**(c)** the change in the Gibbs free energy, Δ G.

**(d)** the change in the entropy, Δ S.

**4. a)** Do problem 5.40 in the text. Assume that the phase boundaries can be represented by straight lines as we did in class.

**b)** Refer to the plot of the Earth's geothermal gradient (see link below) and also note that the earth's pressure gradient is roughly linear in the crust with a mean value of about 30 MPa/km. At a depth of 50 km, will it be more likely that albite or jadeite+quartz would form? Explain.

**Assignment 10: Assigned Mar. 26, Due Apr. 2 (All assignments must be handed in at the beginning of the class -- no late assignments will be accepted -- this is your last assignment!!)**

**1.**This question is based on Prob. 6.18 in the text.

**(a)**Do the first part of Prob. 6.18, showing how Z (the partition function) can be used as a generating function to obtain the mean value of E

^{2}(i.e. derive the first equation in the problem).

**(b)**Write the quantity, (σ

_{E})

^{2}(for which you have an expression in your notes) in terms of Z and its derivatives with respect to β.

**(c)**Now write the specific heat at constant volume, C

_{V}= ∂U/∂T|

_{V}, with a change of variables so that it is expressed in terms of a partial of U with respect to β, rather than T.

**(d)**Since you know how U can be generated from Z (see notes), it is now possible to carry out the differentiation to find an expression for C

_{V}in terms of Z, β, and constants. Write the final result for C

_{V}in terms of (σ

_{E})

^{2}and temperature (rather than β) and verify that it agrees with the second equation in Prob. 6.18.

**(e)**It should now be clear that a wider distribution function corresponds to a higher value of C

_{V}. Look up typical specific heats for steam (water vapour), liquid water, and ice. Do a rough hand sketch of the distribution functions of each. (I am interested in the widths of the distribution function, rather than their amplitudes).

**2.** Do prob. 6.39 in the text. You may omit the repetition for the He atom in part (b). [FYI, at a temperature as high as 1000 K, the height is greater than 100 km.]

**3.** Starting with the Helmholtz Function for a monatomic ideal gas, derive Eqn. 2.49.