NORTH CAROLINA STATE UNIVERSITY

Department of Chemistry Name________________________________

CH 431 Final Exam

Physical Chemistry I December 7, 2001

Given: R = 8.314 J mol^-1 K^-1 = 0.08206 L atm mol^-1 K^-1

1 atm = 1.01325 x 10⁵ Nm^-2 = 760 Torr

Please show your work in the space allocated for each problem if possible. Use addition space on the back and mark it clearly.

I. Short Answer (120 points total)

1. The diagram below shows a plot of the molar free energy vs. temperature for a pure substance.

1. Identify the thermodynamic quantity that represents the slope of the three lines on the diagram below, i.e. what is (¶m/¶t)? (4 points)

The slope (¶m/¶t) = ___________________________.

2. Label the phases solid, liquid, and vapor on the diagram. (6 points)

3. Draw a line that represents the molar free energy of the liquid if a small amount of solute is added. Please sketch and label the line in the diagram above. (5 points)

4. Describe the effect of solute on the chemical potential of the solvent. Include an equation that describes mathematically how the chemical potential of a solvent is affected by the addition of solute. (5 points)

2. The diagram below shows a plot of the free energy vs. extent of reaction.

1. Identify the position where equilibrium occurs and give a mathematical expression of the equilibrium condition. Please label the point on the diagram above. (5 points)

2. If the total pressure on the system is increased, how will the curve shown in the Figure change? To illustrate this you may draw a new diagram as needed. (5 points)

3. A. You are given a partition function for a non-ideal gas. What formula would you use to derive an equation of state from this partition function? (5 points)

2. As an example, please outline the derivation of the ideal gas equation of state.

(5 points)

4. Consider the statement: "The internal energy depends only on the temperature."

1. What formula would you use to calculate the internal energy for an ideal gas from a known partition function. What is the relevant partition function? (5 points)

2. Is this statement true for non-ideal gases? Give a reason for your answer.

(5 points)

5. Give an expression for the relative amount of liquid and vapor present using the tie line in the phase diagram below. Express the result as a ratio in terms of the lengths A and B indicated. (10 points)

6. What is residual entropy? (5 points)

7. Give the appropriate condition for each process or state. Give an expression for an important consequence of this condition. (5 points each)

Example: Consant volume heating: dw = 0. Consequence: dU = dq.

Adiabatic expansion: _____________. Consequence: ____________________.

Isothermal expansion: _______________. Consequence: ____________________.

Chemical equilibrium: _______________. Consequence: ____________________.

8. Sketch a Lennard-Jones potential. Label the well-depth, equilibrium distance, and

zero of energy. Please label the axes. (5 points)

9. Write out the molecular partition function for the three level system shown below.

The molecular partition function q = _____________________. (5 points)

What is the magnitude of the partition function at T = 0 K? (5 points)

What is the magnitude of the partition function at T = ¥ K? (5 points)

10. Give a general formula for the equilibrium temperature if two substances X and Y at initial temperatures T_X and T_Y are brought together and allowed to equilibrate. You may assume that the system (consisting of X and Y) is insulated from the surroundings. There are n_X moles of X with heat capacity C_p,X and n_Y moles of Y with heat capacity C_p,Y. (10 points)

11. A. Given dU = TdS - PdV, please give an expression for dG.

dG = ____________________________. (5 points)

2. Derive an expression for the Maxwell relation for the total derivative

dA = -SdT -PdV.

The Maxwell relation is _______________________________. (5 points)

II. Problems (180 points total)

1. Calculate the entropy change of the system, the surroundings, and the total entropy change for a monatomic ideal gas for a constant pressure expansion of 0.1 moles of gas against 0.492 atm of pressure at 300 K from an initial volume of 2.0 L. (17 points)

V₂ = nRT/P = (0.1 moles)(0.082 L-atm/mole-K)(300K)/0.492 atm) = 5.0 L

DS = nR ln(V₂/V₁) = (0.1 moles)(8.314 J/mole-K) ln(5.0/2.0) = 0.76 J/K for the system

DS = -PDV/T = -(0.492 atm)(5.0 L - 2.0 L)/300 K = -0.498 J/K for the surroundings

DS_sys = ______ ____________ J/K. DS_surr = ________ ____________ J/K.

DS_total = ________ _________ J/K.

2. Given the following data at 35 ^oC determine the activity and activity coefficient for acetone (A) and chloroform (C) at a mole fraction of x_A = 0.60.

xC	PC (torr)	PA (torr)
0	0	347
0.2	35	270
0.4	82	142
0.6	142	102
0.8	219	37
1.0	293	0

The Henry's law constants are K_H,A = 175 torr and K_H,C = 165 torr.

Henry's law standard state

a_A = _________________ g_A = __________________

a_C = _________________ g_C = __________________ (6 points)

Raoult's law standard state

a_A = _________________ g_A = __________________

a_C = _________________ g_C = __________________ (6 points)

Using the data above for 35 ^oC and a mole fraction x_A = 0.6, calculate the free energy and entropy of mixing for a one mole of a regular solution of acetone and chloroform using the Raoult's law standard state activity. Compare the ideal free energy of mixing with non-ideal free energy of mixing.

Ideal

D_mixG = ______________________ J/mole. (4 points)

D_mixS = ______________________ J/mole-K. (4 points)

Non-ideal

D_mixG = ______________________ J/mole. (4 points)

D_mixS = ______________________ J/mole-K. (4 points)

3. A worker at the patent office opens a sealed envelope and reads a new patent disclosure form. The patent describes an engine that can run at 99% efficiency. The patent review scientist scribbles some numbers on a sheet of paper. Her calculation assumes that the engine is an ideal reversible Carnot cycle. After completing the calculation, she shakes her head, smiles, and stamps REJECT on the application.

1. Given that exhaust temperature of the engine must be approximately 300 K, what is the thermodynamic temperature required for the expansion cycle of the engine?

T_hot = _________________ K. (5 points)

2. How much work would be derived for every kJ of heat expelled into the environment by this engine?

|w| = _________________ kJ. (5 points)

4. At 90 ^oC, the vapor pressure of toluene is 400 torr and that of o-xylene is 150 torr. What is the composition of the liquid when the liquid mixture boils at 90 ^oC with a total vapor pressure of 340 torr in an enclosed container? What is the composition of the vapor produced?

x_toluene = _______________ x_o-xylene = _______________ (5 points)

y_toluene = _______________ y_o-xylene = _______________ (5 points)

5. Nitrogen tetroxide, which is present in the equilibrium N₂O₄ (g) Û 2 NO₂ (g), is 18.46 percent dissociated at 25 ^oC and 1.00 bar. Calculate (a) K, (b) D_rG^o, (c) D _rG for the production of NO₂ (g) at 1.00 bar from N₂O₄ at 10.0 bar and 25 ^oC, (d) K at 100 ^oC given that D_rH^o = + 57.2 kJ/mol over the temperature range (you may neglect the temperature dependence of D_rH and D_rS over the temperature range).

Please write your answers for problem 5 below.

K = ___________________. (15 points) D_rG^o = ___________________. (5 points)

D_rG = ___________________. (5 points) K at 100 ^oC = _____________. (10 points)

6. An animal rights group protests against a circus act that involves a 500 kg. skating gorilla. Their expert claims that the melting point of ice will drop below 240 K underneath the mass of gorilla and this is unsafe. Given that the surface area of the skates is 10^-4 m² calculate the melting point of ice underneath the skates. Is the group's expert correct about the melting temperature of ice under the skating gorilla?

You may need the following data

DfusV = -0.00168 L/mole	DvapV = 24.4 L/mole
DfusH = 6,000 J/mole	DvapH = 40,650 J/mole
Tfus = 273.15 K	Tvap = 373.15 K

Temperature of fusion of ice under the gorilla = ____________________. (15 points)

7. A column of air rises adiabatically up the side of Mt. Kilimanjaro. If the initial temperature was 40 ^oC calculate the temperature at 6000 m assuming that the pressure at that elevation is 0.5 atm and that air can be treated as a diatomic ideal gas.

(15 points)

Derivation of temperature/volume relationship for an adiabatic expansion.

_________________________________. (Partial Credit 8 points)

Derivation of temperature/pressure dependence for an adiabatic expansion.

_________________________________. (Partial Credit 13 points)

Numerical answer: T = ___________________ K. (Full Credit)

8. Given the thermodynamic data below:

Compound D_f H^o (kJ mol^-1) S^o (J K^-1 mol^-1) C_p (J K^-1 mol^-1)

Pb (s) 0 64.8 26.44

PbO (s, red) -218.99 66.5 45.81

PbO (s, yellow) -217.32 68.7 45.77

PbO₂ -277.4 68.6 64.64

CO (g) -110.53 197.67 29.14

CO₂ (g) -393.51 213.74 37.11

Calculate the standard Gibbs energy and the equilibrium constant at 400 K for the reaction PbO(s, red) + CO (g) ¾ Pb (s) + CO₂(g).

DG^o = ______________________. (15 points)

K = ________________________. (5 points)

9. The vapor pressure of a 500 g. sample of benzene was 400 torr at 60.6 ^oC, but it fell to 386 torr when 19.0 g. of an involatile compound was dissolved in it. Treating benzene as an ideal solvent, calculate the molar mass of the compound. (15 points)

10. Calculate the temperature of boiling water in equilibrium with the liquid in a pressure cooker at 10 atm. The boiling point of water is 373.15 K and the enthalpy of vaporization is D_vapH^o = 40,650 J/mole. (15 points)