NORTH CAROLINA STATE UNIVERSITY

Department of Chemistry                                                Name________________________________

 

CH 431                                                                                                                              Mid-term 2

Physical Chemistry I                                                                                                                Solutions

 

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

             1 atm = 1.0133 x 105 Nm-2 = 760 Torr


                                   

  

 

  

For H2O, a = 2.1 x 10-4 K-1, k = 49.6 x 10-6 atm-1, DHvap = 40.656 kJ/mol, DHfus = 6.008 kJ/mol

Please answer all questions.  You may assume that all gases are ideal unless stated otherwise.

 

  1. Calculate the Carnot efficiency of a steam engine.  The isothermal expansion occurs at 100 oC with an exhaust at 60 oC.  Compare this efficiency to that of a turbine which operates at 300 oC and discharges at 80 oC. (20 points)

 

Solution:

*

For the steam engine

For the turbine

 

  1. Calculate the increase in entropy when an ideal diatomic gas (Cp = 7R/2) is expanded from 2 L to 20 L and simultaneously heated from 100 oC to 160 oC.  Assume that the initial pressure is 1 atm. (20 points)

 

Solution:

n = PV/RT = (1 atm)(2 L)/(0.08206 L-atm/mol-K)(373 K) = 0.0653 moles

 

 

 

  1. What is the boiling point of water on top of Grandfather Mountain?  The elevation is approximately 2000 m.  You may assume that M = 29 g./mol for the atmosphere. A. Calculate the pressure at an elevation of 2000 m. B. Calculate the boiling point of water at the pressure calculated in A. [If you have trouble in part A. then assume that the pressure is 0.9 atm. for part B.] (25 points)

 

Solution:

The pressure is given by

P = 0.794 atm

 

 

 

  1. We usually assume that the density of water is constant as a function of depth.  In this problem you will test that assumption.  Using the data for H2O available on page 1 of this exam please calculate the density of water in the Philippine Trench at a depth of 5000 m. A. Calculate the pressure at a depth of 5000 m under the sea. For this step you will assume that the density of water is a constant. B. Using the pressure you obtained in part A. calculate the density of water at that depth.  [If you have trouble in part A. then assume that the pressure is 300 atm. for part B.] (25 points)

 

Solution:

 

The pressure is P = rgh = (1000 kg/m3)(9.8 m/s2)(5000 m) = 4.9 x 107 Pa = 483.5 atm

 

The relative volume are density are given by

 

  1. a. Calculate the entropy of mixing of 12 moles of 16O2 with 6 moles of 18O2. (20 points)

 

Solution:

DS = + 95.22 J/K

 

  1. How does the entropy of mixing depend on temperature? (10 points)

 

Answer: The entropy of mixing does not depend on temperature.

 

  1. Calculate the entropy of Argon at 300 K (30 points)

 

Solution: Argon is a monatomic gas.  We treat it as an ideal gas.  The partition function is:

Using Q = qN/N! and the definition of entropy we obtain lnQ = Nlnq – NlnN + N

Thus, the entropy is:

 

 

 

S = 2.5(8.314) + 1.5(8.314)ln(2(3.14159)(39.9)(1.672 x 10-27)(1.38 x 10-23)(300)/(6.626 x 10-34)2) + 8.314ln(0.0224/6.022 x 1023) = 154.1 J/mol –K.

 

7. The standard enthalpy of the reaction C (s) + O2 (g) ® CO2 (g) is – 393.51 kJ/mol of carbon dioxide and the standard Gibb’s energy is –394.36 kJ/mol at 298 K.  Assuming that the entropy and enthalpy are independent of temperature calculate the Gibb’s free energy at 600 K.  Given the heat capacity data below calculate the Gibb’s free energy at 600 K based on the temperature dependence of the enthalpy and entropy (30 points).

 

Gas

Cp (J/mole-K)

CO2

37.11

O2

29.355

C (s)

8.527

 

First, solve for the entropy

 

Then calculate DGo at 600 K.

Taking the temperature dependence into account:

The difference in the heat capacity is:

Putting in these numbers:

and finally

 

8.      We have shown that the chemical potential of Ar and other ideal gases can be calculated from the partition function.  Parts a. and b. can be solved separately, i.e. you do not need to use the result from a. to work part b.

 

a.    Given the fact that A = - kT ln Q derive a formula for m . (10 points)

(Stirling’s approximation is ln N! = N ln N – N)

Solution:

 

b.   The magnitude of mo for Ar is -39.97 kJ/mol.  Given this fact calculate the chemical potential of Ar in a mixture of gases where the partial pressure of Ar is 0.4 atm and the total pressure is 2 atm at a temperature of 400 K. (10 points)

 

Solution: