CHEM 341. Fall 2000. Problem Set #5.
Theoretical Problems in Entropy
1. Prove that two reversible adiabatic paths can never cross. Assume that the energy of
the system under consideration is a function of temperature only. (Hint: Suppose that two such
paths can intersect, and complete a cycle with the two paths plus one isothermal path. Consider
the changes accompanying each stage of the cycle and show that they conflict with the Second
Law.)
 S 
2. Evaluate 
 for a van der Waals gas in terms of n, P, V, and/or T.
 V  T
3. Two blocks of the same metal are of the same size but are different temperatures T1
and T2. These blocks of metal are brought together and allowed to come to the same temperature.
 T  T2 2 
Show that the entropy change is given by S  C p ln  1
 if Cp is constant. How does
 4T1T2 
this equation show that the change is spontaneous?
Entropy Changes for Adiabatic or Isothermal Processes
4. A sample of 1.00 mol of a monatomic perfect gas with Cv,m = 32 R, initially at 298 K
and 10 L, is expanded with the surroundings maintained at 298 K, to a final volume of 20 L, in
different ways:
(a) isothermally and reversibly;
(b) isothermally against a constant external pressure of zero atm;
(c) isothermally against a constant external pressure of 0.50 atm;
(d) adiabatically and reversibly;
(e) adiabatically against a constant external pressure of 0.50 atm
Calculate S, Ssurr, and Stotal for each path.
5. When a perfect gas is allowed to expand isothermally in a piston, U = q + w = 0.
Thus the work done by the system on the surroundings is equal to the heat transferred from the
reservoir to the gas, the efficiency of turning heat into work is 100%. Explain why this is not a
violation of the second law.
Entropy Changes for Reversible Phase Changes
6. The heat capacity of solid iodine between 0C and the melting temperature 113.60C is
represented by the equation (with T in C and Cp in J K1 mol1)
2
C p  54.68  13.4  10 4 T  25
The molar heat of fusion is 15650 J mol1 at the melting point. The entropy of solid
iodine is 117 J K1 mol1 at 25C. What is the entropy of liquid iodine at the melting point?
7. Calculate the change in entropy when 200 g of (a) water at 0C (b) ice at 0C is added
to 200 g of water at 90C in an insulated container at constant P. Note: Cp,m = 75.29 J K mol
and Hfus = 6000 J K1 mol1.
Entropy Changes for Reversible Temperature Changes
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CHEM 341. Fall 2000. Problem Set #5.
8. An ideal, monatomic gas is compressed from 1 bar to 3 bar while being cooled from
500 K to 300 K. Calculate U, H, and S for this process.
9. Calculate the change in entropy (in terms of n, R, and Cv,m) when a monatomic perfect
gas is compressed to half its volume and simultaneously heated to twice its initial temperature.
Entropy Changes at Constant P or V
10. Calculate the entropy change when 1 kg of lead is heated from 315 K to 450 K at
constant P.
0.96  10 5
C p,m  22.13  11.72  10 3 T 
T2
11. One mole of an ideal, monatomic gas undergoes the processes listed below (all
reversible). State if q, w, U, H, and S for each case is negative, zero, positive, or cannot be
determined from the information given. No explanation is needed.
(a) cooling at constant volume;
(b) cooling at constant pressure;
(c) isothermal compression;
(d) adiabatic compression.


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