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.)
2. Evaluate for a van der Waals gas in terms of n, P, V, and/or 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. Show that the entropy change is given by if Cp is constant. How does 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 = (3R)/2 , 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 D S, D Ssurr, and D Stotal for each path.
5. When a perfect gas is allowed to expand isothermally in a piston, DU = 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)
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-1 mol-1 and DHfus = 6000 J K-1 mol-1.
Entropy Changes for Reversible Temperature Changes
8. An ideal, monatomic gas is compressed from 1 bar to 3 bar while being cooled from 500 K to 300 K. Calculate DU, DH, and DS 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.
11. One mole of an ideal, monatomic gas undergoes the processes listed below (all reversible). State if q, w, D U, D H, and D 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.