Assignment 4 (PDF)

Homework 4 Problems – Entropy and Entropy Changes
1. Consider the freezing of 50.0 g of water once it is placed in the freezer compartment of a
refrigerator. Assume the walls of the freezer are maintained at –10ºC. The water, initially
liquid at 0.0ºC, is frozen into ice and cooled to –10ºC. Show that even though the entropy of
the water decreases, the net entropy of the universe increases.
2. For this problem, you will calculate the entropy of aluminum and graphite based on
experimental measurements of the heat capacity.
a. Experimental measurements of the heat capacity of aluminum at low temperatures
(below about 50 K) can be fit to the formula:
𝐶𝑉 = 𝑎𝑇 + 𝑏𝑇 3
where CV is the heat capacity of one mole of aluminum, and the constants a and b are
approximately a = 0.00135 J/K2 and b = 2.48 x 10-5 J/K4. From this data, find a
formula for the entropy of a mole of aluminum as a function of temperature.
b. Experimental measurements of the heat capacity of graphite over a wide range of
temperatures can be fit to the formula:
𝐶𝑉 = 𝑎 + 𝑏𝑇 − 𝑐/𝑇 2
where a = 16.86 J/K, b = 4.77 x 10-3 J/K2, and c = 8.54 x 10-5 J/K4. From this data,
find a formula for the entropy of a mole of graphite as a function of temperature.
3. A solid with heat capacity CA at temperature TA is placed in contact with another solid with
heat capacity CB at a lower temperature TB.
a. Show that the change in entropy due to thermal contact is given by
𝑇
𝑇
∆𝑆 = 𝐶𝐴 ln + 𝐶𝐵 ln
𝑇𝐴
𝑇𝐵
b. Suppose that one of the solids is a heat reservoir such that 𝐶𝐵 → ∞. Show that the
change in entropy due to thermal contact is given by
𝑇𝐵 𝑇𝐴
∆𝑆 = 𝐶𝐴 [ln + − 1]
𝑇𝐴 𝑇𝐵
c. For both (a) and (b), show that ∆𝑆 > 0 for 𝑇𝐴 ≠ 𝑇𝐵
4. Consider a mole of an ideal gas initially at (P1,V1) and finally at (P2,V2).
a. Show that the change in entropy of the gas is given by
𝑇2
𝑃2
𝑆2 − 𝑆1 = 𝐶𝑃 ln ( ) − 𝑅 ln ( )
𝑇1
𝑃1
b. Find an expression for the ∆𝑆(𝑈, 𝑉) and ∆𝑆(𝑇, 𝑉)
c. Based on the results of (a) and (b), determine what types of spontaneous processes
are permissible for an isolated monatomic ideal gas according to the 2nd law.
What types of spontaneous processes are not permissible based on the 2nd law?
5. Potential temperature and static stability
a. Use Poisson’s equation to show that 𝑠 = 𝐶𝑃 ln 𝜃 + 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
b. Show that the vertical gradient of potential temperature can be written as
1 𝜕𝜃 1
= (𝛤 − 𝛤)
𝜃 𝜕𝑧 𝑇 𝑑
where 𝛤𝑑 is the dry adiabatic lapse rate and 𝛤 ≡ 𝑑𝑇/𝑑𝑧 is the environmental lapse
rate.
c. Use (b) to show that if the potential temperature increases with height, the
atmosphere is stable (i.e. dry spontaneous convection ceases). Conversely, show that
if the potential temperature decreases with height, the atmosphere is unstable (i.e. dry
spontaneous convection occurs).

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