Physics 3333, Thermodynamics Fall 1999

Dr. Cox

Test 2 Friday, November 19

Due: Wednesday, November 24, 12:00 noon.. Kingsville students may hand theirs in; West Texas should have work faxed, sometime Wednesday morning.

Number, and put your name on, all pages you turn in. If you use the back of any sheet, that is a different page.

Take-home rules:

Before you sit down to work a problem for credit, you may read it, discuss it, consult texts, notes, or references (except another student's test paper), even work through it with such aids.

But, when you are ready to work a problem for credit, you must begin with only blank paper, writing tool(s), calculator (if needed), the test questions, standard tables (such as, of integrals, of physical constants), and what you have learned. You may decide to abandon such an effort, and study more; if so, you must make a clean restart.

(For elaboration of these rules, see "Hewett's Take-home Exam Instructions.")

Notation: To save complications in e-mail, the following conventions apply:

^ indicates superscripts (either labels or exponents).

_ indicates subscripts (thus, x_1 means x sub 1)

s indicates sigma

Volumes are expressed in liters, L.

Use common fractions to express numerical values when reasonable; otherwise, retain 3 significant digits.

1. A particular Carnot-cycle device uses a sample of a monatomic ideal gas, for which constant-entropy processes obey the equation (P^3) (V^5) = constant. This sample is initially at a pressure of 10^5 N m^(-2) and a temperature of 1000 K, occupying a volume of 1 L. In the first stage of the cycle, the gas expands isothermally to 4 L; in the second stage, the gas expands further to 32 L.

a. Calculate the (signed) work done by the device in each of the four stages of the Carnot cycle.

b. Since this is a monatomic ideal gas for which initial values of P, V, and T are given, N can be determined, and also U as a function of T. Using this information, calculate the (signed) heat input to the device in each of the four stages.

Is this device currently acting as a heat engine or as a heat pump?

c. From the heat transfers, determine the efficiency of this device when used as a heat engine. From the efficiency, determine the highest and lowest temperatures reached in the cycle. (One of those is the initial temperature.)

2. Two particular systems obey the relations 1/T_1 = (5/2) R N_1 / U_1 and 1/T_2 = (7/2) R N_2 / U_2. The systems are set up with N_1 = 2 mole, T_1 = 200 K, and N_2 = 3 mole, T_2 = 300 K. The two systems are then placed in thermal contact. Find the final (equilibrium) temperature, and the final energies U_1 and U_2.

3. The total solar energy available at the Earth's average distance from the Sun (1.5x10^8 km), measured above Earth's atmosphere, is 0.136 W/cm^2.

a. Treating the Sun as a blackbody of radius 7x10^5 km, determine the average temperature of its emitting surface. For this temperature, at what frequency is the maximum of the energy distribution? What color (or, if it does not correspond to visible, what region of the spectrum) corresponds to this frequency?

b. On average, the Earth is in thermal equilibrium, receiving energy from the Sun and radiating it away in all directions. Treating the Earth as a spherical blackbody with a single temperature throughout, determine that temperature. (This averages out day-night and pole-equator variations.) Comparing your result with the world around you, does this treatment seem valid?

4. A type of system has states characterized by the variables energy, volume, and numbers of particles of each of two types. Systems of this type obey the equation

s(U, V, N_1, N_2) = N ln(a U^(3/2) V N^(-5/2) ) - N_1 ln(N_1 / N) - N_2 ln(N_2 / N)

where N = N_1 + N_2 and a is a positive constant.

Two containers, each of 20 L volume, are perfectly isolated from the rest of the world, but separated from each other by a rigid barrier through which heat and type-1 particles can flow, but not type-2 particles. (This is a "semi-permeable" barrier.)

Initially, container A has a sample with N_1^A = 2 mole, N_2^A = 3 mole, T^A = 300 K; container B has a sample with N_1^B = 4 mole, N_2^B = 2 mole, T^B = 200 K. After equilibrium is reached, what are the values of N_1^A, N_1^B, T, P^A, P^B?

(Remember the conditions for equilibrium when energy can be transferred, and when particles can be transferred. Transformation and changing of variables are required. The indicated quantities are not all independent, so one can change variables in several ways.)