Physics 410 – Test #2

30 March – 4 April 2016

Name______

Directions

I.You may use only your textbook, Physics 410 notes, Physics 410 homework, Thermodynamics Processes handout, a calculator, your favorite Mathematics software (like Mathematica), me (Richard Thomas), and our Physics 410 course web site (and the links therein) for this exam. (#3 below is an exception to this rule.)

II.You may not discuss this exam with anyone else but me.

III.You have no time limit for this exam, but it is due at the start of class on Monday, 4April 2016. No late exams will be accepted. Of course you may turn it in earlier.

IV.SHOW YOUR WORK! Incorrect answers without reasoning will receive zero credit. Correct answers without reasoning given might also get docked.

  1. 0.020 moles of a monoatomic ideal gas in a box with a frictionless piston are used to operate a heat engine. Initially, the gas is at 300K, and 0.005 m3. The engine is operated in the following way:
  1. The piston is clamped down so that it cannot move, and the gas is immersed in a 400K thermal reservoir until the gas itself is at 400K.
  2. The gas is removed from the 400K reservoir and insulated so that it cannot exchange heat with the surroundings. The piston is unclamped, and the gas pushes it up until the gas is at 250K.
  3. The piston is clamped down again and the gas is then immersed in a 187.5K reservoir until the gas itself is 187.5K.
  4. The gas is removed from the reservoir and insulated so that no heat is exchanged with its surroundings. The piston is pushed down until the gas is back at 300K.

Notice that there are only two reservoirs: a hot one (400K) and a cold one (187.5K).

a)(10 points) Determine the amount of heat taken from the hot reservoir, the amount of heat expelled to the cold reservoir, and the net work done by the gas. Draw a “cloud and arrows” diagram showing these three things.

b)(5 points) Determine the efficiency of this engine, and compare it to that of a Carnot engine operating between the same two temperatures (400K and 187.5K).

c)(10 points) Determine the ∆S of the 400K reservoir and the 187.5K reservoir. What is the TOTAL ∆S of the gas and two reservoirs (assuming that their temperatures don’t change)? Is this what you expect? Why or why not?

  1. Now run the engine in the previous problem in reverse, so that it operates as a refrigerator! The amount of work the heat engine did in problem 1 is now the amount of work you must do on the gas here. Recall that heat cannot be removed from the inside of the refrigerator unless the gas itself is at a temperature lower than the inside of the refrigerator. Also, the gas cannot expel heat to the surroundings unless it is hotter than the surroundings.

a)(10 points) Determine the amount of heat removed from the inside of the refrigerator, the amount of heat expelled to the surroundings, and the work done on the gas for each cycle.

b)(5 points) Determine the coefficient of performance of this refrigerator and compare it to that of a Carnot refrigerator with the same outside and inside temperatures. Is this what you expect? Why or why not?

  1. (20 points) Describe how a real refrigerator works. Include the relevant P-V diagram, and relate each step in the diagram to what is really going on in the refrigerator. You may use sources in addition to the ones listed above in the instructions to answer this question, but your explanations must be in your own words.

4. (10 points) Why is ice slippery? One theory often quoted in elementary physics textbooks goes like this: When you stand on ice wearing your hockey skates, you increase the pressure on the ice. This increased pressure lowers the melting temperature of the ice, heat comes out of the environment (without changing the temperature of the environment), and some of the ice melts. The thin layer of water on top of the ice makes it slippery, allowing the hockey skater to skate freely. Consider an outdoor ice hockey rink on a cloudy day at 23oF with skate blades at the same temperature. Disprove this theory OR show that it can work this way. The density of ice is 920 kg/m3, you may consider the density of water to be 1000 kg/m3, and the latent heat of fusion of ice is 333,550 J/kg. Assume hockey skate blades are about 1/8 inch wide and about a foot long, and that the mass of a hockey player with equipment is about 100 kg.

  1. Much of this problem is done for you in your textbook. Your answers and explanations here, however, must be your own. You may NOT just copy something.

Consider the differential form of the combined 1stand 2ndlaws:

or

so that and .

Note that this implies that U is a function of S and V, , consistent with an idea we have gone over much of the semester. However, this is only true if the number of particles, N, is constant. In general, though, the number of particles can vary (dN ≠ 0), and changing the number of particles will also change the internal energy. Thus,

or

a)(5 points) What physical quantity does represent? (It is one we’ve covered already in this class.)

b)(5 points) Consider a function f of three variables: . According to Euler, if λ is a constant and , then

.

Note that this condition applies to . Use this to show that .

c)(10 points) Show that .

6. (10 points) Derive an expression for the specific entropy change() of a Dieterici gas. The Dieterici equation of state is

where a and b are constants that depend on the gas. Your final expression should be expressed in terms of cv, T, v, and other constants (like a, b, maybe To, etc.).

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By writing my name above, I affirm that this test represents my work only, without aid from outside sources. In all aspects of this course I perform with honor and integrity.