A proposed ocean power plant will utilize the temperature difference between surface seawater and seawater at a depth of 100 meters. Assume the surface temperature is 26° Celsius and the temperature at the 100-meter depth is 2° Celsius.

a) What is the ideal (Carnot) efficiency of the plant?

b) If the plant generates useful energy at the rate of 100 megawatts while operating with the efficiency found in part (a), at what rate is heat given off to the surroundings?

c) The diagram below represents the Carnot cycle for a simple reversible (Carnot) engine in which a fixed amount of gas, originally at pressure p_{o} and volume V_{o} follows the path ABCDA*. *

In the chart below, for each part of the cycle indicate with +, -, or 0 whether the heat transferred Q and temperature change ΔT are positive, negative, or zero, respectively. (Q is positive when heat is added to the gas, and ΔT is positive when the temperature of the gas increases.)

The pV-diagram below represents the states of an ideal gas during one cycle of operation of a reversible heat engine. The cycle consists of the following four processes.

**Process Nature of Process**

AB Constant temperature ( Th = 500 K)

BC Adiabatic

CD Constant temperature ( Tc = 200 K)

DA Adiabatic

During process AB, the volume of the gas increases from V_{o} to 2V_{o} and the gas absorbs 1,000 joules of heat.

a) The pressure at A is p_{o}. Determine the pressure at B.

b) Using the first law of thermodynamics, determine the work performed on the gas during the process AB.

c) During the process AB, does the entropy of the gas increase, decrease, or remain unchanged? Justify your answer.

d) Calculate the heat Q_{c} given off by the gas in the process CD.

e) During the full cycle ABCDA is the total work the performed on the gas by its surroundings positive, negative, or zero? Justify your answer.

One mole of an ideal gas is initially at pressure *P*_{1}, volume *V*_{1}, and temperature *T*_{1}, represented by point *A *on the *PV *diagram above. The gas is taken around cycle *ABCA *shown. Process *AB *is isobaric, process *BC *is isochoric, and process *CA *is isothermal.

a) Calculate the temperature *T*_{2 }at the end of process *AB *in terms of temperature T_{1}.

b) Calculate the pressure *P*_{2 }at the end of process *BC *in terms of pressure *P*_{1}.

c) Calculate the net work done on the gas when it is taken from *A *to *B *to *C*. Express your answer in terms of *P*_{1} and *V*_{1}.

d) Indicate below all of the processes that result in heat being added to the gas.

____ *AB ____ **BC ____ **CA*

Justify your answer.

An ideal gas initially has pressure p_{o}, volume V_{o}, and absolute temperature T_{o}. It then undergoes the following series of processes:

I. It is heated, at constant volume, until it reaches a pressure 2p_{o}.

II. It is heated, at constant pressure, until it reaches a volume 3 V_{o}.

III. It is cooled, at constant volume, until it reaches a pressure p_{o}.

IV. It is cooled, at constant pressure, until it reaches a volume V_{o}.

a) On the axes below

- draw the p-V diagram representing the series of processes;
- label each end point with the appropriate value of absolute temperature in terms of T
_{o}.

b) For this series of processes, determine the following in terms of p_{o} and V_{o}.

- The net work done on the gas
- The net change in internal energy
- The net heat absorbed

One mole of an ideal gas is taken around the cycle *A→B→C→A *as shown on the PV diagram above.

a) Calculate the temperature of the gas at point *A*.

b) Calculate the net work done on the gas during one complete cycle.

c) i. Is heat added to or removed from the gas during one complete cycle?

_____added to the gas _____removed from the gas

ii) Calculate the heat added to or removed from the gas during one complete cycle.

d) After one complete cycle, is the internal energy of the gas greater, less, or the same as before?

_____greater _____less _____the same

Justify your answer.

e) After one complete cycle, is the entropy of the gas greater, less, or the same as before?

_____greater _____less _____the same

Justify your answer.

A sample of ideal gas is taken through steps I, II, and III in a closed cycle, as shown on the pressure *P *versus volume *V *diagram above, so that the gas returns to its original state. The steps in the cycle are as follows.

I. An isothermal expansion occurs from point *A *to point *B, *and the volume of the gas doubles.

II. An isobaric compression occurs from point *B *to point *C, *and the gas returns to its original volume.

III. A constant volume addition of heat occurs from point *C *to point *A *and the gas returns to its original pressure.

a) Determine numerical values for the following ratios, justifying your answers in the spaces next to each ratio.

; ;

b) During step I, the change in internal energy is zero. Explain why.

c) During step III, the work done on the gas is zero. Explain why.