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262.


Problem 26.41 (RHK)


One mole of a monatomic ideal gas is taken from an initial state of pressure and volume to a final state of pressure by two different processes. (1) It expands isothermally until its volume is doubled, and then its pressure is increased at constant volume to the final state. (2) It is compressed isothermally until its pressure is doubled, and then its volume is increased at constant pressure to the final state. We have to show the path of each process on a pV diagram. For each process we have to calculate in terms of and : (a) the heat absorbed by the gas in each part of the process; (b) the work done on the gas in each part of the process; (c) the change in the internal energy of the gas, ; and (d) the change in entropy of the gas,

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We are considering one mole of a monatomic ideal gas as it undergoes thermodynamic processes as shown in the diagram. Its initial state is at the point a in the diagram. At a pressure of the gas is and its volume is . Its temperature at a will therefore be

(1)

As the gas undergoes isothermal expansion at temperature from volume to from a to b along the path 1, the work done on the gas will be

As the process is isothermal there is no change in the internal energy of the gas, therefore, the heat absorbed by the gas, , during the process 1 will be

The change in entropy

(2)

The process b to d takes place at constant volume. Therefore, no work is done on the gas during this process. Therefore, the heat absorbed by the gas during 2 will be

We have used that for a monatomic gas

and at b

And at d

And

The change in entropy of the gas during the process 2 will be

The change in entropy from a to d in the combined processes 1 and 2 will be

(3)

We consider part 3 process. It is an isothermal compression at temperature from volume to . Therefore, the work done on the gas

and

Therefore,

(4)

In the process 4 the gas is subjected to an isobaric expansion from volume to at pressure .

Therefore, the work done on the gas during the path 4 will be

And the heat absorbed by the gas during 4 will be

We have used that for an ideal monatomic gas

And

The change in entropy

Therefore,