1. there is no change in temperature, and

2. there is no change in internal energy.

We know that

Q1-2 = dU + W1-2 = W1-2

Hence during isothermal expansion of a gas,

Heat added = Work done by the gas

Similarly, during isothermal compression of a gas,

Heat subtracted = Work done on the gas

A little consideration will show that the isothermal process is governed by Boyle's law. Thus the isothermal equation of a perfect gas is pv = Constant.

Now consider a certain quantity of a perfect gas being expanded isothermally, which is shown by the curve 1-2 in Fig. 5.3.

Let

v1 = Initial volume of gas,

p1 = Initial pressure of gas,

v2 = Final volume of gas, and

p2 = Final pressure of gas.

The workdone during isothermal expansion is given by

W1-2 = Area under the curve 1-2

In general,

where r is the expansion ratio (when gas is heated) or compression ratio (when gas is cooled).

**Notes:**

(a) Expansion ratio (r) = (Volume at the end of expansion)/(Volume at the beginning of expansion)

(b) Compression ratio (r) = (Volume at the beginning of compression)/(Volume at the end of compression)

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