It is appropriate to illustrate the relationship between ΔS and q with a consideration of the relationship between entropy and volume. Our consideration will be the simple case of isothermal expansion of a perfect gas, but it is possible to apply the equations to more complicated situations.
Our starting point is the following equation, the definition of entropy change:
Given that the expansion is isothermal, the temperature is a constant that may be taken outside the integral, allowing us to write
In this situation, ΔU = 0 (as there is no temperature change, and the particles of a perfect gas do not interact at all so are unaffected by being more widely separated) , allowing us to use the First Law of Thermodynamics to write
The work of reversible expansion is given by wrev = – ∫ pdV . For a perfect gas, we may use the equation of state pV = nRT to substitute for p and obtain
which integrates to
when Vf > Vi (i.e. when expansion occurs) the ratio of volumes > 1 so the logarithmic term and hence the overall entropy change are both positive, as we expect. Conversely, if Vf < Vi (i.e. contraction) the ratio of volumes < 1 so the logarithmic term and overall entropy change are both negative.