A mass ‘m’ of fluid at temperature T₁ is mixed with an equal amount of the same fluid at temperature T₂. Prove that the resultant change of entropy of the universe is: [2 mc ln (T₁ + T₂) / 2]/[(T₁ T₂)^1/2]and also prove that it is always positive.

Mean temperature of the mixture = (T₁ + T₂) / 2

Thus change in entropy is given by:
∆S = S₂ – S₁)

= mc ^{(T₁ + T₂)/2}∫ _T1 (dT / T) – mc ^T₂∫ _{T1 + T2)/2} (dT / T)

= mc ln (T₁ + T₂)/2 T₁) –
mc ln (2 T₂) / (T₁ + T₂)
= mc ln (T₁ + T₂)/2 T₁) +
mc ln (T₁ + T₂) /2 T₂)

= mc ln (T₁ + T₂) ² / 4 T₁ T₂

= mc ln [(T₁ + T₂) / 2 (T₁ T₂) ^1/2] ²

= 2 mc ln [(T₁ + T₂) / 2 (T₁ T₂) ^1/2]

= 2 mc ln [(T₁ + T₂) / 2 ]/[ (T₁ T₂)^1/2]

Hence, Resultant change of entropy of universe is:
2 mc ln [(T₁ + T₂) / 2]/[ (T₁ T₂)^1/2]

The arithmetic mean (T₁ + T₂) / 2 is greater than the geometric mean (T₁ T₂) ^1/2

Therefore, ln [(T₁ + T₂) / 2]/[ (T₁ T₂)^1/2 ]

is always positive. Hence the entropy of the universe increases.