Explain why two bodies at different temperatures $T_{1}$ and $T_{2}$ if brought in thermal contact do not necessarily settle to the mean temperature $(T_{1} + T_{2}) / 2.$

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Solution

When two bodies at different temperatures $T_{1}$ and $T_{2}$ are brought in thermal contact, heat energy flows from the body at higher temperature to the body at lower temperature till thermal equilibrium is attained.
Let m be the mass of both bodies and $C_{1}$ and $C_{2}$ be the specific heat capacities of bodies at temperatures $T_{1}$ and $T_{2}$ respectively. Also, assume that $T_{1} > T_{2} .$ Since the body at lower temperature absorbs all the energy supplied by a hot body,
$Δ Q_{1} = - Δ Q_{2}$
$∴ m c_{1} Δ T_{1} = - m c_{2} Δ T_{2}$ .........(1)
If 'T' is the equilibrium temperature
$∴ m c_{1} (T_{1} - t) = - m c_{2} (T_{2} - T)$ ...........(2)
Re-arranging equation (2), we get
$T = \frac{c_{1} T_{1} + c_{2} T_{2}}{C_{1} + C_{2}}$
The equilibrium temperature depends on the specific heat capacities of the two bodies and hence cannot be $(T_{1} + T_{2}) / 2$ in general.
(Note: The common temperature will be $(T_{1} + T_{2}) / 2$ only if the specific heat capacities are the same, i.e., if $C_{1} = C_{2},$ then $(T_{1} + T_{2}) / 2.$

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Explain why two bodies at different temperatures T1 and T2 if brought in thermal contact do not necessarily settle to the mean temperature (T1+T2)/2.

Explain why two bodies at different temperatures $T_{1}$ and $T_{2}$ if brought in thermal contact do not necessarily settle to the mean temperature $(T_{1} + T_{2}) / 2.$