Thermal equilibrium: solid + gas

When two bodies at different temperatures are brought into contact, they reach thermal equilibrium: heat flows spontaneously from hot to cold until the temperatures equalise. If the container is isolated and rigid, the problem can be solved with energy conservation alone.

Consider a block of metal immersed in a rigid cylinder containing a gas, the whole thermally isolated. Since the container is isolated ($Q_\text{external}=0$) and rigid ($L=0$), the total internal energy of the system (block + gas) is conserved. The heat given up by the hot body is exactly the heat absorbed by the cold body:

$m_\text{sol}\,c_s\,(T_f - T_\text{sol}) + \frac{f}{2}nR\,(T_f - T_\text{gas}) = 0$

From this single equation the equilibrium temperature $T_f$ can be found.

The two forms of thermal energy

• Solids and liquids: $E_\text{term} = m\,c_s\,T$ (with $T$ in kelvin, $c_s$ = specific heat).
• Ideal gases: $E_\text{int} = \frac{f}{2}nR\,T$. In an isolated container their sum stays constant.

The transformation is clearly irreversible: heat flows spontaneously from hot to cold, never the other way round. Yet energy is perfectly conserved. This is the clearest example of the fact that the first law allows many transformations that in nature only go one way — it is the second law, not the first, that fixes the direction.

The quantitative case (aluminium block in nitrogen) is worked out in Aluminium block in nitrogen.

Links

Topics: Thermodynamics Concepts: First law of thermodynamics · Internal energy · Temperature Skills: Calorimetric equation Objects: Ideal gas · Calorimeter

Related exercises: Nitrogen heated in a rigid cylinder · Heating under the piston · Heat as a substance?