Question

Write the mathematical form of the first law
of thermodynamics for (i) a thermally
insulated system, and (ii) a perfect gas
expanding into vacuum under isothermal
conditions.

Answer 1

Write the mathematical form of the first law of thermodynamics for (i) a thermally insulated system, and (ii) a perfect gas expanding into vacuum under isothermal conditions.

Answer

$W$ - work done by system.

(i) A thermally insulated system.

The insulation prevents heat from flowing into or out of the system, so any change in the system is adiabatic. The first law of thermodynamics:

$$U _ {2} - U _ {1} = \Delta U = - W.$$

When a system expands adiabatically, $W$ is positive (the system doesn't work on its surroundings), $\Delta U$ is negative and the internal energy decreases.

(ii) A perfect gas expanding into vacuum under isothermal conditions.

$$U _ {2} - U _ {1} = \Delta U = 0 \rightarrow U = \text{const}, U (T) = C _ {v} T.$$

$C_v$ - the heat capacity of perfect gas at constant volume.

The gas doesn't work on its surroundings and there is no heat flow through the insulation. So both $Q$ and $W$ are zero and the internal energy is constant.