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

The general form of the first principle of thermodynamics is as follows

$$\delta Q=dU+\delta W$$

(i) in the case when system is thermally insulated $\delta Q=0$, so we get equation

$$dU+\delta W=0$$

(ii) for a perfect gas expanding into vacuum under isothermal conditions we have

$$\delta W=0 \quad {\rm and}\quad dU=0$$

so, the mathematical form of the first law of thermodynamics in this case is as follows

$$\delta Q=0$$