Answer to Question #98807 in Electrical Engineering for Hardi Amiru Sulemana

Question

Show that during the expansion of a quantity of gas according to the law PV^n = constant,the heat energy transfer Q and W are connected by the expression
Q={r - n/r - 1}W .

Answer 1

The work during that process can be expressed as

$$W=\frac{p_bV_b-p_aV_a}{n-1}.$$

According to the first law of thermodynamics, for a gas with degrees of freedom of molecules $i$ we can write:

$$Q=\Delta U+W=\frac{i}{2}\frac{m}{\mu}R\Delta T+\frac{p_bV_b-p_aV_a}{n-1}.$$

Since

$$pV=\frac{m}{\mu}RT,$$

we can now write that

$$Q=\Delta U+W=\frac{i}{2}\frac{m}{\mu}R\Delta T+\frac{\frac{m}{\mu}R\Delta T}{n-1}=\\ \space\\ =\frac{m}{\mu}R\Delta T\bigg(\frac{i}{2}+\frac{1}{n-1}\bigg)=\frac{m}{\mu}R\Delta T\cdot\frac{in-i+2}{2(n-1)}.$$