Question
Internal energy and pressure of a gas per unit volume are related as
A.
$$p = \frac{2}{3}E$$
B.
$$p = \frac{3}{2}E$$
C.
$$p = \frac{1}{2}E$$
D.
$$p = 2E$$
Answer :
$$p = \frac{2}{3}E$$
Solution :
The average translational kinetic energy of a gas molecule is $$\frac{1}{2}m{\overline u ^2}$$ at a temperature $$T$$ . The total energy of the whole of the gas containing $$N$$ molecules is
$$\eqalign{
& {E_k} = \frac{1}{2}m\,N{\overline u ^2}\,\,...{\text{(i)}} \cr
& {\text{The kinetic gas equation is}} \cr
& pV = \frac{1}{3}m\,N{\overline u ^2}\,\,...{\text{(ii)}} \cr
& pV = \frac{2}{3} \times \frac{1}{2}mN{\overline u ^2} \cr
& pV = \frac{2}{3}{E_k}\,\,\,...{\text{(iii)}} \cr
& {\text{so,}}\,\,p = \frac{2}{3}{E_k}\,{\text{per unit volume}} \cr} $$