Question
One $$kg$$ of a diatomic gas is at a pressure of $$8 \times {10^4}\,N/{m^2}.$$ The density of the gas is $$4kg/{m^3}.$$ What is the energy of the gas due to its thermal motion?
A.
$$5 \times {10^4}\,J$$
B.
$$6 \times {10^4}\,J$$
C.
$$7 \times {10^4}\,J$$
D.
$$3 \times {10^4}\,J$$
Answer :
$$5 \times {10^4}\,J$$
Solution :
Volume $$ = \frac{{{\text{mass}}}}{{{\text{density}}}} = \frac{1}{4}{m^3}$$
$$K.E = \frac{5}{2}PV = \frac{5}{2} \times 8 \times {10^4} \times \frac{1}{4} = 5 \times {10^4}\,J$$
Alternatively:
$$\eqalign{
& K.E = \frac{5}{2}nRT = \frac{5}{2}\frac{m}{M}RT = \frac{5}{2}\frac{m}{M} \times \frac{{PM}}{d}\,\,\left[ {\because PM = dRT} \right] \cr
& = \frac{5}{2}\frac{{mP}}{d} = \frac{5}{2} \times \frac{{1 \times 8 \times {{10}^4}}}{4} = 5 \times {10^4}\,J \cr} $$