The amount of heat needed to raise the temperature of 4 moles of rigid diatomic gas from 0$^\circ$ C to 50$^\circ$ C when no work is done is ___________. (R is the universal gas constant).

According to first law of thermodynamics,<br/><br/>$\Delta$Q = $\Delta$U + $\Delta$W ..... (i)<br/><br/>where, $\Delta$Q = quantity of heat energy supplied to the system, $\Delta$U = change in the internal energy of a closed system and $\Delta$W = work done by the system on its surroundings.<br/><br/>As per question, no work is done<br/><br/>$\therefore$ $\Delta$W = 0 ..... (iii)<br/><br/>From Eqs. (i) and (ii), we get<br/><br/>$\Delta$Q = 0 + $\Delta$U $\Rightarrow$ $\Delta$Q = $\Delta$U<br/><br/>or $\Delta$Q = $\Delta$U = nC_V$\Delta$T<br/><br/>where, <br/><br/>C_V = specific heat capacity at constant volume for diatomic gas = ${{5R} \over 2}$<br/><br/>$\Delta$T = change in temperature = (50 $-$ 0) = 50$^\circ$C<br/><br/>n = number of moles = 4<br/><br/>$\Rightarrow$ $\Delta$Q = nC_V$\Delta$T<br/><br/>= $4 \times {{5R} \over 2} \times (50)$ = 500 R = 500 R