One mole of an ideal gas requires $$207\,J$$  heat to rise the temperature by $$10\,K$$  when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by the same $$10\,K,$$  the heat required is (Given the gas constant $$R = 8.3\,J/mol - K$$    )

Solution :
Molar specific heat of a substance is defined as the amount of heat required to raise the temperature of one gram mole of the substance through a unit degree.
$$\eqalign{ & {\text{As}}\,{\left( {dQ} \right)_p} = \mu {C_p}dT\,\,......\left( {\text{i}} \right)\,\left( {{\text{At}}\,{\text{constant pressure}}} \right) \cr & {\text{and}}\,{\left( {dQ} \right)_V} = \mu {C_V}dT\,\,......\left( {{\text{ii}}} \right)\,\left( {{\text{At}}\,{\text{constant volume}}} \right) \cr & {\text{Given,}}\,{\left( {dQ} \right)_p} = 207\,J \cr & R = 8.3\,J/mol - K \cr & dT = 10\,K \cr & {\text{Putting value in Eq}}{\text{. }}\left( {\text{i}} \right) \cr & 207 = 1 \times {C_p} \times 10 \cr & \therefore {C_p} = 20.7\,J/kg \cr & {\text{As}}\,{C_p} - {C_V} = R = 8.3 \cr & {C_V} = 20.7 - 8.3 = 12.4\,J \cr & \therefore {\left( {dQ} \right)_V} = 1 \times 12.4 \times 10 \cr & = 124\,J \cr} $$