JEE Main & Advanced Chemistry States of Matter / पदार्थ की अवस्थाएँ - गैस एवं द्रव Specific And Molar Heat Capacity Of Gases

(1) Specific heat (or specific heat capacity) of a substance is the quantity of heat (in calories, joules, kcal, or kilo joules) required to raise the temperature of 1g of that substance through \[{{1}^{o}}C\]. It can be measured at constant pressure \[({{c}_{p}})\] and at constant volume \[({{c}_{v}})\].

(2) Molar heat capacity of a substance is the quantity of heat required to raise the temperature of 1 mole of the substance by \[{{1}^{o}}C\].

\[\therefore \]          Molar heat capacity = Specific heat capacity ´ Molecular weight, i.e.,

\[{{C}_{v}}={{c}_{v}}\times M\] and \[{{C}_{p}}={{c}_{p}}\times M\].

(3) Since gases upon heating show considerable tendency towards expansion if heated under constant pressure conditions, an additional energy has to be supplied for raising its temperature by \[{{1}^{o}}C\] relative to that required under constant volume conditions, i.e.,

\[{{C}_{p}}>{{C}_{v}}\]or \[{{C}_{p}}={{C}_{v}}+\text{Work done on expansion, }P\Delta V(=R)\]

where, \[{{C}_{p}}=\] molar heat capacity at constant pressure; \[{{C}_{v}}=\] molar heat capacity at constant volume.

(4) Some useful relations of C_p and C_v

(i) \[{{C}_{p}}-{{C}_{v}}=R=2\,calories=8.314J\]

(ii) \[{{C}_{v}}=\frac{3}{2}R\] (for monoatomic gas) and \[{{C}_{v}}=\frac{3}{2}+x\] (for di and polyatomic gas), where x varies from gas to gas.

(iii) \[\frac{{{C}_{p}}}{{{C}_{v}}}=\gamma \] (Ratio of molar capacities)

(iv) For monoatomic gas \[{{C}_{v}}=3\,calories\] whereas, \[{{C}_{p}}={{C}_{v}}+R=5calories\]

(v) For monoatomic gas, \[(\gamma )=\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{\frac{5}{2}R}{\frac{3}{2}R}=1.66\].

(vi) For diatomic gas \[(\gamma )=\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{\frac{7}{2}R}{\frac{5}{2}R}=1.40\]

(vii) For triatomic gas \[(\gamma )=\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{8R}{6R}=1.33\]