# JEE Main & Advanced Chemistry States of Matter Specific And Molar Heat Capacity Of Gases

## Specific And Molar Heat Capacity Of Gases

Category : JEE Main & Advanced

(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 Cp and Cv

(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 $(\gamma )=\frac{{{C}_{p}}}{{{C}_{v}}}=\frac{8R}{6R}=1.33$

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