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The Heat Capacity is defined as the quantity of heat required to raise the temperature of system by 1 degree. $$ \therefore Heat Capacity (C) = \frac { q }{ dT } \qquad .... (1) $$ where q is heat absorbed and \(dT\) is rise in temperature.
Heat Capacity at constant volume is given by, $$ { C }_{ v } = \frac { { q }_{ v } }{ dT } $$ But \({ q }_{ v } = \Delta E\) $$ \boxed { \therefore { C }_{ v } = \frac { \Delta E }{ dT } = { \left( \frac { \partial E }{ \partial T } \right) }_{ v } } \qquad .... (2) $$ Therefore heat capacity at constant volume may be defined as the rate of change of internal energy with temperature at constant volume.
Heat Capacity at constant pressure is given by, $$ { C }_{ p } = \frac { { q }_{ p } }{ dT } $$ But \({ q }_{ p } = \Delta H \) $$\boxed { \therefore { C }_{ p } = \frac { \Delta H }{ dT } = { \left( \frac { \partial H }{ \partial T } \right) }_{ p } } \qquad .... (3) $$ Therefore heat capacity at constant pressure may be defined as the rate of change of enthalpy with temperature at constant pressure.
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