Two Specific Heats of a Gas

A gas has two specific heats. Lets dive into more details. We know that the specific heat of a substance is the amount of heat required to raise the temperature of unit mass of the substance through one degree.

It holds good for solids and liquids. When heat is supplied to a solid or liquid, its temperature changes without causing any practical change in its volume or pressure. So, a definite rise in temperature can be observed when given amount of heat is supplied.

But in case of a gas, its pressure and volume also change with increase in temperature when heat is supplied. So, a gas can be heated in two ways; by keeping its volume constant or by keeping its pressure constant. Hence, a gas has two specific heats which can be defined in the following ways;

Specific Heat Capacity At Constant Volume

It is defined as the amount of heat required to raise the temperature of unit mass of the gas through unit degree at constant volume.
​If $m$ is the mass of a gas and $c_v$ is its specific heat capacity at constant volume, then the amount of heat $Q$ required to raise the temperature of the gas by $dT$ is given by, \[Q=mc_v\;dT\text{ __(1)}\]

Molar Specific Heat Capacity At Constant Volume

It is defined as the amount of heat required to raise the temperature of one mole of a gas through one degree at constant volume.
​If $n$ mole of a gas requires $Q$ amount of heat to raise its temperature through $dT$, then \[Q=nC_V\;dT\text{ __(2)}\] Where, $C_V$ is the molar specific heat of the gas at constant volume.

From $(1)$ and $(2)$, \[mc_v=nC_V\] \[C_V=\frac{m}{n}c_v\] \[C_V=Mc_v\] Where, $M$ is the molar mass of the gas. 

Specific Heat Capacity At Constant Pressure

It is defined as the amount of heat required to raise the temperature of unit mass of the gas through unit degree at constant pressure.
​If $m$ is the mass of a gas and $c_p$ is its specific heat capacity at constant pressure, then the amount of heat $Q$ required to raise the temperature of the gas by $dT$ is given by, \[Q=mc_p\;dT\text{ __(1)}\] 

Molar Specific Heat Capacity At Constant Pressure

It is defined as the amount of heat required to raise the temperature of one mole of a gas through one degree at constant pressure. 
​If $n$ mole of a gas requires $Q$ amount of heat to raise its temperature through $dT$, then \[Q=nC_PdT\text{ __(2)}\] Where, $C_P$ is the molar specific heat of the gas at constant pressure.

From $(1)$ and $(2)$, \[mc_p=nC_P\] \[C_P=\frac{m}{n}c_p\] \[C_P=Mc_p\] Where, $M$ is the molar mass of the gas.

CP is greater than CV

​When volume is kept constant, the gas cannot perform work. The heat supplied will increase the internal energy $(dU)$ i.e. temperature of the gas. Therefore, in case of $C_V$, the supplied heat increases the temperature of the gas only. \[dQ=dU+dW\] \[dQ=dU+P\;dV\]

At constant volume, $dV=0$, \[dQ=dU\] When pressure is kept constant, volume has to be increased and the system will have to do work. So, the system has to increase the temperature by the same amount as before and it has to do work also. Hence, more amount of heat is required. \[\text{Thus, }C_P > C_V\] 

Relation between CP and CV [Mayer’s Relation]

This relation was first obtained by Robert Mayer in 1842. 

Consider one mole of an ideal gas. If the gas is heated at constant volume, so that its temperature increases by $dT$, then, \[\text{Heat Supplied}=nC_V\;dT=C_V\;dT\]

Since the gas is heated at constant volume, it does not perform work. Hence, its internal energy is equal to heat supplied. \[dU=C_V\;dT\text{ __(5)}\] If the gas is heated at constant pressure, so that its temperature increases by $dT$, then, \[dQ=nC_P\;dT\] \[dQ=C_P\;dT\text{ __(6)}\]

Since the gas is heated at constant pressure, its internal energy $(U)$ increases as well as the gas performs work $dW$. Then, according to the first law of thermodynamics, \[dQ=dU+dW\text{ __(7)}\] If $dV$ is the increase in volume, then, \[dW=P\;dV\text{ __(8)}\]

From $(5)$, $(6)$, $(7)$ and $(8)$, \[C_P\;dT=C_V\;dT+P\;dV\text{ __(9)}\] For one mole of an ideal gas, \[PV=RT\] Differentiating with respect to temperature $T$ , \[\frac{d}{dT}(PV)=\frac{d}{dT}(RT)\] \[\frac{d}{dT}(PV)=R\] Heat is supplied at constant pressure. So, \[P\frac{dV}{dT}=R\] \[P\;dV=R\;dT\text{ __(10)}\]

From $(9)$ and $(10)$, \[C_P\;dT=C_V\;dT+R\;dT\] \[C_P=C_V+R\] \[C_P-C_V=R\] This is the relation between $C_P$ and $C_V$. If $M$ is the molar mass of the gas, then, \[\frac{C_P}{M}-\frac{C_V}{M}=\frac{R}{M}\] \[c_p-c_v=r\]

Where, 
$c_p =$ Specific heat capacity at constant pressure 
$c_v =$ Specific heat capacity at constant volume 
​$r =$ Gas constant per unit molar mass 


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