## Specific Heat Capacity of Gas

### Task number: 3947

Determine specific heat capacities *c*_{V} and *c*_{p} of unknown gas provided that at temperature of 293 K and pressure of 100 kPa its density is 1.27 kg m^{−3} and Poisson's constant of the gas is *κ* = 1.4.

#### Hint 1

Use Meyer's relation and realize how you can transform it to a relation between specific heat capacities and not between molar heat capacities at constant pressure and volume.

#### Hint 2

Poisson's constant

\[\kappa = \frac{c_p}{c_V}= \frac{C_p}{C_V},\]*κ*is defined bywhere

*c*_{p}, respectively*C*_{p}is specific, respectively molar heat capacity at constant pressure and*c*_{V}, respectively*C*_{V}is specific, respectively molar heat capacity at constant volume.#### Hint 3

To determine molar mass

*M*_{m}of the gas use the equation of state for ideal gas.#### Analysis

We start the solution from Meyer's relation. We divide it by molar mass of the gas and thus adjust it to the relation between specific heat capacities at constant pressure and volume.

The unknown molar mass can be determined from the equation of state for ideal gas.

Specific heat capacity at constant pressure is determined as the product of Poisson's constant and specific heat capacity at constant volume.

#### Given Values

*T*= 293 Kgas temperature *p*= 100 kPa = 1.00·10^{5}Pagas pressure *ρ*= 1.27 kg·m^{−3}gas density *κ*= 1.4Poisson's constant of the gas *c*_{V}= ?specific heat capacity of the gas at constant volume *c*_{p}= ?specific heat capacity of the gas at constant pressure *Table values:**R*= 8.31 JK^{−1}mol^{−1}molar gas constant #### Solution

We start the calculation with Meyer's relation

*C*_{p}=*C*_{V}+*R*, that relates molar heat capacities at constant volume*C*_{V}and constant pressure*C*_{p}.In this task, however, we need to determine specific heat capacities, not molar heat capacities. This is why we need to divide Meyer's relation by molar mass of the gas

\[\frac{C_p}{M_m} = \frac{C_V}{M_m} + \frac{R}{M_m},\]*M*_{m}resulting in the relation

\[c_p = c_V + \frac{R}{M_m}.\]Now we use the given Poisson's constant

\[\kappa = \frac{C_p}{C_V} = \frac{c_p}{c_V}.\]*κ*defined by the relationWe determine specific heat capacity at constant pressure from this relation

\[c_p=\kappa c_V,\]and substitute it into the above mentioned relationship

\[\kappa c_V=c_V+\frac{R}{M_m}.\]The unknown specific heat capacity at constant pressure

\[c_V=\frac{R}{M_m(\kappa-1)}.\]*c*_{V}is then given byNow we need to determine the unknown molar mass of the gas

\[pV=\frac{m}{M_m}RT. \]*M*_{m}. We use the equation of state for ideal gasWe express molar mass

\[M_m=\frac{m}{V}\frac{RT}{p}\]and the ratio \(\frac{m}{V}\) substitute by the given density ρ of the gas:

\[M_m=\frac{\rho RT}{p}.\]After substituting into the relation for specific heat capacity at constant volume we then obtain:

\[c_V=\frac{R}{M_m(\kappa-1)}=\frac{R}{\frac{\rho RT}{p}(\kappa-1)}=\frac{p}{\rho T(\kappa-1)}.\]The specific heat capacity at constant pressure

\[c_p = \kappa c_V = \frac{\kappa p}{\rho T(\kappa-1)}. \]*c*_{p}then can be directly determined from the equation#### Numerical Solution

\[c_V=\frac{p}{\rho T(\kappa-1)}\] \[c_V= \frac{100\cdot{10^3}}{1.27\cdot{293}\cdot (1.4-1)}\,\mathrm{J\,kg^{-1}K^{-1}}\dot{=}672\,\mathrm{J\,kg^{-1}K^{-1}}\] \[c_p = \kappa c_V =1.4\cdot{671.8}\,\mathrm{J\,kg^{-1}K^{-1}}\dot{=}941\,\mathrm{J\,kg^{-1}K^{-1}}\]#### Answer

Specific heat capacity of unknown gas at constant volume is approximately 672 Jkg

^{−1}K^{−1}.Its specific heat capacity at constant pressure is then approximately 941 Jkg

^{−1}K^{−1}.