## Specific Heat Capacity of Gas

Determine specific heat capacities cV and cp 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 κ is defined by

$\kappa = \frac{c_p}{c_V}= \frac{C_p}{C_V},$

where cp, respectively Cp is specific, respectively molar heat capacity at constant pressure and cV, respectively CV is specific, respectively molar heat capacity at constant volume.

• #### Hint 3

To determine molar mass Mm 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 K gas temperature p = 100 kPa = 1.00·105 Pa gas pressure ρ = 1.27 kg·m−3 gas density κ = 1.4 Poisson's constant of the gas cV = ? specific heat capacity of the gas at constant volume cp = ? specific heat capacity of the gas at constant pressure

Table values:

 R = 8.31 JK−1mol−1 molar gas constant
• #### Solution

We start the calculation with Meyer's relation Cp = CV + R, that relates molar heat capacities at constant volume CV and constant pressure Cp.

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 Mm

$\frac{C_p}{M_m} = \frac{C_V}{M_m} + \frac{R}{M_m},$

resulting in the relation

$c_p = c_V + \frac{R}{M_m}.$

Now we use the given Poisson's constant κ defined by the relation

$\kappa = \frac{C_p}{C_V} = \frac{c_p}{c_V}.$

We 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 cV is then given by

$c_V=\frac{R}{M_m(\kappa-1)}.$

Now we need to determine the unknown molar mass of the gas Mm. We use the equation of state for ideal gas

$pV=\frac{m}{M_m}RT.$

We 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 cp then can be directly determined from the equation

$c_p = \kappa c_V = \frac{\kappa p}{\rho T(\kappa-1)}.$
• #### Numerical Solution

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