Answer :
Answer:
The value is [tex]C_p = 42. 8 J/K\cdot mol[/tex]
Explanation:
From the question we are told that
[tex]\gamma = \frac{C_p }{C_v}[/tex]
The initial volume of the fluorocarbon gas is [tex]V_1 = V[/tex]
The final volume of the fluorocarbon gas is[tex]V_2 = 2V[/tex]
The initial temperature of the fluorocarbon gas is [tex]T_1 = 298.15 K[/tex]
The final temperature of the fluorocarbon gas is [tex]T_2 = 248.44 K[/tex]
The initial pressure is [tex]P_1 = 202.94\ kPa[/tex]
The final pressure is [tex]P_2 = 81.840\ kPa[/tex]
Generally the equation for adiabatically reversible expansion is mathematically represented as
[tex]T_2 = T_1 * [ \frac{V_1}{V_2} ]^{\frac{R}{C_v} }[/tex]
Here R is the ideal gas constant with the value
[tex]R = 8.314\ J/K \cdot mol[/tex]
So
[tex]248.44 = 298.15 * [ \frac{V}{2V} ]^{\frac{8.314}{C_v} }[/tex]
=> [tex]C_v = 31.54 J/K\cdot mol[/tex]
Generally adiabatic reversible expansion can also be mathematically expressed as
[tex]P_2 V_2^{\gamma} = P_1 V_1^{\gamma}[/tex]
=>[tex] [ 81.840 *10^3] [2V]^{\gamma} = [202.94 *10^3] V^{\gamma}[/tex]
=> [tex]2^{\gamma} = 2.56[/tex]
=> [tex]\gamma = 1.356[/tex]
So
[tex]\gamma = \frac{C_p}{C_v} \equiv 1.356 = \frac{C_p}{31.54}[/tex]
=> [tex]C_p = 42. 8 J/K\cdot mol[/tex]