Question

The constant volume heat capacity of a gas can be measured by observing the decrease in temperature when it expands adiabatically and reversibly. If the decrease in pressure is also measured, we can use it to infer the value of γ = Cp/Cv and hence, by combining the two values, deduce the constant-pressure heat capacity. A fluorocarbon gas was allowed to expand reversibly and adiabatically to twice its volume; as a result, the temperature fell from 298.15 K to 248.44 K and its pressure fell from 202.94 kPa to 81.840 kPa. Evaluate Cp

Answer 1

Answer:

The value is  $C_p = 42. 8 J/K\cdot mol$

Explanation:

From the question we are told that  

     $\gamma = \frac{C_p }{C_v}$

The  initial volume of the  fluorocarbon gas is  $V_1 = V$

 The final  volume of the fluorocarbon gas is$V_2 = 2V$

  The initial  temperature of the fluorocarbon gas is  $T_1 = 298.15 K$

  The final  temperature of the fluorocarbon gas is $T_2 = 248.44 K$

   The initial  pressure is $P_1 = 202.94\ kPa$

    The final   pressure is  $P_2 = 81.840\ kPa$

Generally the equation for  adiabatically reversible expansion is mathematically represented as

       $T_2 = T_1 * [ \frac{V_1}{V_2} ]^{\frac{R}{C_v} }$

Here R is the ideal gas constant with the value  

        $R = 8.314\ J/K \cdot mol$

So  

   $248.44 = 298.15 * [ \frac{V}{2V} ]^{\frac{8.314}{C_v} }$

=> $C_v = 31.54 J/K\cdot mol$

Generally adiabatic reversible expansion can also be mathematically expressed as

    $P_2 V_2^{\gamma} = P_1 V_1^{\gamma}$

=>$[ 81.840 *10^3] [2V]^{\gamma} = [202.94 *10^3] V^{\gamma}$    

=>  $2^{\gamma} = 2.56$

=>    $\gamma = 1.356$

So

     $\gamma = \frac{C_p}{C_v} \equiv 1.356 = \frac{C_p}{31.54}$

=>    $C_p = 42. 8 J/K\cdot mol$