Question

An ideal gas (Cp / Cv = γ) is taken through a process in which the pressure and the volume vary as p = aVb. Find the value of b for which the specific heat capacity in the process is zero.

Answer 1

The value of 'b' is - $\gamma$ when the specific heat capacity in the process is zero

Explanation:

As the process has specific heat capacity zero the process is essentially an adiabatic process  

For an Adiabatic  Process

$\frac{C_{P}}{C_{V}}=\gamma$

$\mathrm{C}_{\mathrm{P}}-\mathrm{C}_{\mathrm{V}}=\mathrm{R}$

$c_{v}=\frac{r}{\gamma-1}$

$C_{P}=\frac{\gamma R}{\gamma-1}$

$\mathrm{Pdv}=\frac{1}{\mathrm{b}+1}(\mathrm{Rdt})$

$0=\mathrm{C}_{\mathrm{v}} \mathrm{d} \mathrm{T}+\frac{1}{\mathrm{b}+1}(\mathrm{R} \mathrm{d} \mathrm{t})$

$\frac{1}{b+1}=\frac{-C_{V}}{R}$

$b+1=\frac{-R}{C_{V}} = \frac{-\left(C_{P}-C_{V})}{C_{V}}$

In the above equation, substitute the value as $\frac{C_p}{C_v} = \gamma$

$b = - \gamma + 1$

$b = - \gamma$

Heat capacity:-

• Heat capacity is the ratio of heat that a material absorbs to the change in temperature.
• It is usually defined in terms of actual number of resources being measured, most usually a mole (the atomic weight in grams) as calorie per degree.
• The calorie heat power per gram is called heat specific.
• The calorie measure is dependent on the particular surface heat, described as one calorie per degree Celsius.

Therefore the value of 'b' for which the specific heat capacity in the process is zero is $- \gamma$ where  $\frac{C_p}{C_v} = \gamma$ for an ideal gas taken through an adiabatic process in which the pressure and volume vary as p = aVb