Question

During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute temperature. The ratio $C_p / C_v$ for the gas is

• A. 44654
• B. 2
• C. 44684
• D. 44622

Answer 1

Answer: (D)

44622

Answer 2

Given that, $p \propto T^{3}$
or $p T^{-3}=$ constant .....(i)
Also for adiabatic process
$p V^{\Upsilon}=$ constant
but $pV = RT$ (ideal gas law)
or $V=\frac{R T}{p}$
$=p\left(\frac{R T}{p}\right)^{\Upsilon}=$ constant
$\Rightarrow p^{1-\Upsilon} T^{\Upsilon}=$ constant
or $p T^{\Upsilon /(1-\Upsilon)}=$ constant
Equating Eqs. (i) and (ii), we get
$\frac{\Upsilon}{1-\Upsilon}=-3$
or $\Upsilon=-3+3 \Upsilon$
or $2 \gamma=3$
$\therefore =\Upsilon=3 / 2$
or $\frac{C_{p}}{C_{v}}=3 / 2$