Question

In an adiabatic process, the density of a diatomic gas becomes 32 times its initial value. The final pressure of the gas is found to be n times the initial pressure. The value of n is:

• A. 32
• B. $\frac{1}{32}$
• C. 326
• D. 128

Answer 1

The correct option is D

128

Explanation for the correct option:

The correct option (D)

Adiabatic process is a type of process that occurs without transferring heat between the thermodynamic system and the surroundings. In an adiabatic process, the pressure-volume relation is

1. ${\mathrm{PV}}^{\mathrm{\gamma }}=\mathrm{constant}\mathrm{and}\mathrm{V}\propto 1/⍴,$ , therefore. (γ is the Ratio of specific heats, V is volume of the system, P is the pressure of the system here ⍴ is a constant).
2. $\mathrm{P}\left({⍴}^{-\mathrm{\gamma }}\right)=\mathrm{constant}$
3. $\mathrm{p}/{⍴}^{\mathrm{\gamma }}=\mathrm{constant}$
4. $\mathrm{p}\propto {⍴}^{\mathrm{\gamma }}$
5. $\mathrm{So},{\mathrm{P}}_{\mathrm{f}}/{\mathrm{P}}_{\mathrm{i}}={({⍴}_{\mathrm{f}}/{⍴}_{\mathrm{i}})}^{\mathrm{\gamma }}\mathrm{as}\mathrm{p}={\mathrm{P}}_{\mathrm{f}}/{\mathrm{P}}_{\mathrm{i}}\mathrm{and}{⍴}^{\mathrm{\gamma }}={({⍴}_{\mathrm{f}}/{⍴}_{\mathrm{i}})}^{\mathrm{\gamma }}$(here, P_f is final pressure, and P_i is initial pressure).
6. ${\mathrm{P}}_{\mathrm{f}}/{\mathrm{P}}_{\mathrm{i}}={\left(32\right)}^{7/5}=2^7=128.$

Final answer: So, the correct option is D.