Question

A monatomic gas is suddenly compressed to \[\dfrac{1}{8}th\] of its initial volume adiabatically. The ratio of its final pressure to the initial pressure is:
(Given the ratio of the specific heats of the given gas to be \[\dfrac{5}{3}\])
A. 32
B.\[\dfrac{{40}}{3}\]
C.\[\dfrac{{24}}{5}\]
D. 8

Answer 1

Hint: This is the case in which a type of thermodynamic process is taking place. An adiabatic process is the one in which heat is not exchanged from surroundings during expansion or compression.

Formula Used:
The adiabatic equation is written by using following equation:
\[P{V^\gamma } = \text{constant}\]……(i)
Where P is the pressure of the system, V is the volume of the system and \[\gamma \] is the adiabatic index.
The specific heat capacity for an adiabatic process is given by:
\[\dfrac{{{C_p}}}{{{C_v}}} = \gamma \]
where, \[{C_p}\] is the specific heat at constant pressure and \[{C_v}\] is the specific heat at constant volume.


Complete step by step solution:
Given that \[\dfrac{{{C_p}}}{{{C_v}}} = \dfrac{5}{3}\]
or, \[\gamma = \dfrac{5}{3}\]
Since in an adiabatic process, there is no exchange, so equation (i) can be written as:
\[{P_1}V_1^\gamma = {P_2}V_2^\gamma \]
\[\Rightarrow \dfrac{{{P_2}}}{{{P_1}}} = {(\dfrac{{{V_1}}}{{{V_2}}})^{\dfrac{5}{3}}}\]

Given that gas is compressed \[\dfrac{1}{8}\] of its initial volume, the above equation can be written as
\[\dfrac{{{P_2}}}{{{P_1}}} = {(\dfrac{8}{1})^{\dfrac{5}{3}}}\]
\[\therefore \dfrac{{{P_2}}}{{{P_1}}} = {(8)^{\dfrac{5}{3}}} = 32\]
Therefore, the ratio of its final pressure to initial pressure is 32.

Hence, Option A is the correct answer.

Note: An adiabatic process should be carried out quickly, so that the heat can be exchanged with the surroundings in enough time. Also, the system in which the adiabatic process is to be conducted should be properly and completely insulated from its surroundings. Otherwise, the adiabatic process can not take place. Since, the system is completely isolated, so there will be no pressure from the surroundings and hence work done in an adiabatic process will be zero. As a result, there will be zero change in internal energy and change in the system will be in the form of work only.