Question

Monoatomic, diatomic, and triatomic gases whose initial volume and pressure are same, are compressed till their volume becomes half the initial volume.
(A). If the compression is adiabatic then the monatomic gas will have maximum final pressure.
(B). If the compression is adiabatic then triatomic gas will have maximum final pressure
(C). If the compression is adiabatic then their final pressure will be same
(D). If the compression is isothermic then final pressure will be different

Answer 1

Hint: A gas can undergo compression through various thermodynamic processes. For finding the final pressure we will use the equations of different thermodynamic processes, isothermal and adiabatic, to check the correct option.

Formula used:
For adiabatic process, $P{{V}^{\gamma }}=\text{ constant}$
For isothermal process, $PV=\text{ constant}$

Complete step by step answer:
Thermodynamic process is the passage of a thermodynamic system from an initial state to a final state of thermodynamic equilibrium. The initial and the final states of a system are the defining elements of a thermodynamic process. The four types of thermodynamic process are: Isothermal, Isochoric, Isobaric and Adiabatic.
In an isothermal process, the temperature of a system remains constant. Thermal equilibrium is maintained during the process. Equation of isothermal process is given by, $PV=\text{ constant}$
In an isochoric process, the volume of the closed system undergoing the process remains unchanged. The Isochoric process is an Isovolumetric process. Equation of Isochoric process is given by, $\dfrac{P}{T}=\text{ constant}$
In the Isobaric process, the pressure of the system stays constant. Equation of Isobaric process is given by, $\dfrac{V}{T}=\text{ constant}$
In adiabatic processes, the heat and mass transfer between system and surroundings remains zero. Equation of adiabatic process is given by, $P{{V}^{\gamma }}=\text{ constant}$
Where,
$P$ is the pressure of the system
$V$ is the volume of the system
$T$ is the temperature of the system
$\gamma $ is the ratio of specific heats $\left( \gamma =\dfrac{{{C}_{P}}}{{{C}_{V}}} \right)$
In the question we are given that the initial volume and pressure of Monatomic, diatomic, and triatomic gases are the same. Final volume is half the initial volume.
For an adiabatic process, we have,
$P{{V}^{\gamma }}=\text{ constant}$
$\dfrac{{{P}_{1}}}{{{P}_{2}}}={{\left( \dfrac{{{V}_{2}}}{{{V}_{1}}} \right)}^{\gamma }}$
As given final volume is half the initial volume
${{V}_{2}}=\dfrac{{{V}_{1}}}{2}$
It gives, ${{P}_{2}}={{2}^{\gamma }}{{P}_{1}}$
Now, for a monatomic gas, the value of $\gamma $ is the highest
Thus, for the same change in volume, the monatomic gas will have the maximum pressure.
For isothermal process, we have,
$PV=\text{ constant}$
${{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}}$
The initial and final volume of gases are the same, it means for the same initial pressure, the final pressure of all the gases will be the same.

Hence, the correct option is A.

Note: Students need to remember the equations of thermodynamic processes to solve the above equation. Also, keep in mind that in an Adiabatic process, heat and mass transfer between system and surroundings remain zero while in Isothermal process, the temperature of the system remains the same while going through the process.