Question

For an adiabatic compression the quantity PV
A. Increases
B. Decreases
C. Remains constant
D. Depends on $\gamma $

Answer 1

Hint: We are given an adiabatic compression of a gas. We have to find what happens to PV during this compression process. We have the ideal gas equation and adiabatic compression equation. We know during the process of compression volume of the gas decreases. With the inference using the two equations we can find change in PV during adiabatic compression.

Formula used: Ideal gas equation,
$PV=nRT$
Adiabatic compression equation
$T{{V}^{\gamma -1}}=\text{constant}$

Complete step by step answer:
We are given a situation of adiabatic compression.
We have a gas which is adiabatically compressed.
We have the ideal gas equation,
$PV=nRT$, Where ‘P’ is pressure of the gas, ‘V’ is volume of the gas, ‘n’ is number of moles gas being compressed, ‘R’ is ideal gas constant and ‘T’ is the temperature.
We know that ‘n’ and ‘R’ are constants.
Therefore we can rewrite the equation as,
$PV\propto T$
We know, for an adiabatic compression process,
$T{{V}^{\gamma -1}}=\text{constant}$
From this equation, we can say that
$T\propto \dfrac{1}{{{V}^{\gamma -1}}}$
We know, value of$\gamma >1$ , therefore we can say that $\gamma -1>0$
For an adiabatic compression, since the gas is getting compressed, the volume of the gas decreases.
Therefore, form the relation$T\propto \dfrac{1}{{{V}^{\gamma -1}}}$, since volume and temperature are inversely proportional, when volume decreases we can say that temperature increases.
Now that we know volume decreases and so temperature increases, we have the relation between ‘PV’ and ‘T’ to be,
$PV\propto T$ , i.e. pressure and temperature is directly proportional.
Hence when temperature increases, PV also increases for an adiabatic compression.

So, the correct answer is “Option A”.

Note: An adiabatic process is a thermodynamic where there is no heat exchange between the system and surrounding.
Adiabatic compression is a thermodynamic process in which no heat is added or subtracted from the gas and the internal energy of the gas is increased to the same amount as the external work is done on the gas. During this compression process volume of the gas gets decreased.