Question

A gas at NTP is suddenly compressed to one-fourth of its original volume. If $\lambda $ is supposed to be $\dfrac{3}{2}$, then the final pressure is
(A) 4 atmosphere
(B) $\dfrac{3}{2}$ atmosphere
(C) 8 atmosphere
(D) $\dfrac{1}{4}$ atmosphere

Answer 1

Hint: When a gas is suddenly compressed, it does not have enough time to release heat to its surroundings. When there is no heat transfer to or from a system, the process is known as an adiabatic process.

Complete step-by-step answer:
Now since we know that the sudden compression of gas is an adiabatic process, the relation between pressure and volume can be expressed by
\[{{P}_{0}}V_{0}^{\gamma }=P{{V}^{\gamma }}=\text{constant}\]
Here, ${{P}_{0}}$= pressure of gas before compression
${{V}_{0}}$= volume of gas before compression
P= pressure of gas after compression
V= volume of gas after compression
$\gamma $= adiabatic index of the gas
It is given to us that $\gamma =\dfrac{3}{2}$, $V=\dfrac{1}{4}{{V}_{0}}$, and since the gas is at NTP ${{P}_{0}}=1atm$. So, by substituting these values in the expression, we can calculate the final pressure of the gas.
\[
  \Rightarrow P={{P}_{0}}{{\left( \dfrac{{{V}_{0}}}{V} \right)}^{\gamma }} \\
 \Rightarrow P=1\times {{\left( \dfrac{{{V}_{0}}}{\dfrac{1}{4}{{V}_{0}}} \right)}^{\dfrac{3}{2}}} \\
 \Rightarrow P={{4}^{\dfrac{3}{2}}} \\
  \Rightarrow P=8atm
\]
So, when a gas at NTP is suddenly compressed to one-fourth of its volume and $\gamma =\dfrac{3}{2}$, the final temperature of the gas will be equal to option (C) 8 atmosphere.

Note: It is easy to confuse between the isothermal and the adiabatic processes.
The main difference between the two is that while in the adiabatic process there is no transfer of heat to or from the surroundings, to make the overall temperature constant, it is accompanied by a change in the temperature of the system.
On the other hand, in an isothermal process, while there is no change in temperature of the system, to make the overall temperature constant, there is a transfer of heat to and from the surrounding.