Question

A gas can be liquified at temperature T & pressure P provided:
(A) $T = T_c{\text{ }}and{\text{ }}P < {\text{ }}P_c$
(B) $T < T_c{\text{ }}and{\text{ }}P > P_c$
(C) $T > T_c{\text{ }}and{\text{ }}P > P_c$
(D) $T > T_c{\text{ }}and{\text{ }}P < P_c$

Answer 1

To solve this question, we need to know about the following concepts:
a) Critical Temperature $\left( {T_c} \right):$ Every gas has a certain characteristic temperature above which the gas can not be liquified even if the pressure is very high. This particular temperature is known as the critical temperature of the gas.
b) Critical Pressure $\left( {P_c} \right):$ The minimum pressure required to liquefy a gas at its critical temperature is termed as the critical pressure of the gas.

Complete answer:

The molecules in the gaseous state are widely separated from each other but in liquid state, they are comparatively close to each other. So, to liquefy a gas, the molecules must be brought closer to each other. It is possible by increasing the pressure on the gas molecules or by decreasing the temperature of the gas.
Scientist Andrews discovered the critical conditions necessary for the liquefaction of a gas, i.e., above its critical temperature $\left( {Tc} \right)$ a gas can never be liquified.
So, our first condition is to check the temperature of the gas. If it is below its $T_c$, it can be liquified & if it is above its $T_c$, it cannot be liquified. Upon this condition, our last two options i.e. $\left( C \right){\text{ }}\& {\text{ }}\left( D \right)$ are eliminated.
$[In{\text{ }}\left( C \right){\text{ }}and{\text{ }}\left( D \right)$ the temperature is above $T_c$ ${\text{ }}so,$ the gas can never be liquified according to these two options.]
So, now we have only two options in our hand i.e., $\left( A \right){\text{ }}\& {\text{ }}\left( B \right).$
In $\left( A \right){\text{ }}\& {\text{ }}\left( B \right)$ the temperature is at & below $T_c$ respectively. Now we have to check the next condition, i.e. condition of critical pressure $\left( {P_c} \right).$
According to the definition of $P_c$, if the gas is at $T_c$, the minimum pressure should be equal to $P_c$, not less than that.
Now, look out the option $\left( A \right).$ it states that the gas is at $T_c$ but the pressure is less than $P_c$. Thus, according to the definition of $P_c$, it can not be liquified when the pressure is less than $P_c$. Therefore, $\left( A \right)$ is also eliminated.
Now, we have only $\left( B \right).$ it states that the gas is below its $T_c$ (fulfils the liquefaction condition) & the pressure is more than $P_c$ (it also fulfils the liquefaction condition, since, the minimum pressure to be exerted to liquify is $P_c$, i.e., pressure should be at least equal to $P_c$ or more than $P_c$).

So, the correct answer is (B).

Note: The conditions for liquefaction of a gas are –

Temperature of the gas $\left( T \right)$ is at $T_c$ or other than $T_c$, then the pressure on the gas $\left( P \right)$ should be equal to or greater than $P_c$.
If the temperature of the gas is less than $T_c{\text{ }}$ the gas can not be liquified.