Q.Using the relation $T_c = \dfrac{8a}{27Rb}$, estimate the critical temperature of carbon dioxide given its van der Waals constants $a = 3.59\ \text{atm L}^2\text{mol}^{-2}$ and $b = 0.0427\ \text{L mol}^{-1}$ ($R = 0.0821\ \text{L atm K}^{-1}\text{mol}^{-1}$). Compare your result with the accepted value of $304.2\ \text{K}$.

Every real gas has a characteristic critical temperature ($T_c$) above which it is physically impossible to

liquefy the gas by applying pressure alone, no matter how great that pressure is. Below its critical temperature,

sufficient pressure will condense a gas into a liquid; above it, the gas simply becomes more compressed without

ever separating into two distinct phases. This means cooling a gas below its critical temperature is a necessary

prerequisite before liquefaction by compression can succeed — pressure by itself is never sufficient if the gas

starts out above $T_c$.

Two related quantities complete the description: the critical pressure ($P_c$) is the minimum pressure needed

to liquefy the gas when it is held exactly at its critical temperature, and the critical volume ($V_c$) is the

molar volume of the gas under these critical conditions. Together, $(T_c, P_c, V_c)$ define the critical point,

the unique condition at which the distinction between the liquid and gaseous states of a substance disappears

entirely — the densities of the liquid and vapour phases become identical.

Critical temperature values vary enormously between gases and directly determine how readily each can be

liquefied under ordinary laboratory or industrial conditions. Carbon dioxide, with $T_c = 31.1^\circ\text{C}$ —

only slightly above normal room temperature — can be liquefied simply by compressing it at or below room

temperature, exactly how $\text{CO}_2$ fire extinguishers and carbonated-beverage cylinders store liquid

$\text{CO}_2$ under moderate pressure. Ammonia, with $T_c = 132.4^\circ\text{C}$, liquefies even more readily at …