Question

Define the critical temperature of a gas. Give its expression.

Answer 1

Hint: The molecules of gas are in a random motion as compared to the liquid ones as the molecules of a liquid are densely packed. But the gases can be liquefied with the help of a combination of pressure and temperature, but after a certain temperature gas cannot be liquified further, no matter how much pressure you increase.

Complete Solution :
The critical temperature $\left( {{T}_{c}} \right)$ is that temperature after which gases cannot be liquified further. Different gases have a different critical temperature., for example, Ammonia has 132.4 degree Celsius and Carbon Dioxide have 31 degree Celsius as their Critical Temperature.-

The equation can be derived through Van-Der Waals equation:
$\Rightarrow \left( P+\dfrac{a}{{{V}^{2}}} \right)\left( V-b \right)=RT$
By a little rearrangement,
$\Rightarrow $ $P=\dfrac{RT}{V-b}-\dfrac{a}{{{V}^{2}}}$
At Critical Temperature
$\Rightarrow $ $\dfrac{\partial P}{\partial V}=0$ $\dfrac{{{\partial }^{2}}P}{\partial {{V}^{2}}}=0$
P is a function of V, hence making it use here
$\Rightarrow \dfrac{\partial P}{\partial V}=-\dfrac{RT}{{{\left( V-b \right)}^{2}}}+\dfrac{2a}{{{V}^{3}}}=0$ (1)
$\Rightarrow $ $\dfrac{2a}{{{V}^{3}}}=\dfrac{RT}{{{\left( V-b \right)}^{2}}}$ (2)
$\Rightarrow \dfrac{a}{{{V}^{4}}}=\dfrac{RT}{2V{{(V-b)}^{2}}}$ (3)

Differentiating equation (1) concerning V
$\Rightarrow \dfrac{{{\partial }^{2}}P}{\partial {{V}^{2}}}=\dfrac{2RT}{{{\left( V-b \right)}^{3}}}-\frac{6a}{{{V}^{4}}}=0$
Dividing both sides by 2, and rearranging
$\Rightarrow \dfrac{RT}{{{\left( V-b \right)}^{3}}}=\dfrac{3a}{{{V}^{4}}}$
Putting the value of the equation of 2 in the above equation
$\Rightarrow \dfrac{RT}{{{\left( V-b \right)}^{3}}}=\dfrac{3RT}{2V{{\left( V-b \right)}^{2}}}$
Dividing both the sides by $\dfrac{RT}{{{\left( V-b \right)}^{2}}}$
$3V - 3b = 2V$
$\Rightarrow {{V}_{c}}=3b$

This obtained expression is called Critical Volume, putting this in equation (1) we get,
$\Rightarrow \dfrac{RT}{4{{b}^{2}}}=\dfrac{2a}{27{{b}^{3}}}$
$\Rightarrow {{T}_{c}}=\dfrac{8a}{27Rb}$
This is the expression for Critical Temperature, where a and b are Van-Der Waals constants
R= Gas Constant.

Note: The reason behind that gases do not liquefy beyond a critical temperature is because of the impact of increasing temperature and pressure on the density of the gas. As the temperature and pressure rises, the density of the gases increases too and at the critical point the density of the gas and liquid are the same. Beyond this point, the molecules will gain more kinetic energy where they will not be bound by intermolecular forces.