Question

(A) Derive a relation between partial pressure and mole fraction of gas.
(B) Write the Vander waals equation for one mole.

Answer 1

Hint: Ideal gases are the gases which have elastic collisions between their molecules and there are no intermolecular forces of attraction. The gases just show ideal behavior under certain conditions of temperature and pressure. Moreover, Vander waals forces are weak intermolecular forces that are dependent on the distance between atoms or molecules.

Complete step by step answer:
The ideal gases are made up of molecules which are in constant motion in random directions. The pressure occurs due to the collision between the molecules with the walls of the container.
The mole fraction of a specific gas in a mixture of gases is equal to the ratio of the partial pressure of that gas to the total pressure exerted by the gaseous mixture.
Now, the ideal gas equation for individual gases is given by:
 $PV = nRT$ …………………………. (1)
The ideal gas equation for the mixture is given by:
 ${P_{total}}V = {n_{total}}RT$ …………………. (2)
Now, dividing equation 1 by 2 we get,
 $\dfrac{{PV}}{{{P_{total}}V}} = \dfrac{{nRT}}{{{n_{total}}RT}}$
 $\dfrac{P}{{{P_{total}}}} = \dfrac{n}{{{n_{total}}}}$ ………………………. (3)
Now, the mole fraction of the gas is:
 $X = \dfrac{n}{{{n_{total}}}}$ ……………………………. (4)
Now, from equation $3$and$4$, we get
 $P = X{P_{total}}$
The partial pressure exerted by each gas in a gas mixture is independent of the pressure exerted by all other gases present.
Hence, the relation between the partial pressure and mole fraction of a gas is:
 $P = X{P_{total}}$
Now, in the second part, we have to write the Vander Waals equation for one mole.
The Vander waals equation is given as:
 $(P + \dfrac{{a{n^2}}}{{{V^2}}})(V - nb) = nRT$ Where, T is the temperature, R is the universal gas constant, V is volume and P is pressure.
Now, we have to derive to equation for one mole
So, for $1$mole, substitute $n = 1$ in the above equation
So, by substituting the value of n, we obtain the following equation,
 $(P + \dfrac{a}{{{V^2}}})(V - b) = RT$
So, this is the Van Der Waals equation for one mole.

Note: Although the ideal gas equation has many limitations, this equation holds well as long as the density is kept low. This equation is applicable for single gas or even a mixture of multiple gases where ‘n’ will stand for the total moles of gas particles in the given mixture.