Name: Graham's Diffusion Law
Uploaded: 2019-07-19
Duration: 9 min 43 s
Description: Gases diffuse at a different rate. Watch this video to understand Graham's diffusion law, which deals with the diffusion of gases. Have fun learning!

Question

Derive van der Waals equation of state. Explain the importance of van der Waals gas equation.

Accepted Answer

Derivation of van der Waals equation:

van der Waals equation has been obtained by modifying the ideal gas equation $PV = nRT$ by applying corrections for the volume and pressure.

(i) Correction for volume. Suppose the volume occupied by the gas molecules is $V$ . When the molecules are moving, their effective volume is four times the actual volume, i.e., $4 V$ . Let us call it $b$ , i.e., $b = 4 V$ (called excluded volume or co-volume). Thus, the free volume available to the gas molecules for movement, i.e., Corrected volume $=$ ( $V - b$ ) for one mole
$= (V - nb)$ for $n$ moles --------(i)

(ii) Correction for pressure: A molecule ( $A$ ) lying within the vessel is attracted equally by other molecules on all sides but a molecule near the wall ( $B$ ) is attracted (pulled back) by the molecules inside as shown in the figure. Hence, it exerts less pressure. In other words, observed pressure is less than the ideal pressure.

Hence, Corrected pressure $= P + p$ . Evidently, the correction term $p$ is proportional to density of the gas near the wall and the density of the gas inside, i.e., $p \propto {(density)}^{2}$ or $p \propto d^{2}$
But, $d \propto \frac{1}{V}$ for one mole, $d \propto \frac{n}{V}$ for $n$ moles hence, $p \propto \frac{n^{2}}{V^{2}}$ for $n$ moles or $p = \frac{{an}^{2}}{V^{2}}$ for $n$ moles.

Therefore, corrected pressure $= (P + \frac{a}{V^{2}})$ for one mole or $(P + \frac{{an}^{2}}{V^{2}})$ ---------(ii) for $n$ moles. Where $a$ is a constant depending upon the nature of the gas.

Substituting the corrected values of volume and pressure in the ideal gas equation, we get $(P + \frac{{an}^{2}}{V^{2}}) (V - nb) = nRT$ for $n$ moles. This equation is called van der Waals equation of state for $n$ moles.

Importance of van der Waals gas equation:

(i) Able to predict the behaviour of gases better than the ideal gas equation.

(ii) Applicable not only to gases but for all fluids. Able to calculate the critical conditions of liquefaction and derive an expression of the Principle of Corresponding States.

(iii) The arrangement of the equation in a cubic equation in volume. The cubic equation gives three volumes that are useful for calculating the volume at and below critical temperatures.

Derive van der Waals equation of state. Explain the importance of van der Waals gas equation.

Important Questions on States of Matter : Gases and Liquids

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