## Van der Waals Equation Derivation

**Van der Waals** derivation in 1873 modified the ideal gas equation of state by incorporating the size correction of gas molecules and the intermolecular attraction effect of the real gases. These two constant effects or factors in Van der Waals calculation are arises because the ideal gas law formula has been derived from the kinetic theory where gas molecules consider unit or point masses and force of attraction on the gas surface are ignored by kinetic postulates. The molecule occupies a certain volume as seen from the fact that gases can be liquified or solidified at low temperatures and high pressure. On decreasing pressure, the thermal energy or specific heat of gas molecules decreased which favors liquefaction, diffusion, or solidification.

In crystalline solids, the atoms, ions, or molecules are considerable resistance to any further attempt of compression but gas molecules are compressible. Therefore, Van der Waals was first to introduce mathematical calculation or correction formula to the above kinetics assumption in the ideal gas equation, P_{i}V_{i} = nRT.

### Volume Correction Formula for Gases

In learning chemistry or physics, real gas molecules are assumed to be a hard rigid sphere. Therefore the available space for free movement of the molecules becomes less than the original molar volume. Hence the available space for free movement of 1-mole real gas molecules (V_{i}) = V – b, where V_{i} and V = molar volume of ideal and real gases respectively and b = volume correction factor. Let us take a gas molecule with radius r and diameter σ = 2r.

When two molecules encounter each other the distance between the centers of the two gas molecules would be the diameter (σ ) of the molecule. Therefore, the space indicated by the dashed circle below the picture will be unavailable space for the pair of molecules.

The solution of the volume correction factor (b) helps for the calculation of the diameter or radius of the gas molecule. Therefore, after this volume correction formula of gas, the ideal gas equation is written as, P_{i} (V – b) = RT.

### Pressure Correction Formula for Gases

The pressure of the gas is developed due to the wall collision of the gas molecules. But due to intermolecular attraction, the colliding molecules will experience an inward pull. Therefore, the pressures exerted by the molecules in real gases will be less than the ideal gases. Since the ideal gases have no intermolecular attraction. Therefore, P_{ideal} > P_{real}, or, P_{ideal} = P_{real} + P_{a}, where P_{a} = pressure correcting term for real gases.

Higher the intermolecular attraction in the gas molecules greater is the magnitude of pressure correction term. Therefore, the pressure correction term depends on the frequency of molecular collisions. The average pressure exerted by the molecules decreased by P_{a}, which is proportional to the square of the density of gas molecules.

Therefore, P_{a} ∝ 1/V^{2}, since density ∝ 1/V

∴ P_{a} = a/V^{2}

where a = Van der Waals constant for gases.

### Van der Waals Equation for Real Gases

Using pressure and volume corrections factor formula, Van der Waals equation for 1-mole real gas molecules, (P + a/V^{2})(V – b) = RT.

For n moles real gases, the volume has to change because the volume is a thermodynamics extensive property in this equation. Therefore, the Van der Waals equation of state for n-mole real gases, (P + an^{2}/V^{2})(V – nb) = nRT.

### Van der Waals Gases Intermolecular Attraction

- For Van der Waals gases which has no intermolecular attraction (a = 0) but size effect considered (b ≠ 0). For such cases, P
_{real}> P_{real}= nRT/V-nb, since P_{ideal}= nRT/V only. This means that the molecular size effect of the molecules (repulsive interaction) creates higher pressure than that observed from the ideal gas law and mass or volume gas molecules are negligible. - For the real gases which has attractive force (a ≠ 0) but size effect not considered (b = 0). For such cases, P
_{real}< P_{real}. Therefore, the intermolecular attraction reduces the pressure of real gases.

### Unit Dimension of Van der Waals Constants

Van der Waals equation uses for the calculation of unit and dimensions of constant a and b. From the VDW equation P_{a} = an^{2}/V^{2} for n-mole real gases, where P_{a} = unit of internal pressure. Therefore, the unit of a = atm lit^{2} mol^{-2}. Again nb = unit of volume, hence unit of b = lit mol^{-1}.

In SI system, the unit of a = (newton meter^{-2}) meter^{6} mol^{-2} = N meter^{4} mol^{-2} and b = meter^{3} mol^{-1}. Therefore, the dimensions of a and b = [M L^{5} T^{-2} mol^{-2}] and [L^{3} mol^{-1}] respectively.

### Significance of Van der Waals Constant

- The term ‘a’ originates from the intermolecular attraction and P
_{a}= an^{2}/V^{2}. Therefore internal pressure or attractive force of the gas measured by a. But higher the value of a grater is the intermolecular attraction and more easily the gas liquefied. Therefore Van der Waals constant (a) for greenhouse gas carbon dioxide gas = 3.95 but hydrogen gas = 0.22. - Another constant b measures the molecular size and also the repulsive forces. Grater, the value of b larger the size of the gas molecule. Therefore the constant b for carbon dioxide = 0.04 but Hydrogen = 0.02.

### Compressibissibility and Van der Waals Equation

Van der Waals equation uses for the calculation of Boyle temperature, critical temperature and explains the compressibility factor in Amagat curves for n-mole real gases. Mathematical calculation of Boyle temperature for gases, T_{B} = a/Rb. This shows the compressibility factor is the function of temperature and pressure only.

Now, when b > a/RT, the initial slope is positive and the size-effect or factor (b) dominates the base properties of the gas. However, b < a/RT, the initial slope is negative and the effect of attractive forces (a) dominate. Therefore, Van der Waals equation balance both size effect and intermolecular forces in the positive and negative plot of compressibility vs pressure.

From the above slope, we can say that at 0°C, the effect of attractive forces dominated (carbon dioxide, nitrogen, and hydrocarbons like methane, ethane, acetylene), while the molecular size effect dominates for the behavior of hydrogen gas. The unit value of Van der Waals constant hydrogen, helium, nitrogen, etc for study chemistry or physics is extremely small as these gases are difficult to liquefy.