Question

How can the Van Der Waals equation of state be used to determine whether a substance will be a solid, liquid or gas at a certain temperature and pressure?

Answer 1

Dear Student,

The van der Waals equation state is an equation relating the density of gases and liquids (fluids) to the pressure (p), volume (V), and temperature (T) conditions.

A Vander Waals Equation of state is a formula describing the interconnection between various macroscopically measurable properties of a system.

For physical states of matter, this equation usually relates the thermodynamic variables of pressure, temperature, volume and number of atoms to one another.

It is available via its traditional derivation (a mechanical equation of state), or via a derivation based in statistical thermodynamics, the latter of which provides the partition function of the system and allows thermodynamic functions to be specified.

Gas - There are several types of gases with slightly different behaviours. These are ideal gasses, real gasses. The ideal gas law is often used as the first order description of any gas.

Ideal Gas - Although no gas is truly ideal, many gasses follow the ideal gas law very closely at sufficiently low pressures. The ideal gas law was originally determined empirically and is simply

               p V = n R T

        p = absolute pressure (not gage pressure

        V = volume

        n = amount of substance (usually in moles)

        R = ideal gas constant

        T = absolute temperature (not F or C)

where some values for R are

        8.3145 J mol^-1 K^-1

        0.0831451 L bar K^-1 mol^-1

        82.058 cm³ atm mol^-1 K^-1

        0.0820578 L atom mol^-1 K^-1

        1.98722 cal mol^-1 K^-1

        62.364 L Torr K^-1 mol^-1

Real Gas - Real gas laws try to predict the true behaviour of a gas better than the ideal gas law by putting in terms to describe attractions and repulsions between molecules. These laws have been determined empirically or based on a conceptual model of molecular interactions or from statistical mechanics.

A well known real gas law is the van der Waals equation

                ( P + a / Vm² )( Vm - b ) = R T

        P = pressure

        Vm = molar volume

        R = ideal gas constant

        T = temperature

where a and b are either determined empirically for each individual compound or estimated from the relations.

        a = 27 R² Tc² / 64 Pc

        b = R Tc / 8 Pc

        Tc = critical temperature

        Pc = critical pressure

The first parameter, a, is dependent upon the attractive forces between molecules while the second parameter, b, is dependent upon repulsive forces.

Liquid - Liquids are much less compressible than gasses. Even when a liquid is described with an equation similar to a gas equation, the constants in the equation will result in much less dramatic changes in volume with a change in temperature. Like wise at constant volume, a temperature change will give a much larger pressure change than seen in a gas.

Solid - The volume of a solid will generally change very little with a change in temperature. However, most solids are very incompressible so a constant volume heating will give a very large pressure change for even a small change in temperature. Crystals, glasses and elastomers are all types of solids.

A common equation of state for both liquids and solids is

                Vm = C1 + C2 T + C3 T² - C4 p - C5 p T

        Vm = molar volume

        T = temperature

        p = pressure

        C1, C2, C3, C4, C5 = empirical constants

where the empirical constants are all positive and specific to each substance.

For constant pressure processes, this equation is often shortened to

                Vm = Vmo ( 1 + A T + B T² )

        Vm = molar volume

        Vmo = molar volume at 0 degrees C

        T = temperature

        A, B = empirical constants

where A and B are positive.

Therefore, the above explanation helps in determining the three physical states of matter solid,liquid and gas by using Vander Waals equation of state at certain temperature and pressure.