Question

derive vander woals equation of state and also give the significance of constants a and b?​

Answer 1

Answer:

Van der Waals equation is also known as Van der Waals equation of state for real gases which do not follow ideal gas law. According to ideal gas law, PV = nRT where P is the pressure, V is the volume, n is the number of moles, T is the temperature and R is the universal gas constant. The Van der Waals Equation derivation is explained below.

Derivation of Van der Waals equation

For a real gas, using Van der Waals equation, the volume of a real gas is given as (Vm – b), where b is the volume occupied by per mole.

Therefore, ideal gas law when substituted with V = Vm – b is given as:

P(Vm−b)=nRT

Because of intermolecular attraction P was modified as below

(P+aV2m)(Vm−b)=RT (P+an2V2)(V−nb)=nRT

Where,

Vm: molar volume of the gas

R: universal gas constant

T: temperature

P: pressure

V: volume

Thus, Van der Waals equation can be reduced to ideal gas law as PVm = RT.

Van der Waals Equation Derivation for one mole of gas

Following is the derivation of Van der Waals equation for one mole of gas that is composed of non-interacting point particles which satisfies ideal gas law:

p=RTVm=RTv p=RTVm−b C=NaVm (proportionality between particle surface and number density)

a′C2=a′(NAVM)2=aV2m p=RTVm−b−aV2m⇒(p+aV2m)(Vm−b)=RT (p+n2aV2)(V−nb)=nRT (substituting nVm = V)

Van der Waals equation applied to compressible fluids

Compressible fluids like polymers have varying specific volume which can be written as follows:

(p+A)(V−B)=CT

Where,

p: pressure

V: specific volume

T: temperature

A,B,C: parameters