Question

Derivation of van der waals equation for real gases​

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.

Equation:

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

Answer 2

Answer:

The Van Der Waals equation was derived to define the physical state of a real gas.

Explanation:

Van Der Waals equation is an equation that is used to relate the relationship existing between the pressure, volume, temperature, and amount of real gases. For a real gas containing n  moles, the real gas equation derivation is as follows.

$\left(\mathrm{P}+\left(\frac{a n^{2}}{V^{2}}\right)\right) \mathrm{V}-\mathrm{nb}=\mathrm{nRT} \\Where,\\ P is the pressure,\\ V is the volume,\\ \mathrm{T} is the temperature,\\ \mathrm{n} is the number of moles of gases$

'a' and 'b' are the constants that are specific to each gas.

van der waals equation for real gases​.

Ideal gas equation,PV=nRT

In the case of a real gas when students are using Van Der Waals equation,

The volume of a real gas =  (Vm - b),

where b can be considered as the volume occupied by per mole.

So the ideal gas law gets substituted with V = Vm  - b,

it is given as :

$P(V_m- b) = nRT.$

Due to the presence of intermolecular attraction P was modified by,

$\begin{aligned}&\left(\frac{P+a}{V^{2}}\right)(V m-b)=R T \\&\left(\frac{P+a n^{2}}{V^{2}}\right)(V-n b)=n R T\end{aligned}$

Where,

$V_m: molar volume of the gas\\R: universal gas constant\\T:Temperature\\P: Pressure\\V: Volume$

Thus, it is possible to reduce Van Der Waals equation to ideal gas law as $P V m=R T .$