**Real Gases Definition:
**Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law.

Real gases deviate slightly from ideal gas laws because

- Real gas molecules attract one another.
- Real gas molecules occupy a finite volume.

We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.

## Real Gases | Definition, Formula, Units – Kinetic Theory of Gases

**Real or van der Waals’ Gas Equation**

\(\left(p+\frac{a}{V^{2}}\right)\) (V – b) = RT

where, a and b are called van der Waals’ constants.

**Dimension** [a] = [ML^{5}T^{-2}] and [b] = [L^{3}]

**Units** a = N-m^{4} and b = m^{3}.

**Note:**

Real gases obey this equation at high pressure and low temperature

**Pressure of a gas**

Pressure due to an ideal gas is given by

p = \(\frac{1}{3} \frac{m n}{V}\)v^{2} = \(\frac{1}{3} \rho \bar{v}^{2}\)

For one mole of an ideal gas, where, m = mass of one molecule, n = number of molecules,

V = volume of gas, \(\bar{v}=\sqrt{\frac{\bar{v}_{1}^{2}+\bar{v}_{2}^{2}+\ldots+\bar{v}_{n}^{2}}{n}}\)

is called root mean square (rms) velocity of the gas molecules and M = molecular weight of the gas.

If p is the pressure of the gas and E is the kinetic energy per unit volume is E, then

p = \(\frac{2}{3}\)E

**Note:**

Effect of mass, volume and temperature on pressure

- when volume and temperature of a gas are constant, then pressure ∝ mass of gas.
- when mass and temperature of a gas are constant, then pressure ∝ \(\frac{1}{\text { volume }}\)
- when mass and volume of gas are constant, then pressure ∝ temperature ∝ c
^{2}.

### Kinetic Energy of a Gas and Speed of Gas Molecules

(i) Average kinetic energy of translation per molecule of a gas is given by

E = \(\frac{3}{2}\) kT

where, k = Boltzmann’s constant.

(ii) Average kinetic energy of translation per mole of a gas is given by

E = \(\frac{3}{2}\) RT

where, R = universal gas constant.

(iii) For a given gas kinetic energy

E ∝ T

⇒ \(\frac{E_{1}}{E_{2}}=\frac{T_{1}}{T_{2}}\)

(iv) Root mean square (rms) velocity of the gas molecules is given by

\(v\) = \(\sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 p}{\rho}}\)

(v) For a given gas, \(v\) ∝ \(\sqrt{T}\)

(vi) For different gases, \(v\) ∝ \(\frac{1}{\sqrt{M}}\)

(vii) Boltzmann’s constant, k = \(\frac{R}{N}\)

where, R is an ideal gas constant and N = Avogadro number.

Value of Boltzmann’s constant is 1.38 × 10^{-28} J/K.

(viii) The average speed of molecules of a gas is given by

\(\bar{v}=\sqrt{\frac{8 k T}{\pi m}}=\sqrt{\frac{8 R T}{\pi M}}\)

(ix) The most probable speed of molecules u of a gas is given by

\(v_{\mathrm{mp}}=\sqrt{\frac{2 k T}{m}}=\sqrt{\frac{2 R T}{M}} \Rightarrow v_{\mathrm{rms}}>\bar{v}>v_{\mathrm{mp}}\)

(x) With rise in temperature rms speed of gas molecules increases as

\(v_{\mathrm{rms}}\) ∝ \(\sqrt{T}\)

(xi) With the increase in molecular weight rms speed of gas molecule decrease as

\(v_{\mathrm{rms}}\) ∝ \(\frac{1}{\sqrt{M}}\)

(xii) Rms speed of gas molecules is of the order of km/s, e.g. at NTP for hydrogen gas

\(v_{\mathrm{rms}}\) = \(\sqrt{\frac{3 R T}{M}}=\sqrt{\frac{3 \times 8.31 \times 273}{2 \times 10^{3}}}\) = 1.84

(xiii) Rms speed of gas molecules does not depend on the pressure of gas (if temperature remains constant) because p ∝ ρ (Boyle’s law). If pressure is increased n times, then density will also increase by n times but υ_{rms} remains constant.