**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.

**Kinetic Theory of Gases:**

In this concept, it is assumed that the molecules of gas are very minute with respect to their distances from each other. The molecules in gases are in constant, random motion and frequently collide with each other and with the walls of any container.

In this portion, you will learn about the properties of gases, based on density, pressure, temperature and energy. Continue reading here to learn more.