Kinetic Theory of Gases
(NEET – Medical Physics – Study Notes)
1. Simple Explanation
The kinetic theory explains the macroscopic properties of gases (pressure, temperature, volume) from the microscopic motion of their molecules.
- Gases consist of a huge number of tiny particles that move randomly, collide elastically, and are separated by large gaps.
- The pressure exerted on a container wall is produced by molecules striking the wall.
- Temperature is a measure of the average kinetic energy of the molecules.
2. Key Assumptions
| # | Assumption | Physical Meaning |
|---|---|---|
| 1 | Molecules are point‑like (volume ≪ container volume). | (V_{\text{molecules}} \ll V_{\text{container}}) |
| 2 | No intermolecular forces (except during brief elastic collisions). | Ideal gas behaviour |
| 3 | Collisions are perfectly elastic. | Kinetic energy conserved in each collision |
| 4 | The motion is random and isotropic. | No preferred direction |
| 5 | The number of molecules (N) is huge ((\sim 10^{23})). | Enables statistical treatment |
3. Derivation of the Ideal Gas Equation from Kinetic Theory
Consider a cubic box of side (L) containing (N) identical molecules of mass (m).
-
Collision with one wall (x‑direction)
- A molecule with velocity component (v_x) collides elastically with the wall.
- Change in momentum: (\Delta p = 2 m v_x).
-
Time between successive impacts on the same wall by the same molecule
- (\Delta t = 2L / |v_x|).
-
Average force from one molecule
[ \bar{F}_x = \frac{\Delta p}{\Delta t} = \frac{2 m v_x}{2L/|v_x|} = \frac{m v_x^2}{L} ] -
Total force on the wall (sum over all molecules)
[ F = \frac{m}{L}\sum_{i=1}^{N} v_{x,i}^{,2} ] -
Pressure
( P = \dfrac{F}{A}) with (A=L^2):
[ P = \frac{m}{L^3}\sum_{i=1}^{N} v_{x,i}^{,2} ] -
Isotropy
(\langle v_x^2\rangle = \langle v_y^2\rangle = \langle v_z^2\rangle \Rightarrow \langle v^2\rangle = 3\langle v_x^2\rangle).
Therefore, [ \sum v_{x,i}^{,2} = \frac{N}{3}\langle v^2\rangle ] -
Ideal Gas Law
[ P = \frac{N m \langle v^2\rangle}{3V} \quad\Longrightarrow\quad PV = \frac{N m \langle v^2\rangle}{3} ] Using (N = nN_A) (number of molecules = moles × Avogadro’s number) and defining the average kinetic energy: [ \frac{1}{2}m\langle v^2\rangle = \frac{3}{2}kT ] we arrive at the familiar equation: [ PV = nRT ]
4. Important Relations
| Quantity | Formula | Meaning |
|---|---|---|
| Mean kinetic energy | (\tfrac{1}{2}m\langle v^2\rangle = \tfrac{3}{2}kT) | Energy per molecule |
| Root‑mean‑square speed | (v_{\text{rms}} = \sqrt{\langle v^2\rangle} = \sqrt{\dfrac{3kT}{m}}) | Typical speed |
| Average speed | (\langle v \rangle = \sqrt{\dfrac{8kT}{\pi m}}) | Mean of speed distribution |
| Mean free path | (\lambda = \dfrac{kT}{\sqrt{2}\pi d^2 P}) | Average distance between collisions |
| Viscosity of ideal gas | (\eta = \frac{1}{3} \rho v_{\text{rms}} \lambda) | Resistance to flow |
| Thermal conductivity | (k_{\text{th}} = \frac{1}{3} \rho c_p v_{\text{rms}} \lambda) | Heat transfer capability |
| Diffusion coefficient | (D = \frac{1}{3} v_{\text{rms}} \lambda) | Rate of mixing |
5. Worked Example
Problem:
Calculate the root‑mean‑square speed ((v_{\text{rms}})) of a nitrogen ((N_2)) molecule at (25^\circ\text{C}) (298 K).
Given:
- Molar mass of (N_2): (28.02,\text{g mol}^{-1}).
- Mass of one molecule (m = \dfrac{28.02 \times 10^{-3},\text{kg}}{N