## Kinetic Theory of Gas

# Specific heat capacity and Mean free path

- The distance travelled by a gas molecule between two successive collisions is known as free path.

\tt \lambda = \frac{Total\ distance\ travelled\ by\ a\ gas\ molecule\ between\ collision}{total\ number\ of\ collision} - Mean free path \tt \lambda = \frac{1}{\sqrt{2}\pi nd^2} where "d" is diameter of the molecule n = number of molecules per unit volume.
- The average speed of molecules is v then \tt \lambda = \frac{vt}{N}=v\times T
- (λ) mean free path = \tt \frac{KT}{\sqrt{2}\pi d^2p} for constant volume and hence constant number density 'n' of gas molecules.
- For Diffusion of gases \tt \lambda=\frac{3\eta}{V_{av}}

where η = coefficient of viscosity

V_{av}= Average speed of molecules

### View the Topic in this video From 1:02 To 54:23

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1. **For monoatomic gas**

f = 3 (only translational)

The total internal energy of a mole of such a gas is

\tt U = \frac{3}{2}k_{B}T \ N_{A} = \frac{3}{2}RT

The molar spacific heat at constant volume

\tt C_{v} = \frac{dU}{dt} = \frac{3}{2}RT

C_{p} = C_{v} + R

\tt \Rightarrow \gamma = \frac{C_{p}}{C_{v}} = \frac{5/2 R}{3/2 R} = \frac{5}{3} = 1.67

2. **For diatomic gas**

f = 5 (3 translational + 2 rotational)

Similarly, \tt U = \frac{5}{2}k_{B} T \times N_{A} = \frac{5}{2}RT

\tt \gamma = \frac{7/2 R}{5/2 R}

If the diatomic molecules is not rigid but has in addition a vibrational mode.

\tt U = \frac{5}{2}k_{B}T + k_{B}T = \frac{7}{2} RT

\tt C_{v} = \frac{7}{2}R \ and \ C_{p} = \frac{9}{2}R

\tt \gamma = \frac{9}{7}

3. **For polyatomic gas**

C_{v} = (3 + f)R and C_{p} = (4 + f) R

\tt \gamma = \left[\frac{4 + f}{3 + f}\right]

Where, f = number of vibrational modes

4. Specific heat capacity of solid is

\tt C = \frac{\Delta Q}{\Delta T} = \frac{\Delta U}{\Delta T} = 3R

Specific heat capacity of water is

\tt C = \frac{\Delta Q}{\Delta T} = \frac{\Delta U}{\Delta T} = 9R

5. **Mean Free Path:** The mean free path of a molecule is the average distance by which the molecule travels between successive collisions.

Mean free path,

\tt \lambda = \frac{1}{\sqrt{2}\pi d^{2}n}

where, n = number of molecules per unit volume, d = diameter of gas molecule.