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
    Vav = 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
Cp = Cv + 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
Cv = (3 + f)R and Cp = (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.