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

Disclaimer: may from time to time provide links to third party Internet sites under their respective fair use policy and it may from time to time provide materials from such third parties on this website. These third party sites and any third party materials are provided for viewers convenience and for non-commercial educational purpose only. Compete does not operate or control in any respect any information, products or services available on these third party sites. makes no representations whatsoever concerning the content of these sites and the fact that has provided a link to such sites is NOT an endorsement, authorization, sponsorship, or affiliation by with respect to such sites, its services, the products displayed, its owners, or its providers.

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.