Shortcut Methods
Mean Free Path Numerical for JEE and CBSE Board Exams
A typical numerical for calculating the mean free path of a gas molecule in the JEE and CBSE board exams can involve finding the mean free path of molecules in air at room temperature and pressure (NTP). The following steps outline the general procedure:
Given Information:
- Temperature: T
- Pressure: P
- Boltzmann constant: k = 1.38 × 10^-23 J/K
Formula for Mean Free Path (λ):
$$λ = \frac{kT}{\sqrt{2}\pi d^2P}$$
- λ is the mean free path in meters (m)
- k is the Boltzmann constant
- T is the temperature in Kelvin (K)
- d is the diameter of the gas molecules in meters (m)
- P is the pressure in Pascals (Pa)
Calculating the Diameter of Gas Molecules (d):
- Approximate mean diameter of air molecules using density (assuming a spherical shape.):
$$V = \frac{4}{3}π\left(\frac{d}{2}\right)^3$$
- Calculate volume of a single molecule.
- Use the density of gas and the mass of an individual molecule to find the diameter:
$$d = \sqrt[3]{\frac{3m}{4 \pi \rho}}$$
Calculating the Mean Free Path (λ):
- Substitute the values of T, P, k, and d into the formula for λ.
Remember, the provided values for density, temperature, and pressure may vary in different numerical problems.
Additionally, numericals on diffusion, viscosity, and the equation of state for non-ideal gases can follow similar principles, with specific formulas related to each concept.
