Question

Define mean free path. Assuming that the
collision probability is a function of distance,
obtain the Survival Equation.

Answer 1

The mean free path is the average distance travelled by a moving particle between successive collisions which modify its direction or energy or other particle properties.

The mean free path is equal to

$$\lambda = \frac { 1}{\sqrt2 \pi d^2 n_V}$$

Since each collision removes a particle from the group of uncollided particles, the change in Nc and dNc can be written as

$$dN_C = -P_C(T, P)N_C dz (1)$$

where P_C(T, P) is constant defining that a collision occurs.

Integrating equation (1) using the boundary condition that when z=0, Nc=N, gives the fraction of particles at any position, z, that have yet to collide

$$\frac { N_C}{N}=exp (-P_C(T, P) z) (2)$$

The equation is called as survival equation it describes the number of particles that have survived travelling distance z without colliding.