See attached file.
We can compute the number of molecules that strikes the sphere using the formula 1/4 n v for the flux of particles distributed according to an isotropic velocity distrbution. The area of the sphere is 4 pi r^2, therefore the total number of collisions per unit time is:
1/4 n v 4 pi r^2 = pi r^2 n v (1)
We can also derive this from first principles assuming only an isotropic velocity distribution. Suppose the velocity distribution is given by f(v), which means that the number of molecules per unit volume within a volume d^3 v in velocity space is given by f(v) d^3 v. The fact that the velocity disribution is isotropic means that f(v) is actually a functon of the magnitude of v, which we denote by |v|. The volume element d^3 v can be written as |v|^2 d Omega d|v|, with d Omega the solid angle element. So, inside the gas at a particular point, the number of particles with speeds in the range between |v| and |v| + d|v| which are moving into (of from) some solid angle d Omega is given by:
f(|v|)|v|^2 d Omega d|v| (2)
If we integrate this over Omega and |v|, we obtain the total number of molecules per unit volume, which is the density n. To evaluate the number of molecules that strike the sphere, we imagine that the sphere isn't really there. We just focus on a spherical region in the gas with radius r. Then all the molecules that would hit the surface of the sphere if the sphere occupied that region, now move straight ahead. Consider molecules that are coming from some directions that are within a small range solid angle range d Omega (if you specify the direction using spherical coordinates theta and phi and look at molecules coming from a direction within theta and theta + dtheta and within phi and phi + d phi, then d Omega =
sin(theta) dphi d theta).
To compute the number of molecules intercepted by a surface you have to multiply the component of the flux in the normal direction of the surface by the surface area (we can also write this as the inner product of the flux and the area of the surface, where we then consider the area as a vector oriented in the normal direction). In the case of the sphere, we can avoid computing this by evaluating an integral by noting that the flux would also be intercepted at right angles by the surface that cuts the sphere into half which has its normal in the direction opposite to the flux. This surface is, of course, just a circle with radius r and thus has an area of pi r^2
So, this means that the number of molecules with speeds between |v| and |v| + d|v| coming from within a solid angle
d Omega from some ...
A detailed solution is derived from first principles.