Question

The speed of a molecule in a uniform gas at equilibrium is a random variable V whose pdf is given by f(v)={kv2e−bv2,v>00, else where,where k is an appropriate constant and b depends on the absolute temperature and mass of the molecule, m, but we will consider b to be known.(a) Calculate k so that f(v) forms a proper pdf.(b) Find the pdf of the kinetic energy of the molecule W, where W=mV2/2.

Answer 1

a) To calculate k so that f(v) forms a proper pdf, we need to make sure that the integral of f(v) over all values of v is equal to 1. This means that we need to solve the following equation:

∫f(v)dv = 1

We can do this by substituting in the values for f(v) and b, and then solving for k:

∫kv2e−bv2dv = 1

After solving this equation, we can find that k = 1/(2b3/2).

b) To find the pdf of the kinetic energy of the molecule W, we need to use the following equation:

f(w)=f(v)*|dv/dw|

We can then substitute in the values for f(v) and b, and then solve for the pdf of W:

f(w) = kv2e−bv2*|2mv/2w|

After solving this equation, we can find that the pdf of W is given by f(w) = k/2b3/2w2e−b(m/2w)2.