As we know how the speeds of the helium atoms are

distributed. This distribution of speeds of the atoms or molecules in an ideal

gas is called the Maxwell distribution of velocities

We shall derive this

expression in starting from the Boltzmann factor

Here is the distribution, in terms of a continuous

probability distribution function.

The probability P(v)

dv that an atom or molecule has a speed between v and v + dv is given by

P(v)

dv = A v^2 e ^(?mv^2 /2kT )dv

(1)

where A is a normalization constant to ensure that ?

A=

a. Make

the substitution u = (m/2kT) ^1/2 v in Eq. 1 above to show that this simplifies

the probability distribution function:

u is a scaled, unitless speed, introduced to make the

probability distribution function universal, that is, independent of either the

particle mass or the temperature. P(u) du is the probability that a particle

has a scaled speed u between u and u + du.

b.

Show that in terms of this probability

distribution function, the root mean square speed u is just . (Note there are no units)

c.

Make a graph of P(u) vs u, that is, make a graph

of Eq. 2. Show (by differentiating to find the maximum) that the maximum value

of P(u) occurs at u = 1, and also that P(1) = .

d.

Finally, show that the value of u corresponding

to the escape velocity is approximately u (escape) = 10.2. The “area” under the

tail of the curve in the graph of P(u) vs u, which would have to be calculated

by computing this numerical integral.

is

therefore the answer , the fraction of helium atoms having a speed greater than

the escape velocity (11.2 km/s).

The value of this integral is extremely small.