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
dv = A v^2 e ^(?mv^2 /2kT )dv
where A is a normalization constant to ensure that ?
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.
Show that in terms of this probability
distribution function, the root mean square speed u is just . (Note there are no units)
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) = .
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.
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.