Question 8.2: Calculate the average speed of particles of an ideal gas at ...
Calculate the average speed of particles of an ideal gas at temperature T .
Following our discussion of average values in Section 2.2 of Chapter 2, the average speed is obtained by multiplying each speed v by the probability P(v)dv and integrating over all possible speeds giving
v_{av}=\int_{0}^{\infty } vp(v)dv.Using Eqs. (8.22) and (8.23), the integrand in this last equation can be expressed in terms of the dimensionless variable u. We have
P(v)dv=4\pi v^{2}\left(\frac{m}{2\pi k_{B}T} \right)^{3/2} e^{-mv^{2}/2k_{B}T}dv. (8.22)
u=v\left(\frac{m}{2\pi k_{B}T} \right)^{1/2} . (8.23)
v_{av}=\frac{4}{\sqrt{\pi } } \left(\frac{2k_{B}T}{m} \right)^{1/2} \int_{0}^{\infty }u^{3} e^{-u^{2}}du.We may now use Eq. (8.18) to identify the above integral as I_{3}(1) and express the average velocity as
I_{n}(a)=\int_{0}^{\infty }u^{n}e^{-au^{2}}du . (8.18)
v_{av}=\frac{4}{\sqrt{\pi } } \left(\frac{2k_{B}T}{m} \right)^{1/2}I_{3} (1).According to the formula for I_{3}(a) given in Appendix G, I_{3}(1) has the value 1/2. We thus have
v_{av}=\frac{2}{\sqrt{\pi } } \left(\frac{2k_{B}T}{m} \right)^{1/2}.The average speed of the particles may be expressed in terms of the most probable speed v_{p} by using Eq. (8.24). We obtain
v_{p}=\left(\frac{2 k_{B}T}{m} \right) ^{1/2}. (8.24)
v_{av}=\frac{2}{\sqrt{\pi } } v_{p}.