Answer
The proofs are below.
Work Step by Step
a) Equation 14.9 states:
$ v =\sqrt{\frac{\gamma P}{\rho}}$
The ideal gas law states:
$ P =\frac{nRT}{V}$
Thus:
$ v =\sqrt{\frac{\gamma \frac{nRT}{V}}{\rho}}$
Finally, we know that $\rho=\frac{m}{V}$. Thus:
$ v =\sqrt{\frac{\gamma nRT}{m}}$
Therefore, we find:
$v_{sound} =\sqrt{\frac{\gamma kT}{m}}$
b) Equation 17.4 states:
$v_{th}=\sqrt{3kt}{m}$
We simplify the above equation to find:
$v_{sound} =\sqrt{\frac{1.4 kT}{m}}$
Thus, we find that the percentage is:
$\frac{\sqrt{1.4}}{\sqrt3}\times 100 = \fbox{68 percent}$
