Answer
We divide Eq. $19-31$ by Eq. $19-22$ :
$$
\frac{v_{\text {avg } 2}}{v_{\text {mos } 1}}=\frac{\sqrt{8 R T / \pi M_{2}}}{\sqrt{3 R T / M_{1}}}=\sqrt{\frac{8 M_{1}}{3 \pi M_{2}}}
$$
which, for $v_{\text {arg } 2}=2 v_{\text {rms } 2},$ leads to
$$
\frac{m_{1}}{m_{2}}=\frac{M_{1}}{M_{2}}=\frac{3 \pi}{8}\left(\frac{v_{\text {arg } 2}}{v_{\text {rms }}}\right)^{2}=\frac{3 \pi}{2}=4.7
$$
Work Step by Step
We divide Eq. $19-31$ by Eq. $19-22$ :
$$
\frac{v_{\text {avg } 2}}{v_{\text {mos } 1}}=\frac{\sqrt{8 R T / \pi M_{2}}}{\sqrt{3 R T / M_{1}}}=\sqrt{\frac{8 M_{1}}{3 \pi M_{2}}}
$$
which, for $v_{\text {arg } 2}=2 v_{\text {rms } 2},$ leads to
$$
\frac{m_{1}}{m_{2}}=\frac{M_{1}}{M_{2}}=\frac{3 \pi}{8}\left(\frac{v_{\text {arg } 2}}{v_{\text {rms }}}\right)^{2}=\frac{3 \pi}{2}=4.7
$$
