Question 6.14: Calculating a Root-Mean-Square Speed Which is the greater sp...
Calculating a Root-Mean-Square Speed
Which is the greater speed, that of a bullet fired from a high-powered M-16 rifle (2180 mi/h) or the root-mean-square speed of H_2 molecules at 25°C?
Analyze
This is a straightforward application of equation (6.20). We must use SI units: R = 8.3145 J K^{-1} mol^{-1} and M = 2.016 × 10^{-3} kg mol^{-1}. Recall that 1 J = 1 kg m² s^{-2}.
u_{rms} = \sqrt{\overline{u^2}}=\sqrt{\frac{3 R T}{M}} (6.20)
Determine u_{rms} of H_2 with equation (6.20).
u_{ rms } = \sqrt{\frac{3 \times 8.3145 kg m ^2 s ^{-2} mol^{-1} K ^{-1} \times 298 K }{2.016 \times 10^{-3} kg mol^{-1}}}
= \sqrt{3.69 \times 10^6 m ^2 / s ^2} = 1.92 \times 10^3 m / s
The remainder of the problem requires us either to convert 1.92 × 10³ m/s to a speed in miles per hour, or 2180 mi/h to meters per second. Then we can compare the two speeds. When we do this, we find that 1.92 × 10³ m/s corresponds to 4.29 × 10³ mi/h. The root-mean-square speed of H_2 molecules at 25 °C is greater than the speed of the high-powered rifle bullet.
Assess
The cancellation of units yields a result for u_{rms} with the correct units (m/s). Also, Figure 6-15 shows that u_{rms} for H_2 is a bit greater than 1500 m/s at 273 K. At 298 K, u_{rms} should be slightly greater than it is at 273 K.
