Six molecules of an ideal gas have velocities 1, 3, 5, 5, 6 and 5 m/s respectively. At any given temperature, if \(\bar{V}\) and Vrms represent average and rms speed of the molecules, then
Vrms > \(\bar{V}\)
The average speed is the arithmetic mean of the speeds, while the rms speed is the square root of the mean of the squared speeds. Because squaring gives more weight to larger values, the rms speed is never smaller than the average speed and equals it only when all speeds are the same.
Step 1 — Average speed: \(\bar{V} = \dfrac{1+3+5+5+6+5}{6} = \dfrac{25}{6} \approx 4.17\ \text{m s}^{-1}\).
Step 2 — Mean square speed: \(\dfrac{1+9+25+25+36+25}{6} = \dfrac{121}{6} \approx 20.17\ \text{m}^{2}\text{s}^{-2}\), so \(V_{\text{rms}} \approx 4.49\ \text{m s}^{-1}\).
Step 3 — Comparison: \(V_{\text{rms}}^{2} - \bar{V}^{2} = 20.17 - 17.36 \gt 0\), so \(V_{\text{rms}} \gt \bar{V}\).
The average speed is about 4.17 m/s, not 5 m/s. The square of the rms speed (about 20.17) is larger, not smaller, than the square of the average speed (about 17.36), and the two speeds cannot be equal because the molecules do not all move with the same speed.
Vrms > \(\bar{V}\) option is correct.
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