Let \(\overline{V}\), Vrms, Vp denotes the mean speed, root mean square speed and most probable speed of the molecules each of mass m in an ideal monoatomic gas at absolute temperature T Kelvin. Which statement(s) is/are correct?
In an ideal gas the molecular speeds follow the Maxwell distribution, which is non-zero for every speed from zero to arbitrarily large values. Its three characteristic speeds and the average kinetic energy follow from kT/m.
Step 1 — Characteristic speeds: \(V_p=\sqrt{\dfrac{2kT}{m}},\ \ \overline{V}=\sqrt{\dfrac{8kT}{\pi m}},\ \ V_{rms}=\sqrt{\dfrac{3kT}{m}}\).
Step 2 — Their ratio \(\sqrt{2}:\sqrt{8/\pi}:\sqrt{3}\approx1.41:1.60:1.73\) gives \(V_p\lt\overline{V}\lt V_{rms}\), so that statement holds.
Step 3 — Average kinetic energy of a molecule of a monoatomic gas: \(\dfrac{3}{2}kT=\dfrac{3}{4}m\left(\dfrac{2kT}{m}\right)=\dfrac{3}{4}mV_p^{2}\), so that statement holds too.
Step 4 — Because the distribution has a tail at high speeds and extends down to zero speed, some molecules are faster than √2Vrms and some are slower than \(V_p/\sqrt{2}\). Both statements that set an upper or a lower limit on molecular speed fail.
\(V_p\lt\overline{V}\lt V_{rms}\) and Average kinetic energy of a molecule is \(\dfrac{3}{4}mV_p^{2}\) options are correct.
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