516.82
m/s
1,860.55
km/h
—
m/s
—
m/s
6.2145
× 10⁻²¹ J
0.028014
kg/mol
516.82
m/s
1,860.55
km/h
—
m/s
—
m/s
6.2145
× 10⁻²¹ J
0.028014
kg/mol
The Root Mean Square (RMS) Velocity Calculator determines the characteristic speed of gas molecules at a given temperature using the kinetic molecular theory. The RMS velocity is defined as the square root of the average of the squared velocities of all molecules in a gas sample:
$$v_{rms} = \sqrt{\frac{3RT}{M}}$$
where R = 8.314 J/(mol·K) is the universal gas constant, T is the absolute temperature in kelvin, and M is the molar mass in kg/mol. This formula emerges directly from the equipartition theorem, which assigns $$\frac{1}{2}k_BT$$ of kinetic energy to each translational degree of freedom.
The calculator also provides two related velocity measures from the Maxwell-Boltzmann distribution: the mean velocity $$v_{avg} = \sqrt{\frac{8RT}{\pi M}}$$ and the most probable velocity $$v_{mp} = \sqrt{\frac{2RT}{M}}$$. These three speeds satisfy the inequality $$v_{mp} < v_{avg} < v_{rms}$$ at all temperatures, reflecting the asymmetric shape of the Maxwell-Boltzmann speed distribution.
Enter the temperature in kelvin and the molar mass in g/mol. The calculator converts molar mass to kg/mol (dividing by 1000) and applies the kinetic theory formulas:
RMS velocity: $$v_{rms} = \sqrt{\frac{3RT}{M}}$$
Mean velocity: $$v_{avg} = \sqrt{\frac{8RT}{\pi M}}$$
Most probable velocity: $$v_{mp} = \sqrt{\frac{2RT}{M}}$$
The average translational kinetic energy per molecule is: $$\langle KE \rangle = \frac{3}{2}k_BT$$
where $$k_B = 1.381 \times 10^{-23}$$ J/K is Boltzmann's constant. Notice that kinetic energy depends only on temperature, not on mass — all gases at the same temperature have the same average kinetic energy. However, lighter molecules move faster to achieve this energy, which is why hydrogen effuses faster than oxygen (Graham's law).
RMS velocity tells you how fast molecules are moving on average (weighted by energy). At room temperature (300 K), nitrogen molecules (M = 28) travel at ~517 m/s — faster than the speed of sound (343 m/s). Hydrogen at the same temperature reaches ~1920 m/s. The ratio $$v_{rms} \propto 1/\sqrt{M}$$ explains why lighter gases diffuse and effuse more rapidly. Temperature dependence $$v_{rms} \propto \sqrt{T}$$ means doubling the absolute temperature increases molecular speed by a factor of $$\sqrt{2} \approx 1.41$$.
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N₂ molecules at room temperature move at ~517 m/s (1860 km/h), yet the gas appears still because motion is random.
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Light helium atoms at 500 K reach ~1767 m/s — relevant for helium leak detection and atmospheric escape.
The most probable velocity $$v_{mp}$$ is the peak of the Maxwell-Boltzmann distribution. The mean velocity $$v_{avg}$$ is the arithmetic average of all molecular speeds. The RMS velocity $$v_{rms}$$ is the square root of the average squared speed. They relate as $$v_{mp} : v_{avg} : v_{rms} = 1 : 1.128 : 1.225$$.
RMS velocity depends only on temperature and molar mass. Increasing pressure at constant temperature adds more molecules but doesn't change individual molecular speeds. Pressure affects collision frequency and mean free path, but not the average speed.
The speed of sound in a gas is related to RMS velocity: $$v_{sound} = \sqrt{\frac{\gamma RT}{M}}$$, where γ is the heat capacity ratio. For a diatomic gas (γ = 7/5), $$v_{sound} \approx 0.68 \times v_{rms}$$.
All gases at the same temperature have the same average kinetic energy: $$\frac{1}{2}mv^2 = \frac{3}{2}k_BT$$. Since KE = ½mv², lighter molecules must have higher velocities to store the same energy. This is why $$v_{rms} \propto 1/\sqrt{M}$$.
Graham's Law states that the rate of effusion is inversely proportional to the square root of molar mass: $$\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$$. This follows directly from the RMS velocity relationship and is used in isotope separation (e.g., uranium enrichment).
Classical kinetic theory predicts zero velocity at 0 K (absolute zero). However, quantum mechanics shows that molecules retain zero-point energy even at absolute zero. In practice, absolute zero is unattainable (Third Law of Thermodynamics).
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