Maxwell-Boltzmann Speed Distribution Calculator
Find the probability density f(v) of the Maxwell-Boltzmann speed distribution, plus the most probable and mean molecular speeds, for any gas.
📊 What is the Maxwell-Boltzmann Speed Distribution Calculator?
This Maxwell-Boltzmann speed distribution calculator finds the probability density f(v) for molecular speed in a gas, using f(v) = 4π(M/(2πRT))^(3/2) v² e^(-Mv²/(2RT)). Enter a temperature, a molar mass, and a speed, and it returns the density at that speed along with the distribution's three characteristic speeds: the most probable speed, the mean speed, and the RMS speed.
This calculator is useful for chemistry and physics students studying the kinetic theory of gases, for anyone checking where a chosen speed falls relative to a gas's characteristic speeds, and for visualizing how temperature and molar mass reshape the distribution curve. It complements the sibling RMS Speed Calculator, Most Probable Speed Calculator, and Mean Speed Calculator, which each compute one characteristic speed on its own.
A common misconception is that f(v) itself is a probability. It is not, f(v) is a probability density, its value only becomes a probability once multiplied by a speed interval dv, or integrated over a range of speeds. The peak of the curve at v_p simply marks where molecular speeds are most concentrated, not where any single molecule is guaranteed to be.
Behind the scenes, this calculator uses the practical form of the distribution written in terms of molar mass M (g/mol, converted to kg/mol) and the universal gas constant R, so it never needs Boltzmann's constant or Avogadro's number as separate inputs, R and M already absorb them.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Below the peak
N₂ at 298.15 K (M=28.014 g/mol), evaluated at v=200 m/s
Example 2 — At the most probable speed
N₂ at 298.15 K, evaluated at v=vp=420.69 m/s
Example 3 — At the mean speed
N₂ at 298.15 K, evaluated at v=vmean=474.70 m/s
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Maxwell-Boltzmann speed distribution?
The Maxwell-Boltzmann speed distribution is a probability density function f(v) that describes how molecular speeds are spread out in a gas at thermal equilibrium. It gives the relative likelihood of finding a molecule with a particular speed v, and its shape depends only on the gas's temperature and molar mass.
What is the formula for the Maxwell-Boltzmann speed distribution?
f(v) = 4π(M/(2πRT))^(3/2) v² e^(-Mv²/(2RT)), where R is the universal gas constant (8.314 J/(mol·K)), T is the absolute temperature in Kelvin, M is the molar mass in kg/mol, and v is the molecular speed in m/s.
What units does f(v) have?
f(v) has units of seconds per metre (s/m), since it is a probability density with respect to speed. Multiplying f(v) by a small speed interval dv gives the fraction of molecules with speeds between v and v+dv, a dimensionless probability.
What is the most probable speed?
The most probable speed v_p = √(2RT/M) is the speed at the peak of the Maxwell-Boltzmann curve, where f(v) reaches its maximum value. More molecules have speeds near v_p than near any other single speed.
What is the mean speed?
The mean speed v_mean = √(8RT/(πM)) is the average of all molecular speeds in the gas. Because the distribution has a long tail toward high speeds, the mean speed is always slightly higher than the most probable speed.
How is the Maxwell-Boltzmann distribution related to RMS speed?
RMS (root-mean-square) speed, v_rms = √(3RT/M), is the square root of the average of the squared speeds. It is the largest of the three characteristic speeds because squaring before averaging weights fast-moving molecules more heavily than the plain mean speed does.
Why isn't the Maxwell-Boltzmann curve symmetric?
The v² term in the formula suppresses very low speeds (few molecules move at nearly zero speed) while the exponential term suppresses very high speeds, but not as sharply, so the curve rises steeply from zero, peaks at v_p, and then tails off more slowly toward high speeds, making it right-skewed.
Does the area under the Maxwell-Boltzmann curve always equal 1?
Yes, integrating f(v) over all speeds from 0 to infinity always gives exactly 1, since every molecule must have some speed. This calculator's practical formula, using molar mass M and the gas constant R, automatically preserves this normalization.
How does temperature affect the distribution?
Raising the temperature shifts the entire curve toward higher speeds and flattens its peak, since all three characteristic speeds scale with √T. A hotter gas has a wider spread of molecular speeds and a lower, broader peak than a cooler gas of the same molar mass.
How does molar mass affect the distribution?
Heavier gases (higher M) produce a narrower, taller peak shifted toward lower speeds, since all three characteristic speeds scale with 1/√M. At the same temperature, a light gas like helium has a much wider, flatter distribution shifted to higher speeds than a heavy gas like carbon dioxide.
What real-world phenomena depend on the Maxwell-Boltzmann distribution?
It underlies atmospheric escape (whether a planet's gravity can retain a gas, since only the fast tail of the distribution can exceed escape velocity), reaction rate theory (only molecules above a certain speed have enough energy to react), and effusion rates through small openings.
What units does this calculator use?
Temperature is entered in Kelvin, molar mass in grams per mole (the standard periodic-table units, converted internally to kg/mol), and speed in metres per second. The output density f(v) is given in seconds per metre.