Calculate thermal velocity (rms, mean or most probable speed) of gas molecules or electrons from temperature, using a preset gas or your own molar mass.
Thermal Velocity Formula
The following particle-mass equation gives root-mean-square thermal speed for a classical dilute ideal gas in Maxwell–Boltzmann equilibrium at positive absolute temperature. Characteristic speeds must be much smaller than light speed; interactions, quantum effects and relativistic motion can invalidate this model.
- Where vrms is the RMS thermal speed (m/s)
- k is the Boltzmann constant (1.380649×10−23 J/K)
- T is the absolute temperature (K)
- m is the mass of one particle (kg)
For RMS speed, take the square root of 3kT/m using particle mass m. This calculator instead accepts molar mass M = NAm in kg/mol, giving vrms = √(3RT/M). Here R = kNA = 8.31446261815324 J/(mol·K), from the exact Boltzmann and Avogadro constants. The other choices are mean speed √(8RT/(πM)) and most-probable speed √(2RT/M).
What is a Thermal Velocity?
Definition:
In this calculator, thermal velocity refers to a selected characteristic molecular speed: RMS, mean speed or most-probable speed. These are velocity magnitudes, not the average velocity vector. In an isotropic equilibrium gas with no bulk drift, the mean velocity vector is zero even though these characteristic speeds are positive.
How to Calculate Thermal Velocity?
Example Problem:
The following example outlines the steps and information needed to calculate thermal velocity (RMS thermal speed).
First, determine the absolute temperature. In this example, the absolute temperature is found to be 300 K.
Next, determine the mass of one particle. For this problem, m = 4.651734×10−26 kg for a nitrogen molecule, N2. The corresponding calculator input is molar mass M = NAm ≈ 28.0134 g/mol. Select RMS speed and enter 300 K and that molar mass.
Finally, calculate the thermal velocity using the formula above:
vrms = SQRT ( 3 * k * T / m )
vrms = SQRT ( 3 * 1.380649×10−23 * 300 / (4.651734×10−26) )
vrms ≈ 516.8 m/s
FAQ
What is the Boltzmann constant and why is it important in calculating thermal velocity?
The Boltzmann constant (k) is a physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. It is crucial in calculating thermal velocity because it provides a link between the macroscopic temperature (a measure of thermal energy) and the microscopic motion of particles, allowing for the calculation of the speed at which particles move due to thermal energy.
How does the mass of a particle affect its thermal velocity?
At a fixed temperature, each of these characteristic speeds scales as the inverse square root of particle mass or molar mass, not as its simple inverse. A gas with four times the molar mass has half the characteristic speed under the same classical model. This is a statistical distribution relation, not a statement that all particles travel at one speed.
Can thermal velocity be applied to all states of matter, or is it specific to gases?
These specific Maxwell–Boltzmann formulas model classical dilute-gas molecular speeds at equilibrium. They are not a general calculation of vibrations, collective motion or transport in liquids and solids. Low-temperature quantum behavior, strong interactions and relativistic motion require other models. Exact absolute zero is outside this calculator’s positive-temperature classical distribution domain.
