Most Probable Speed
Speed at which Maxwell-Boltzmann distribution peaks.
Variables
Derivation
Set df/dv = 0 for Maxwell-Boltzmann distribution f(v) ∝ v²e^(-mv²/2kT). v_p < v_mean < v_rms.
Your notes
Constants in this formula
Check value
Round inputs run through this page's calculator, with each constant at its tabulated value. Use it to test your own working or a rearranged form. It is a test case, not a real-world example.
- R = 8.314 462 618… J/(mol·K) (Gas constant)
- T = 2 K (Temperature)
- M = 3 kg/mol (Molar mass)
Result: vp = 3.3295571 m/s
Other thermodynamics formulas with the same quantities
- Ideal Gas Law (Kinetic Theory) shares Gas constant, Temperature
- Ideal Gas Law (Molecular) (Kinetic Theory) shares Boltzmann constant, Temperature
- Mean Free Path (Kinetic Theory) shares Boltzmann constant, Temperature
- Work (Isothermal Process) (First Law) shares Gas constant, Temperature
- Internal Energy (Ideal Gas) (First Law) shares Gas constant, Temperature
- Stefan-Boltzmann Law (Heat & Temperature) shares Temperature
Where this formula is listed
- Thermodynamics formulas, under Kinetic Theory, alongside Pressure (Kinetic Theory).
- In the thermodynamics list it comes after Pressure (Kinetic Theory) and before Mean Speed.
- Thermodynamics formula sheet, one of 34 formulas printed with symbol keys and SI units.
- The full formula index, grouped by category and topic.
Common questions
What is the Most Probable Speed formula?
Most Probable Speed is written v_p = √((2k_B T)/m) = √(2RT/M). Speed at which Maxwell-Boltzmann distribution peaks. It belongs to Thermodynamics, under Kinetic Theory.
What are the units in v_p = √((2k_B T)/m) = √(2RT/M)?
In SI units, v_p is most probable speed, measured in m/s; k_B is boltzmann constant, measured in kg·m²/(s²·K); T is temperature, measured in K; m is molecular mass, measured in kg; R is gas constant, measured in kg·m²/(s²·mol·K); M is molar mass, measured in kg/mol.
How is the Most Probable Speed formula derived?
Set df/dv = 0 for Maxwell-Boltzmann distribution f(v) ∝ v²e^(-mv²/2kT). v_p < v_mean < v_rms. The result is v_p = √((2k_B T)/m) = √(2RT/M).