Library / Foundations: Physics / FPH-48
Foundations: Physics
Gas pressure from molecular motion
Pressure of an ideal gas explained by molecules striking the walls: one third of the density times the mean square molecular speed.
- Ref
- FPH-48
- Rev
- v1
- Field
- Foundations: Physics
- Result
- p [Pa]
Pressure comes from molecules striking the container walls. Averaging over random directions gives p = ⅓ ρ vrms², where ρ is the gas density and vrms the root-mean-square molecular speed. It links a large-scale quantity (pressure) to a microscopic one (speed). Together with the ideal-gas law it gives the rms speed in terms of temperature.
Pressure of an ideal gas explained by molecules striking the walls: one third of the density times the mean square molecular speed. It links the large-scale pressure to the microscopic motion.
Calculator
- Pressure in bar1.034 bar
- Mean kinetic energy per unit volume, ½ρvrms²155,000 J/m³
- Ratio p / (½ρvrms²) (always 2/3)0.6667
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A gas of density 1.16 kg/m³ with an rms speed of 517 m/s exerts 103,400 Pa, about 1.034 bar: close to atmospheric pressure, as expected for nitrogen near 300 K.
- Doubling the speed at the same density quadruples the pressure (to 413,400 Pa), which is why heating a gas at constant volume raises its pressure steeply.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ρ Gas density | 0.05 to 20 kg/m³ | 4,455 | 1,782,000 | 1,777,000 | +10% | +10 % |
| vrms Root-mean-square molecular speed | 100 to 2,000 m/s | 3,867 | 1,547,000 | 1,543,000 | +10% | +21 % |
Sweep of Root-mean-square molecular speed
| Root-mean-square molecular speed (m/s) | Pressure (Pa) |
|---|---|
| 100 | 3,867 |
| 337.5 | 44,040 |
| 575 | 127,800 |
| 812.5 | 255,300 |
| 1,050 | 426,300 |
| 1,288 | 641,000 |
| 1,525 | 899,200 |
| 1,763 | 1,201,000 |
| 2,000 | 1,547,000 |
Sweep one variable
Variables and how each one contributes
Pressure
What the equation gives you. Everything else feeds it.
Gas density
The density of the gas, in kg/m³. For air at room conditions it is about 1.2. Raises the pressure in direct proportion: more molecules per unit volume strike the wall more often. Raising it by 0.5 (to 1.66) moves the result by +43.1 %, from 103,400 to 147,900 Pa.
Root-mean-square molecular speed
The root-mean-square molecular speed, in m/s. Nitrogen at 300 K has about 517 m/s. Raises the pressure with its square: faster molecules hit harder and more often. Raising it by 50 m/s (to 567 m/s) moves the result by +20.3 %, from 103,400 to 124,300 Pa.
Worked example with the values above
- Square the speed: vrms² = 517² = 267,300 m²/s².
- Multiply by the density: ρ vrms² = 1.16 × 267,300 = 310,100 Pa.
- Divide by 3: p = 103,400 Pa.
Assumptions and limits
- An ideal gas: negligible molecular volume and no intermolecular forces except in collisions.
- Random, isotropic translational motion, with the gas at a single absolute temperature in kelvin.
Common mistakes
- Using the mean speed instead of the rms speed.
- Forgetting the factor ⅓.
Statistics and uncertainty
The relation is exact for an ideal gas, and the quantity measured in practice is the pressure. Because pressure depends on vrms², a 1 % error in the speed gives a 2 % error in the pressure.
Where it is used
- Estimating gas pressure from density and the typical molecular speed.
- Checking kinetic-theory results against the ideal-gas law.
Maths you need
- Ideal gas: molecules with negligible size and no attraction
- Absolute temperature: kelvin scale, starting at absolute zero