Root Mean Square Speed Calculator for Ideal Gas
Calculate RMS speed, kinetic energy, and molecular velocities for ideal gases using kinetic theory
Calculate RMS Speed for Ideal Gas
Select a common gas or choose custom to enter your own molar mass
Temperature of the gas
Universal gas constant (default: 8.314 J/(K·mol))
RMS Speed Results
Gas: Air (dry)
Formula used: v_rms = √(3RT/M)
Temperature: 293.15 K
Molar mass: 28.965 g/mol
Speed Comparison
Physical Interpretation
Example Calculations
Air at Room Temperature
Gas: Air (dry)
Temperature: 20°C = 293.15 K
Molar mass: 28.965 g/mol = 0.028965 kg/mol
RMS speed: v_rms = √(3 × 8.314 × 293.15 / 0.028965) ≈ 502 m/s
Hydrogen at Standard Temperature
Gas: Hydrogen (H₂)
Temperature: 273.15 K (0°C)
Molar mass: 2.016 g/mol = 0.002016 kg/mol
RMS speed: v_rms = √(3 × 8.314 × 273.15 / 0.002016) ≈ 1838 m/s
Common Gas Properties
Speed Relationships
RMS Speed
v_rms = √(3RT/M)
Root mean square velocity
Average Speed
v_avg = √(8RT/πM)
Mean molecular speed
Most Probable Speed
v_mp = √(2RT/M)
Peak of distribution
Physical Constants
Understanding Root Mean Square Speed
What is RMS Speed?
Root Mean Square (RMS) speed is a measure of the average speed of gas molecules. It represents the square root of the mean of the squares of individual molecular velocities. RMS speed is particularly useful because it relates directly to the kinetic energy of gas molecules.
Kinetic Theory Assumptions
- •Gas molecules are point particles with negligible volume
- •Collisions are perfectly elastic
- •No intermolecular forces except during collisions
- •Molecules are in constant, random motion
Mathematical Derivation
Starting from Ideal Gas Law
PV = nRT
Combined with kinetic energy: KE = ½mv² = (3/2)kT
RMS Speed Formula
v_rms = √(3RT/M)
Where R = gas constant, T = temperature, M = molar mass
Note: RMS speed increases with temperature and decreases with molar mass.
Applications and Significance
Diffusion and Effusion
RMS speed determines how quickly gases diffuse and effuse. Lighter gases like hydrogen diffuse faster than heavier gases.
Atmospheric Escape
Molecules with speeds exceeding escape velocity can leave planetary atmospheres, explaining atmospheric composition.
Chemical Kinetics
Reaction rates depend on molecular speeds and collision frequency, which are related to RMS speed.
Speed Distribution
Maxwell-Boltzmann Distribution
Gas molecules follow a Maxwell-Boltzmann speed distribution. The three characteristic speeds are:
- • Most probable speed (peak of curve)
- • Average speed (mean of distribution)
- • RMS speed (related to kinetic energy)
Temperature Effects
As temperature increases, the distribution curve flattens and shifts to higher speeds. All three characteristic speeds increase proportionally to the square root of temperature.