PV = nRT
Pressure times volume equals number of moles times the gas constant times absolute temperature.
Equation of state of a perfect gas relating pressure, volume, moles and temperature.
- P
- pressure (Pa)
- V
- volume (m^3)
- n
- number of moles (mol)
- R
- universal gas constant (J/(mol.K))
- T
- absolute temperature (K)
Use when — The gas is treated as ideal, at conditions where molecular size and intermolecular attraction are negligible.
Common trap — Using this equation unmodified at high pressure or low temperature where deviation from ideal behaviour is significant.
P = (1/3) rho v_rms^2
Pressure equals one third of density times root mean square speed squared.
Pressure expressed in terms of gas density and mean square molecular speed.
- P
- pressure (Pa)
- rho
- gas density (kg/m^3)
- v_rms
- root mean square speed (m/s)
Use when — Applying the kinetic-theory derivation of pressure from molecular collisions.
Common trap — Substituting average speed or most probable speed in place of rms speed here.
v_rms = sqrt(3RT/M) = sqrt(3kT/m)
Root mean square speed equals the square root of three R T over M, or equivalently three k T over m.
Root mean square speed of gas molecules in terms of temperature and molar or molecular mass.
- M
- molar mass (kg/mol)
- m
- mass of one molecule (kg)
- k
- Boltzmann constant (J/K)
- T
- absolute temperature (K)
Use when — The question asks for rms speed or a quantity that traces back to mean square speed, such as pressure or kinetic energy.
Common trap — Treating this as the average speed of molecules; it is not.
v_avg = sqrt(8RT/(pi M))
Average speed equals the square root of eight R T over pi M.
Mean speed of gas molecules from the Maxwell speed distribution.
- v_avg
- average speed (m/s)
Use when — The question specifically asks for the mean speed of molecules, not rms or most probable speed.
Common trap — Assuming v_avg equals v_rms; the coefficients differ.
v_mp = sqrt(2RT/M)
Most probable speed equals the square root of two R T over M.
Speed at which the Maxwell speed distribution has its maximum.
- v_mp
- most probable speed (m/s)
Use when — The question asks for the speed at the peak of the molecular speed distribution.
Common trap — Confusing this with the average or rms speed; the three are always in the order v_mp < v_avg < v_rms.
(1/2) m v_rms^2 = (3/2) kT per molecule
One half m v rms squared equals three halves k T, per molecule.
Kinetic interpretation of temperature: average translational kinetic energy per molecule is proportional to absolute temperature.
- T
- absolute temperature (K)
Use when — Relating molecular translational kinetic energy directly to temperature, independent of gas identity.
Common trap — Applying this translational-only relation to total internal energy of a diatomic or polyatomic gas without adding the other degrees of freedom.
U = (f/2) nRT
Internal energy equals f over two times n times R times T.
Internal energy of an ideal gas from the equipartition of energy over its degrees of freedom.
- f
- degrees of freedom per molecule (dimensionless)
- U
- internal energy (J)
Use when — The degrees of freedom for the gas have already been fixed, for example 3 for monatomic or 5 for diatomic at moderate temperature.
Common trap — Using f = 3 for a diatomic gas by habit instead of checking whether rotational modes are active.
Cv = (f/2) R
Molar specific heat at constant volume equals f over two times R.
Molar specific heat at constant volume from degrees of freedom.
- Cv
- molar specific heat at constant volume (J/(mol.K))
Use when — Deriving Cv purely from the kinetic model, before using it inside a thermodynamic process.
Common trap — Quoting a numeric Cv value for a gas without stating which degrees-of-freedom count it assumes.
Cp = Cv + R = ((f+2)/2) R
Molar specific heat at constant pressure equals molar specific heat at constant volume plus R.
Molar specific heat at constant pressure, obtained from Cv using the ideal gas relation.
- Cp
- molar specific heat at constant pressure (J/(mol.K))
Use when — An ideal gas undergoes a constant-pressure process and Cv is already known.
Common trap — Adding R twice, once here and again inside a process calculation.
gamma = Cp/Cv = (f+2)/f
Gamma equals Cp over Cv, which equals f plus two over f.
Ratio of specific heats, fixed once the degrees of freedom are fixed.
- gamma
- ratio of specific heats (dimensionless)
Use when — The degrees-of-freedom count for the gas is settled and gamma is needed, including for use inside an adiabatic process elsewhere.
Common trap — Using a gamma value appropriate to a monatomic gas for a diatomic gas or vice versa.
lambda = 1/(sqrt(2) pi d^2 n)
Mean free path equals one over root two pi d squared n.
Average distance a molecule travels between successive collisions.
- lambda
- mean free path (m)
- d
- molecular diameter (m)
- n
- number density of molecules (1/m^3)
Use when — Estimating molecular-scale transport behaviour, such as how far a molecule travels before colliding again.
Common trap — Forgetting that number density n depends on pressure and temperature through the gas equation, so lambda is not a fixed constant for a gas.
N = n N_A
Total number of molecules equals number of moles times Avogadro's number.
Total number of molecules from number of moles and Avogadro's number.
- N
- total number of molecules (dimensionless)
- N_A
- Avogadro's number (1/mol)
Use when — Converting between a mole-based and a molecule-count-based form of a kinetic-theory relation.
Common trap — Mixing per-mole and per-molecule constants, such as R and k, without converting through N_A.
