Physics · Thermal Physics and Properties of Matter

Kinetic Theory of Gases formulas for JEE

Every Kinetic Theory of Gases formula you need for JEE, grouped by concept.

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29 formulas1 concepts
01Kinetic Theory of Gases
01

Kinetic Theory of Gases

29 formulas

Average Speed

vavg=8kBTπm=8RTπM0v_{avg} = \sqrt{\frac{8 k_B T}{\pi m}} = \sqrt{\frac{8 R T}{\pi M_0}}

The mean molecular speed in a Maxwell-Boltzmann distribution.

applies whenMaxwellian velocity distribution.
speedaveragejee-advanced

Collision Frequency

ν=1τ=2nπd2⟨v⟩\nu = \frac{1}{\tau} = \sqrt{2} n \pi d^2 \langle v \rangle

Rate at which a molecule undergoes collisions.

applies whenAssuming Maxwellian speed distribution.
collisionfrequencykinetics

Combined Gas Law

P1V1T1=P2V2T2\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Relates states of a fixed amount of gas undergoing a change.

applies whenConstant number of moles.
ideal_gasstate_change

Dalton's Law of Partial Pressures

Ptotal=P1+P2+⋯=∑μiRTVP_{total} = P_1 + P_2 + \dots = \sum \frac{\mu_i R T}{V}

Total pressure of a mixture of non-reacting gases is the sum of their individual partial pressures.

applies whenNon-reacting ideal gas mixture.
partial_pressuremixture

Equipartition of Energy

EDOF=12kBTE_{DOF} = \frac{1}{2} k_B T

Average energy associated with each quadratic term (degree of freedom) in thermal equilibrium.

applies whenClassical limits, high enough temperature to activate modes.
equipartitiondegrees_of_freedom

Specific Heat Ratio (Gamma)

γ=CpCv=1+2f\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}

Ratio of specific heats related to the degrees of freedom.

applies whenIdeal gas with f active degrees of freedom.
specific_heatgammaadiabatic_index

Graham's Law of Diffusion

r1r2=M2M1\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}

Ratio of diffusion or effusion rates for two different gases.

applies whenGases at identical temperatures and pressures.
diffusioneffusiongrahamjee-advanced

Ideal Gas Equation (Molar Form)

PV=μRTPV = \mu RT

Standard equation of state for an ideal gas relating pressure, volume, temperature, and moles.

applies whenIdeal gas behavior (low pressure, high temperature limit).
ideal_gasthermodynamicsstate

Ideal Gas Equation (Number Density Form)

P=nkBTP = n k_B T

Ideal gas equation expressed in terms of molecular number density.

applies whenIdeal gas.
ideal_gasdensityboltzmann

Ideal Gas Equation (Mass Density Form)

P=ρRTM0P = \frac{\rho R T}{M_0}

Ideal gas equation expressed in terms of mass density.

applies whenIdeal gas.
ideal_gasmass_density

Total Internal Energy of Ideal Gas

U=f2μRTU = \frac{f}{2} \mu RT

Total internal energy calculated from degrees of freedom.

applies whenIdeal gas, f = total active degrees of freedom.
internal_energydegrees_of_freedom

Maxwell-Boltzmann Speed Distribution

dNv=4πN(m2πkBT)3/2v2e−mv22kBTdvdN_v = 4\pi N \left( \frac{m}{2\pi k_B T} \right)^{3/2} v^2 e^{-\frac{mv^2}{2k_B T}} dv

Number of molecules with speeds between v and v + dv.

applies whenGas in thermal equilibrium.
distributionmaxwellboltzmannjee-advanced

Mayer's Relation

Cp−Cv=RC_p - C_v = R

Relationship between molar specific heat at constant pressure and constant volume.

applies whenIdeal gas.
specific_heatmayer

Mean Free Path

l=12πnd2l = \frac{1}{\sqrt{2} \pi n d^2}

Average distance a molecule travels between two successive collisions.

applies whenAssuming Maxwellian speed distribution.
mean_free_pathcollisions

Mean Free Path (P, T Dependence)

l=kBT2πd2Pl = \frac{k_B T}{\sqrt{2} \pi d^2 P}

Mean free path expressed in terms of macroscopic pressure and temperature.

applies whenIdeal gas.
mean_free_pathpressuretemperaturejee-advanced

Average Translational Kinetic Energy

ϵt‾=12mv2‾=32kBT\overline{\epsilon_t} = \frac{1}{2} m \overline{v^2} = \frac{3}{2} k_B T

Average translational kinetic energy per molecule of a gas.

applies whenDepends only on absolute temperature.
energytemperaturemicroscopic

Specific Heat (Cv) of Mixture

Cv,mix=μ1Cv1+μ2Cv2μ1+μ2C_{v, mix} = \frac{\mu_1 C_{v1} + \mu_2 C_{v2}}{\mu_1 + \mu_2}

Equivalent molar specific heat at constant volume for a gas mixture.

applies whenNon-reacting ideal gas mixture.
mixturespecific_heatjee-advanced

Gamma of Mixture

μ1+μ2γmix−1=μ1γ1−1+μ2γ2−1\frac{\mu_1 + \mu_2}{\gamma_{mix} - 1} = \frac{\mu_1}{\gamma_1 - 1} + \frac{\mu_2}{\gamma_2 - 1}

Equivalent specific heat ratio (gamma) for a gas mixture.

applies whenNon-reacting ideal gas mixture.
mixturegammajee-advanced

Equivalent Molar Mass of Mixture

Mmix=μ1M1+μ2M2μ1+μ2M_{mix} = \frac{\mu_1 M_1 + \mu_2 M_2}{\mu_1 + \mu_2}

Effective molar mass of a non-reacting gas mixture.

applies whenNon-reacting gas mixture.
mixturemolar_massjee-advanced

Number of Moles

μ=MM0=NNA\mu = \frac{M}{M_0} = \frac{N}{N_A}

Calculation of moles from total mass or total number of molecules.

molesavogadromass

Momentum Transfer to Wall

Δp=2mvx\Delta p = 2mv_x

Momentum imparted to a wall during a perfectly elastic molecular collision in 1D.

applies whenPerfectly elastic collision with a stationary wall.
momentumcollisionderivation

Rebound Speed (Moving Wall)

vrebound=u+2Vv_{rebound} = u + 2V

Speed of a gas molecule after an elastic collision with a massive wall (like a piston) moving towards it.

applies whenElastic collision; V is wall speed, u is initial molecular speed.
collisionpistonkinematics

Most Probable Speed

vmp=2kBTm=2RTM0v_{mp} = \sqrt{\frac{2 k_B T}{m}} = \sqrt{\frac{2 R T}{M_0}}

The speed possessed by the largest fraction of molecules in a gas.

applies whenMaxwellian velocity distribution.
speedmost_probablejee-advanced

Pressure-Energy Relation

PV=23EPV = \frac{2}{3} E

Relationship between the pressure of an ideal gas and its total translational kinetic energy.

applies whenIdeal gas.
pressureenergytranslational

Kinetic Pressure Equation

P=13nmv2‾P = \frac{1}{3} n m \overline{v^2}

Macroscopic pressure derived from microscopic kinetic theory.

applies whenIsotropic gas in thermal equilibrium.
pressurekinetic_theorymicroscopic

Root Mean Square (RMS) Speed

vrms=3kBTm=3RTM0v_{rms} = \sqrt{\frac{3 k_B T}{m}} = \sqrt{\frac{3 R T}{M_0}}

The square root of the mean squared speed of gas molecules.

applies whenThermal equilibrium.
speedrmstemperature

Specific Heat Capacity of Solids

C=3RC = 3R

Dulong-Petit law prediction for molar specific heat of solids.

applies whenHigh temperatures where quantum effects are negligible.
solidspecific_heatdulong_petit

Vibrational Mode Energy

ϵv=12m(dydt)2+12ky2\epsilon_v = \frac{1}{2} m \left(\frac{dy}{dt}\right)^2 + \frac{1}{2} k y^2

Energy of a 1D vibrational mode consisting of both kinetic and potential energy components.

applies whenHigh temperatures where vibrational modes are active.
vibrationenergydiatomic

Molecular Volume Fraction

f=N43πr3Vtotalf = \frac{N \frac{4}{3} \pi r^3}{V_{total}}

Ratio of the actual volume of the molecules to the total volume of the gas.

applies whenHard sphere model approximation.
volumemoleculesfraction
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