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Key Equations

Ideal gas law in terms of molecules pV=NkBTpV=NkBT
Ideal gas law ratios if the amount of gas is constant p1V1T1=p2V2T2p1V1T1=p2V2T2
Ideal gas law in terms of moles pV=nRTpV=nRT
Van der Waals equation [p+a(nV)2](Vnb)=nRT[p+a(nV)2](Vnb)=nRT
Pressure, volume, and molecular speed pV=13Nmv2pV=13Nmv2
Root-mean-square speed vrms=3RTM=3kBTmvrms=3RTM=3kBTm
Mean free path λ=V42πr2N=kBT42πr2pλ=V42πr2N=kBT42πr2p
Mean free time τ=kBT42πr2pvrmsτ=kBT42πr2pvrms
The following two equations apply only to a monatomic ideal gas:
Average kinetic energy of a molecule K=32kBTK=32kBT
Internal energy Eint=32NkBT.Eint=32NkBT.
Heat in terms of molar heat capacity at constant volume Q=nCVΔTQ=nCVΔT
Molar heat capacity at constant volume for an ideal gas with d degrees of freedom CV=d2RCV=d2R
Maxwell–Boltzmann speed distribution f(v)=4π(m2kBT)3/2v2emv2/2kBTf(v)=4π(m2kBT)3/2v2emv2/2kBT
Average velocity of a molecule v¯=8πkBTm=8πRTMv¯=8πkBTm=8πRTM
Peak velocity of a molecule vp=2kBTm=2RTMvp=2kBTm=2RTM
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