University Physics Volume 2

# Key Equations

### 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 $[p+a(nV)2](V−nb)=nRT[p+a(nV)2](V−nb)=nRT$ Pressure, volume, and molecular speed $pV=​13Nmv2–pV=​13Nmv2–$ 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 $f(v)=4π(m2kBT)3/2v2e−mv2/2kBTf(v)=4π(m2kBT)3/2v2e−mv2/2kBT$ Average velocity of a molecule $v¯=8πkBTm=8πRTMv¯=8πkBTm=8πRTM$ Peak velocity of a molecule $vp=2kBTm=2RTMvp=2kBTm=2RTM$