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7. Show that the pressure `P` of a gas can be written `P=1/3ρv^2`, where `ρ` is the density of the gas and `v` is the rms speed of the molecules.
7. Show that the pressure P of a gas can be written P=1/3ρv^2, where ρ is the density of the gas and v is the rms speed of the molecules. `PV = NkT`
`v_(rms) =sqrt((3kT)/m)`
`ρ = M/V`
Equations:
`PV = NkT`
`v_(rms) =sqrt((3kT)/m)`
`ρ = M/v`
Solution:
Total mass `M = Nm`
`ρ = M/V => V = M/ρ =(Nm)/ρ`
`v_(rms) =sqrt((3kT)/m) = >T = (mv_(rms))^2/(3k)`
`PV = NkT => P = (NkT)/V`
`P = (Nk((mv_(rms))^2/(3k)))/((Nm)/ρ)`
`P = 1/3ρv_(rms)^2`
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