Relation between kinetic energy and degree of freedom
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First of all, if we have an ideal gas then pV=nRTpV=nRT, so p=(nRT)/Vp=(nRT)/V. Then, W=(nRT)/VdVW=(nRT)/VdV. Your Work equation is wrong.
To clarify on how to develop from dU=dWdU=dW in an alternative way, we can obtain some interesting properties.
We have dU=dW→U=WdU=dW→U=W , for an adiabatic process. (It is also known that Cp=(f/2)nR+nRCp=(f/2)nR+nR and Cv=(f/2)nRCv=(f/2)nR ; Heat capacities for constant pressure and volume, respectively)
So:
dU=(f/2)nRdT=(Cv)dT,asCv=(f/2)nRdU=(f/2)nRdT=(Cv)dT,asCv=(f/2)nR
Now we can develop the equation:
(Cv)dT=−((nRT)/V)dV<−>(Cv)dT=−((nRT)/V)dV<−>
(Cv)dT/T=−((Cp−Cv)/V)dV(Cv)dT/T=−((Cp−Cv)/V)dV
Integrating both terms, (assuming T0T0,V0V0as initial TT and VV, respectively)
Cv∗ln(T/T0)=−(Cp−Cv)∗ln(V/V0)<−>Cv∗ln(T/T0)=−(Cp−Cv)∗ln(V/V0)<−>
ln((T/T0)Cv)=ln((V/V0)(Cv−Cp))<−>ln((T/T0)Cv)=ln((V/V0)(Cv−Cp))<−>
T/T0=(V/V0)((Cv−Cp)/Cv)T/T0=(V/V0)((Cv−Cp)/Cv)
And as γ=(Cv/Cp)γ=(Cv/Cp): T/T0=(V/V0)(1−γ)T/T0=(V/V0)(1−γ) ; where γγ is a greek letter, which represents the adiabatic constant.
We can say that, TV(γ−1)=T0V0=constantTV(γ−1)=T0V0=constant We can also obtain PVγPVγ = constant applying the Ideal gas law in TV(γ−1)=constantTV(γ−1)=constant.
To clarify on how to develop from dU=dWdU=dW in an alternative way, we can obtain some interesting properties.
We have dU=dW→U=WdU=dW→U=W , for an adiabatic process. (It is also known that Cp=(f/2)nR+nRCp=(f/2)nR+nR and Cv=(f/2)nRCv=(f/2)nR ; Heat capacities for constant pressure and volume, respectively)
So:
dU=(f/2)nRdT=(Cv)dT,asCv=(f/2)nRdU=(f/2)nRdT=(Cv)dT,asCv=(f/2)nR
Now we can develop the equation:
(Cv)dT=−((nRT)/V)dV<−>(Cv)dT=−((nRT)/V)dV<−>
(Cv)dT/T=−((Cp−Cv)/V)dV(Cv)dT/T=−((Cp−Cv)/V)dV
Integrating both terms, (assuming T0T0,V0V0as initial TT and VV, respectively)
Cv∗ln(T/T0)=−(Cp−Cv)∗ln(V/V0)<−>Cv∗ln(T/T0)=−(Cp−Cv)∗ln(V/V0)<−>
ln((T/T0)Cv)=ln((V/V0)(Cv−Cp))<−>ln((T/T0)Cv)=ln((V/V0)(Cv−Cp))<−>
T/T0=(V/V0)((Cv−Cp)/Cv)T/T0=(V/V0)((Cv−Cp)/Cv)
And as γ=(Cv/Cp)γ=(Cv/Cp): T/T0=(V/V0)(1−γ)T/T0=(V/V0)(1−γ) ; where γγ is a greek letter, which represents the adiabatic constant.
We can say that, TV(γ−1)=T0V0=constantTV(γ−1)=T0V0=constant We can also obtain PVγPVγ = constant applying the Ideal gas law in TV(γ−1)=constantTV(γ−1)=constant.
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