I am working through the derivation of an adiabatic process of an ideal gas pVγ and I can't see how to go from one step to the next. Here is my derivation so far which I understand:
dE=dQ+dW dW=−pdV dQ=0 dE=CVdT
therefore
CVdT=−pdV
differentiate the ideal gas equation pV=NkBT
pdV+Vdp=NkBdT
rearrange for dT and substitute into the 1st law:
CVNkB(pdV+Vdp)=−pdV.
The next part is what I am stuck with I can't see how the next line works specifically how to go from CVCp−CV=1γ−1
using the fact that Cp−CV=NkB and γ=CpCV it can be written
CvNkB=CVCp−CV=1γ−1.
If this could be explained to me, I suspect it is some form of algebraic rearrangement that I am not comfortable with that is hindering me.
Answer
I think the last line does not follow from the previous steps. It is used to show how γ comes in place, so I extrapolated a bit and show the next few steps:
Since CVNkB=CVCp−CV=CVCVCpCV−CVCV=1γ−1 Therefore, CVNkB(pdV+Vdp)=1γ−1(pdV+Vdp)=−pdV Dividing both sides with pdV: 1γ−1(1+VpdpdV)=−1 Continue to simplify the expressions and you will reach your result of pVγis constant.
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