Monday, 15 October 2018

homework and exercises - Adiabatic process of an ideal gas derivation


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 CVCpCV=1γ1



using the fact that CpCV=NkB and γ=CpCV it can be written


CvNkB=CVCpCV=1γ1.


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=CVCpCV=CVCVCpCVCVCV=1γ1 Therefore, CVNkB(pdV+Vdp)=1γ1(pdV+Vdp)=pdV Dividing both sides with pdV: 1γ1(1+VpdpdV)=1 Continue to simplify the expressions and you will reach your result of pVγis constant.


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