Fosheimdet
- 15
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I have been tearing my hair out for a while over a step in the proof of the relation pV^{\gamma}=constant. The textbook has assumed that we are dealing with an ideal gas undergoing an adiabatic process. Therefore dQ=0 and we get
$$C_vdT + (c_p-c_V)\left(\frac{\partial T}{\partial V}\right)_pdV=0$$
which gives
$$dT=-(\gamma-1)\left(\frac{\partial T}{\partial V}\right)_pdV$$
Where $$\gamma=\frac{C_p}{C_V}$$
Now comes the part I don't get. They say that because we are dealing with an ideal gas, we have $$T=pV/nR$$ which gives $$\left(\frac{\partial T}{\partial V}\right)_p = \frac{T}{V}$$
Why isn't \left(\frac{\partial T}{\partial V}\right)_p=p/nR? Is there something obvious I'm missing? Would love to get this cleared up so I can get some sleep tonight.
$$C_vdT + (c_p-c_V)\left(\frac{\partial T}{\partial V}\right)_pdV=0$$
which gives
$$dT=-(\gamma-1)\left(\frac{\partial T}{\partial V}\right)_pdV$$
Where $$\gamma=\frac{C_p}{C_V}$$
Now comes the part I don't get. They say that because we are dealing with an ideal gas, we have $$T=pV/nR$$ which gives $$\left(\frac{\partial T}{\partial V}\right)_p = \frac{T}{V}$$
Why isn't \left(\frac{\partial T}{\partial V}\right)_p=p/nR? Is there something obvious I'm missing? Would love to get this cleared up so I can get some sleep tonight.