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$$C_vdT + (c_p-c_V)\left(\frac{\partial T}{\partial V}\right)_pdV=0$$

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 [itex]\left(\frac{\partial T}{\partial V}\right)_p=p/nR[/itex]? Is there something obvious I'm missing? Would love to get this cleared up so I can get some sleep tonight.