Stuck on the derivation of pV^gamma=c

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The discussion centers on the derivation of the relation pV^γ = constant for an ideal gas undergoing an adiabatic process. Participants clarify that the assumption of constant pressure allows for the relationship T = pV/nR to lead to the conclusion that (∂T/∂V)_p = T/V. This connection resolves confusion regarding why (∂T/∂V)_p is not equal to p/nR, emphasizing that T is a function of both V and P, but pressure is held constant in this context. The conversation highlights the importance of understanding the roles of temperature and volume in the derivation. Ultimately, the clarification provides relief to those struggling with the concept.
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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.
 
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Fosheimdet said:
T=pV/nR
What's dT? Remember, T is function of both V and P.
 
Yes, but p is constant as indicated by the subscript in \frac{\partial T }{\partial V}_p.
 
Oops, dragged a "red herring" in front of you. Maybe it's too obvious.
Fosheimdet said:
p/nR \left(\frac{\partial T}{\partial V}\right)_p=p/nR? Is there something obvious I'm missing?
What's p/nR? Ideal gas. Rearrange things any way you wish, and p/nR is also equal to ____ ?
 
T/V. You have my gratitude. I will now shed a tear for all the sleep this trivial thing has cost me.
 
Been there, done that.
 
So I know that electrons are fundamental, there's no 'material' that makes them up, it's like talking about a colour itself rather than a car or a flower. Now protons and neutrons and quarks and whatever other stuff is there fundamentally, I want someone to kind of teach me these, I have a lot of questions that books might not give the answer in the way I understand. Thanks
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