Partial derivatives of Gas Law

  • Thread starter peter2108
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  • #1
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In James Stewart's Calculus exercise 82 page 891 asks you to show that:

[itex]\frac{\partial P}{\partial V}\frac{\partial V}{\partial T}\frac{\partial T}{\partial P} = -1[/itex]

I can do this by noting that [itex]V = \frac{nRT}{P}[/itex] so that:

[itex]\frac{\partial V}{\partial T}[/itex] = [itex]\frac{\partial}{\partial T}\left(\frac{nRT}{P}\right)[/itex] = [itex]\frac{nR}{P}[/itex]

and then doing likewise to find the other terms.

But this confuses me because I have treated P as a constant like n and R while it is actually a function of T and V. If I make it a function of T and V I get stuck and anyway the thing gets a lot more complicated than the exercise intends. So how can I just treat P as a constant when I know its not?

Thanks for any help, Peter
 
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Answers and Replies

  • #2
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When you take a partial derivative of a function of several variables, everything except the variable you are taking the derivative of are considered a constant. All I think the exercise is doing is asking you to take the three partials and multiply them together recognizing that PV in the denominator equals nRT so they cancel leaving -1.
 

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