Partial derivatives with dependent variables (fixed) question.

[Curl]

In statistical mechanics we express partial derivatives of functions, keeping some variables fixed. But these variables are functions of the other variables (which are not fixed).

I'm just confused by this, what is the convention for taking these derivatives? For example, if we have S as a function of T, V and P, or S=S(T,V,P)

we want to find partial of S with respect to T, holding P and V constant. Put P and V are functions of T, i.e. P*V=constant*T. How does this work?

If I have, say, S=log(T*V*P), then is the partial derivative mentioned above just equal to (ds/dt)V,P = 1/T ?

What if I write it as S=log(T*(constant*T)) using the relation P*V=constant*T. Shouldn't the derivative be the same thing? Why is there a contradiction?

Does anyone have any good way of explaining this or some good links? I tried to look through books/wikipedia/websites and found nothing.

Thanks.

 

[mathman]

You need to distinguish between partial derivative and total derivative when the"constant" parameters are functions of the "variable" parameter. That's what getting you into trouble in your example.

 

[Curl]

Okay, I think I know what you mean. But can you give me an example or point me to some examples elsewhere?

My old calc book does not cover this, I already looked.

 

[Edgardo]

As mathman already mentioned you have to know the difference between partial and total derivative. Have a look at this excellent video by David Metzler in which he gives an intuitive explanation with a bug on a hot plate.

 

[CellCoree]

I can understand your confusion with this concept. Let me try to explain it in a simple way.

In statistical mechanics, we often encounter situations where we need to find the rate of change of a function with respect to one of its independent variables while keeping the other independent variables fixed. This is known as a partial derivative.

Now, in the scenario you have described, we have a function S that is dependent on three variables - T, V, and P. Let's say we want to find the partial derivative of S with respect to T while keeping V and P constant. This means that we are only interested in how S changes with respect to T, while V and P remain fixed.

However, in this particular case, V and P are not truly independent variables. They are functions of T, as indicated by the relation P*V=constant*T. This means that as T changes, V and P also change accordingly to maintain the constant value of P*V.

So, when we take the partial derivative of S with respect to T, holding V and P constant, we are essentially looking at the effect of T on S while taking into account the indirect effect of T on V and P through their relationship.

Now, to answer your specific questions - if we have S=log(T*V*P), then the partial derivative with respect to T, holding V and P constant, is indeed 1/T. This is because the derivative of log(T) is 1/T, and the other terms (V and P) are treated as constants in this case.

If we write S=log(T*(constant*T)), the derivative would still be the same because we are essentially multiplying by a constant and taking the log of it, which does not change the derivative.

I hope this explanation helps to clarify your doubts. If you need further assistance, I would suggest consulting a textbook or an online resource specifically on partial derivatives in statistical mechanics. Good luck!