How Do I Take Partial Derivatives of the Dieterici Equation of State?

[cashmerelc]

I have to find the expansivity of a substance obeying the Dieterici equation of state using the cyclical relation.

I understand what I need to do, but I'm having a problem with the partial derivatives of the equation of state. I was wondering if anyone could refresh me on how to take a partial derivative of this equation:

P=(RT/v-b)e^(-a/RTv)

Taking the partial derivative of P with respect to T is simply the product rule, correct?

But when I have to take the derivative of T wrt v, or the der. of v wrt P, I have to solve the equation for T and v respectively, right? That is where I'm having the problem, with the variables in the exponent of e.

Any help someone could give me would be greatly appreciated.

 

[malawi_glenn]

this is in wrong section, someone will move this thread.

also show the work you have done, so we can see if you have done right and what it is that you don't understand.

 

[cashmerelc]

Well, that's what I'm having trouble with.

I have to take dT/dv keeping P constant.

So I need to get all of the Ts on one side of the equation. The problem I'm having deals with the T in the exponent. I know to solve for a variable in an exponent of e, you take the ln of both sides. But in this problem, there is a T not in the exponent also, so you'd end up with a T and a ln |T| on one side. So there's got to be another way to do it unless I'm missing something obvious.

And sorry about posting in the wrong thread.

 

[malawi_glenn]

Well, that's what I'm having trouble with.

I have to take dT/dv keeping P constant.

So I need to get all of the Ts on one side of the equation. The problem I'm having deals with the T in the exponent. I know to solve for a variable in an exponent of e, you take the ln of both sides. But in this problem, there is a T not in the exponent also, so you'd end up with a T and a ln |T| on one side. So there's got to be another way to do it unless I'm missing something obvious.

And sorry about posting in the wrong thread.

Can you write the equation and how you obtained it?

The rules of this forum is that you show work done before getting help.

 

[HallsofIvy]

You don't have to solve for T or v. You could just use "implicit differentiation".

 

[cashmerelc]

This is the original equation, which is given:

P=(RT/v-b)e^(-a/RTv)

I have to find the partial derivative of T with respect to v. This is what I'm saying my problem is. I am trying to separate T out so I can write the equation as a function of T.

Here is what I have...

T= (v-b)P/Re^(-a/RTv)

So there is still a T on both sides of the equation.. So even if I get them on the same side, I have this:

Te^(-a/RTv)=(v-b)P/R

I can't figure out how to get the T out of the exponent without messing up the other T.

 

[cashmerelc]

/facepalm

I always forget implicit differentiation. Thanks a million, HallsofIvy!