What is the implicit differentiation of the van der Waals equation?

[ABearon]

Summary:: van der waals

I have a pretty good understanding of implicit differentiation. However I'm stuck on a homework problem and could use some help.

[P + (an^2)/V^2][V - nb] = nRT a,n,b,R are constants

My professor wants me to take the implicit differentiation of P wrt T. She then says solve for P then take the derivative to check your answer. I understand the chain rule and product rule. Could someone show the steps to solving using these rules. I'm used to having 3 intermediate variables as functions of t and seeing it this way is confusing me.

 

[Svein]

Do \frac{d}{dT} on both sides.

 

[ABearon]

Do \frac{d}{dT} on both sides.

I got {nR - [P+(an^2)/V^2][dV/dT]}/(V-nb) - d/dt[(an^2)/V^2]

 

[PeroK]

I got {nR - [P+(an^2)/V^2][dV/dT]}/(V-nb) - d/dt[(an^2)/V^2]

Using Latex might help:

https://www.physicsforums.com/help/latexhelp/

Although, to be honest, I have no idea what you are trying to do. How does differentiating an equation check the answer? Check it in what way?

 

[Mark44]

She then says solve for P then take the derivative to check your answer.

Although, to be honest, I have no idea what you are trying to do. How does differentiating an equation check the answer? Check it in what way?

I believe what is meant here is to do these operations:

1. Find the derivative dP/dT using implicit differentiation.
2. Solve for P, and then differentiate normally to find dP/dT.

Both techniques should result in the same derivative.

 

[Mark44]

@ABearon, you have posted several threads in sections other than the homework sections. In the future, please post homework questions in one of the forum sections under Homework & Coursework.

 

[PeroK]

I believe what is meant here is to do these operations:

1. Find the derivative dP/dT using implicit differentiation.
2. Solve for P, and then differentiate normally to find dP/dT.

Both techniques should result in the same derivative.

I guess the objective is to find $\frac{dP}{dT}$. That seems like an obvious thing to say in the OP.

 

[Mark44]

I guess the objective is to find $\frac{dP}{dT}$. That seems like an obvious thing to say in the OP.

From post #1:

My professor wants me to take the implicit differentiation of P wrt T.

A better way to say this probably would be "Find dP/dT using implicit differentiation."

 

[RPinPA]

My professor wants me to take the implicit differentiation of P wrt T. She then says solve for P then take the derivative to check your answer. I understand the chain rule and product rule. Could someone show the steps to solving using these rules. I'm used to having 3 intermediate variables as functions of t and seeing it this way is confusing me.

Wait. Are you differentiating with respect to temperature (T) or time (t)? Don't keep switching back and forth between them, choose one symbol. it's very confusing.

I got {nR - [P+(an^2)/V^2][dV/dT]}/(V-nb) - d/dt[(an^2)/V^2]

Adjusting the formatting to what I think that says based on the brackets, I think that line says this:
$$\frac{nR - \frac {P + an^2} {V^2} \frac {dV}{dT}}{V - nb} - \frac {d}{d?} \frac {an^2}{V^2}$$

Since V is apparently treated as a function of t or T, whatever you're differentiating with respect to, then you can apply the Chain Rule to that last term.

I'm not saying whether this is a correct differentiation or not, just what I think you wrote. Actually I can see one thing clearly wrong with it at a glance: Implicit differentiation of an equation should give you an equation.

 

[vanhees71]

I'm a bit puzzled by the question. In thermodynamics it's important to say which quantities are kept constant when calculating derivatives. For a gas you have three independent variables (e.g., $n$, $V$, and $T$).