Calculating Derivative of P with Respect to V for Constant T and R

[PrudensOptimus]

If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form

P = (nRT/(V - nb)) - ((an^2)/V^2)

in which a, b, n, and R are constants. Find dP/dV

I tried to solve it by knowing that a, b, n, and R are constants, so only V, T are variables.

So I did this:

P = nR(dT/d(V-nb)) - ((an^2)*(-2V^-3))

but I still didn't get the correct answer. I believe I did something wrong, could someone help me out?

 

[KLscilevothma]

The question says "If gas in a cylinder is maintained at a constant temperature T". So I don't think T is a variable. Does the answer contain somthing like dT/dV? I don't think so because T isn't a variable.

 

[PrudensOptimus]

not only does the answer including T, it has a, n, in it too.

 

[KLscilevothma]

Is the answer
-nRTV/(V-nb)² + (2an²)/V³ ?

 

[KLscilevothma]

Originally posted by PrudensOptimus
not only does the answer including T, it has a, n, in it too.

If T isn't a constant but a variable, I would expect (dT/dV) as part of the answer. (chain rule)

By the way, remember you need to use quotient rule when differentiate (nRT/(V - nb)) with respect to V as V is in the denominator

 

[plus]

Originally posted by KL Kam
If T isn't a constant but a variable, I would expect (dT/dV) as part of the answer. (chain rule)

By the way, remember you need to use quotient rule when differentiate (nRT/(V - nb)) with respect to V as V is in the denominator

dT/dV will not be in the answer, as T is assumed to be constant, so therefore does not depend upon V.

 

[PrudensOptimus]

Originally posted by KL Kam
Is the answer
-nRTV/(V-nb)² + (2an²)/V³ ?

Yep how did you get that?

 

[KLscilevothma]

T is constant in this question

dP/dV
=d/dV [nRT/(V - nb) - an²/V²]
=d/dV [(nRT/(V - nb)] - d/dV (an²/V²)
now take all the constants out to the left hand side of d/dV
=nRT*d/dV [1/(V-nb)][/color] - an²* d/dV (1/V²)[/color] ......(1)

The blue part:
[1/(V-nb)] = (V-nb)^-1
d/dV [1/(V-nb)] = -1*(V-nb)^-2 = - 1/(V-nb)²
(the power rule)

Alternately,
d/dV [1/(V-nb)]
= [(V-nb)d/dV (1) - 1*d/dV (V-nb)]/(V-nb)²
(the quotient rule)
= (0-1)/(V-nb)²
= - 1/(V-nb)² [/color]

the green part
d/dV (1/V²)
= -2V^-3[/color]
I think you can do it because you got it right in your first post

Substitute the blue part and green part back to (1), then you'll get the answer.

 

[PrudensOptimus]

awesome!:)

 

[heisenberg]

Is that yoda guy smart or what? WOW