# A text problem

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?

## Answers and Replies

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.

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

KLscilevothma
-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

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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
-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)] - an2* d/dV (1/V2) ......(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

the green part
d/dV (1/V2)
= -2V-3

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!:)

Is that yoda guy smart or what? WOW