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

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Discussion Overview

The discussion revolves around calculating the derivative of pressure P with respect to volume V for a gas in a cylinder, under the condition of constant temperature T. Participants explore the implications of constants and variables in the given formula and the differentiation process involved.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents the formula for pressure P and seeks help in finding dP/dV, indicating that only V and T are variables.
  • Another participant clarifies that T is maintained as a constant, suggesting that dT/dV should not appear in the answer.
  • A different participant argues that if T were a variable, dT/dV would be expected in the answer, referencing the chain rule.
  • Some participants discuss the need to apply the quotient rule for differentiation, particularly for the term involving nRT/(V - nb).
  • One participant proposes a potential answer for dP/dV, prompting others to confirm or challenge the correctness of this result.
  • Another participant provides a detailed breakdown of the differentiation process, including the application of the power rule and quotient rule.

Areas of Agreement / Disagreement

There is disagreement regarding the treatment of T as a constant versus a variable, with some participants asserting T is constant and others suggesting it could be variable under certain interpretations. The discussion on the correct application of differentiation techniques also reflects differing viewpoints.

Contextual Notes

Participants express uncertainty about the inclusion of certain variables in the derivative, particularly regarding the constants a, n, and R, and the implications of T being constant or variable. The differentiation steps also reveal potential missing assumptions about the application of rules.

Who May Find This Useful

This discussion may be useful for students or individuals interested in thermodynamics, calculus, and the application of differentiation in physical contexts, particularly in gas laws.

PrudensOptimus
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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?
 
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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.
 
Last edited:
not only does the answer including T, it has a, n, in it too.
 
Is the answer
-nRTV/(V-nb)2 + (2an2)/V3 ?
 
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
 
Last edited:
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.
 
Originally posted by KL Kam
Is the answer
-nRTV/(V-nb)2 + (2an2)/V3 ?

Yep how did you get that?
 
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.
 
awesome!:)
 
  • #10
Is that yoda guy smart or what? WOW
 

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