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

In summary, the formula for pressure in a cylinder with a constant temperature T is P = (nRT/(V - nb)) - ((an^2)/V^2). To find dP/dV, use the quotient rule to differentiate (nRT/(V - nb)) and (-an^2/V^2), resulting in the answer of -nRTV/(V-nb)^2 + (2an^2)/V^3.
  • #1
PrudensOptimus
641
0
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?
 
Mathematics news on Phys.org
  • #2
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:
  • #3
not only does the answer including T, it has a, n, in it too.
 
  • #4
Is the answer
-nRTV/(V-nb)2 + (2an2)/V3 ?
 
  • #5
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:
  • #6
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.
 
  • #7
Originally posted by KL Kam
Is the answer
-nRTV/(V-nb)2 + (2an2)/V3 ?

Yep how did you get that?
 
  • #8
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.
 
  • #9
awesome!:)
 
  • #10
Is that yoda guy smart or what? WOW
 

1. What does "Calculating Derivative of P with Respect to V for Constant T and R" mean?

This refers to finding the rate of change of pressure (P) with respect to volume (V), while keeping temperature (T) and gas constant (R) constant. It is a fundamental concept in thermodynamics and is used to analyze gas behavior.

2. Why is it important to calculate the derivative of P with respect to V for constant T and R?

This calculation allows us to understand how a gas will behave when changes are made to its volume while keeping temperature and gas constant. It also helps us determine the relationship between pressure and volume, which is described by the ideal gas law.

3. How is the derivative of P with respect to V for constant T and R calculated?

It is calculated using the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature. By taking the derivative of this equation with respect to V while keeping T and R constant, we can obtain the expression dP/dV = -nRT/V^2.

4. What does the value of the derivative of P with respect to V for constant T and R tell us?

This value tells us the slope of the pressure-volume curve for a given gas at a specific temperature and with a constant number of moles and gas constant. A positive value indicates that pressure increases as volume increases, while a negative value indicates the opposite.

5. How is the derivative of P with respect to V for constant T and R used in real-world applications?

This concept is used in various industries, such as in designing and understanding the behavior of engines, refrigeration systems, and other devices that use gases. It is also used in the study of atmospheric pressure and weather patterns.

Similar threads

Replies
3
Views
2K
Replies
1
Views
535
Replies
10
Views
580
  • Calculus and Beyond Homework Help
Replies
24
Views
673
  • Introductory Physics Homework Help
Replies
2
Views
775
  • Classical Physics
Replies
6
Views
731
  • Introductory Physics Homework Help
Replies
4
Views
507
  • Introductory Physics Homework Help
Replies
2
Views
664
Replies
3
Views
911
Replies
2
Views
504
Back
Top