1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculation of work (P.dV) in an isothermal system

  1. Oct 25, 2012 #1
    Here's the textbook way of calculating the work done by an ideal gas in an isothermal case.

    PV=nRT
    P.dV=(nRT/V).dV
    ∴ ∫P.dV=nRT∫dV/V
    → W2-W1=nRT*ln(V2/V1)

    My question.
    Consider a cylindrical (or of any other shape) container of surface area A and a frictionless movable piston attached.
    Let the coordinate x run along the length of this container, with origin at the bottom of the container.
    Let the piston be at some x at some point of time.

    At this x, P.V= constant = nRT
    This holds for all x.
    Now since V(x)=A.x
    P(x)=constant/A.x=K/x .... for some contant k (=nRT/A)

    Now P and V are both functions of x.
    While calculating the infinitesimal work P.dV, how can we treat P(x) constant over the range dx?

    Instead, since P(x)V(x)=constant
    differentiating both sides, d[P(x)V(x)]=0
    → V(x).dP + P(x).dV=0
    ∴ P(x).dV= -V(x).dP

    Shouldn't this relation hold?

    I ask this because I was solving a similar problem, although which wasn't isothermal, involved P and V that depended on x and the approach used there was the second one.
     
  2. jcsd
  3. Oct 25, 2012 #2
    The relationship holds and if you integrate -VdP instead of pdV you get the same result, don't you?
    I am not sure what is the question...
     
  4. Oct 28, 2012 #3
    Usually, in calculus, functions of x are taken as being constant along any differential displacement dx.
     
  5. Oct 28, 2012 #4
    Any rigorous mathematical explanation you could point me to? I know that if dx is infinitesimally small, f(x) will not change much in dx, but it will change neverthless. So i am looking for a rigorous mathematical explanation of why we can treat it as constant.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Calculation of work (P.dV) in an isothermal system
  1. Work calculations (Replies: 6)

Loading...