Results 1 to 7 of 7

Math Help - [SOLVED] Proof using calculus?

  1. #1
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539

    [SOLVED] Proof using calculus?

    Bulk modulus of a material is given by B = -dP/(dV/V). Prove that Bulk modulus for adiabatic process = γP. (For adiabatic process PV^γ = K)

    Can anybody explains all the steps?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Hi
    Quote Originally Posted by fardeen_gen View Post
    Bulk modulus of a material is given by B = -dP/(dV/V). Prove that Bulk modulus for adiabatic process = γP. (For adiabatic process PV^γ = K)

    Can anybody explains all the steps?
    The hint is to work with \ln \left(PV^{\gamma}\right) :


    <br />
\ln \left(PV^{\gamma}\right)=\ln K \Longleftrightarrow \ln P+\gamma\ln V=\ln K

    Let's compute the differential of \ln P+\gamma\ln V :

    <br />
\begin{aligned}<br />
\mathrm{d} \left(\ln P+\gamma\ln V\right) &= \frac{\partial}{\partial P}\left(\ln P+\gamma\ln V\right)\mathrm{d}P  + \frac{\partial}{\partial V}\left(\ln P+\gamma\ln V\right)\mathrm{d}V \\<br />
&= \left(\frac{\partial \ln P}{\partial P}+\gamma\frac{\partial \ln V}{\partial P}\right)\mathrm{d}P  +\left( \frac{\partial \ln P}{\partial V}+\gamma  \frac{\partial \ln V}{\partial V}\right)\mathrm{d}V \\<br />
\end{aligned}<br /> <br />

    Since \frac{\partial \ln V}{\partial P}=\frac{\partial \ln P}{\partial V}=0 and \mathrm{d}\ln u=\frac{\mathrm{d}u}{u}

    <br />
\begin{aligned}<br />
\mathrm{d} \left(\ln P+\gamma\ln V\right)&=\frac{\mathrm{d}\ln P}{\mathrm{d}P}\mathrm{d}P+\gamma\frac{\mathrm{d}\  ln V}{\mathrm{d}V}\mathrm{d}V\\<br />
&=\frac{\mathrm{d}P}{P}+\gamma\frac{\mathrm{d}V}{V  }\\<br />
\end{aligned}<br />

    hence \frac{\mathrm{d}P}{P}=-\gamma\frac{\mathrm{d}V}{V}. Can you conclude ?
    Last edited by flyingsquirrel; June 29th 2008 at 07:41 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539
    I cant. I started with calculus just two days ago after finishing trigo two days ago. Pls conclude.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539
    Is d ln u = ln u / u a property of derivatives?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539
    I couldnt understand ur second step at all! Please explain!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member fardeen_gen's Avatar
    Joined
    Jun 2008
    Posts
    539
    How is d ln K = zero?

    P.S. I dont seem to know a thing. Sigh...
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    I've edited my previous post.

    Quote Originally Posted by fardeen_gen View Post
    I cant. I started with calculus just two days ago after finishing trigo two days ago. Pls conclude.
    You don't know how to do a division ?

    Quote Originally Posted by fardeen_gen View Post
    Is d ln u = ln u / u a property of derivatives?
    Yes. For t\mapsto u(t), \mathrm{d}\ln u is defined by \mathrm{d}\ln u=\frac{\mathrm{d}(\ln u)}{\mathrm{d}t}\cdot \mathrm{d}t and as \frac{\mathrm{d}(\ln u)}{\mathrm{d}t}=\frac{1}{u}\cdot \frac{\mathrm{d}u}{\mathrm{d}t} you get the expected equality. (Note that in this exercise we're in the particular case where u(t)=t : we're working with u(P)=P and u(V)=V )

    Quote Originally Posted by fardeen_gen View Post
    I couldnt understand ur second step at all! Please explain!
    See my previous post. (edited)

    Quote Originally Posted by fardeen_gen
    How is d ln K = zero?
    K is a constant which depends neither on V nor on P : \frac{\partial K}{\partial V}=\frac{\partial K}{\partial P}=0 hence

    <br />
\mathrm{d}\ln K = \frac{\partial K}{\partial P}\mathrm{d}P+\frac{\partial K}{\partial V}\mathrm{d}V=0\cdot \mathrm{d}P+0\cdot\mathrm{d}V=0
    Last edited by flyingsquirrel; June 29th 2008 at 08:20 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] direct proof and proof by contradiction
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 27th 2010, 11:07 PM
  2. [SOLVED] Calculus Help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 4th 2010, 03:27 PM
  3. [SOLVED] Calculus 1
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 16th 2006, 04:20 PM
  4. [SOLVED] [SOLVED] MV Calculus Help
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 30th 2005, 12:08 PM
  5. [SOLVED] [SOLVED] Calculus help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 29th 2005, 02:11 PM

Search Tags


/mathhelpforum @mathhelpforum