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Calculating the ratio between heat capacities of a gas

  1. Mar 1, 2010 #1
    Hi, my task is to calculate the ratio (gamma) between the specific heats of a gas (Cp and Cv). The only information I have is a table of data for the pressures and volumes of the gas at different temperatures. I dont know if its monatomic, diatomic etc. (it's a later task to determine this).

    My attempt so far. I've been researching this for hours and I've discovered that the ratio I need to find is gamma = Cp / Cv. I've learned that Cp = Cv + R where R is the gas constant and that Cv = 3R/2 for a monatomic gas, 5R/2 for a diatomic gas and that Cp = 5R/2 for a monatomic gas, 7R/2 for a diatomic gas but have been unable to find where these numbers come from! I also found a useful formula that
    gamma = 1 + (R/Cv) so I only really need to find either Cv or Cp using my data to solve this question.

    Thanks for any help.
     
  2. jcsd
  3. Mar 1, 2010 #2
    What kind of transformation is the gas doing?

    If it is diabatic, then you can find [tex]\gamma[/tex] from [tex]PV^{\gamma}=K[/tex], and then compare that value with the theoretical values of [tex]\frac{C_p}{C_v}[/tex] and find out what kind of molecule the gas is made of.

    Here's an example for a monoatomic gas: It has 3 translational degrees of freedom, no rotation and no vibration, so [tex]N=3[/tex], and by the energy equipartition theorem you have a total energy of
    [tex]U=n(3+0+0)\frac{1}{2}RT[/tex]
    where n is the mole number.

    In order to calculate [tex]C_v[/tex] you can prove that
    [tex]C_v=\frac{1}{n}(\frac{dU}{dT})_{v=const}[/tex]
    and because
    [tex]C_p=C_v+R[/tex] you know the theoretical [tex]\gamma[/tex].
     
  4. Mar 1, 2010 #3
    After reading your reply I read through the question again and noticed that it mentions the the gas is undergoing adiabatic expansion in a later part of the question, I never read this far into the question before so I didn't realise. Would the relation you mentioned hold true for an adiabatic expansion? I'm going to do more research with this new found information!

    Thanks a lot!
     
  5. Mar 1, 2010 #4
    It holds for any adiabatic process,in other words, any process in wich there is no heat exchange between the system and the surrounding universe.
     
  6. Mar 3, 2010 #5
    OK so I had a go at using the method you told me and, I did:
    [tex]\frac{C_p}{C_v}[/tex] = [tex]\frac{C_v + R}{C_v}[/tex]
    = [tex]\frac{\frac{1}{n}(\frac{dU}{dT}) + R}{\frac{1}{n}(\frac{dU}{dT})}[/tex]

    And assumed (for now) that it is a monatomic gas to calculate a set of U values which I plotted against the T values and drew a trendline through these points to get the gradient. Now, I (wrongly) cancelled the top and bottom n to get

    [tex]\frac{C_p}{C_v}[/tex] = [tex]\frac{(\frac{dU}{dT}) + R}{(\frac{dU}{dT})}[/tex]

    and substituted the values in to get 1.66 which relates to H or He. However, as I stated I wrongly cancelled the ns I think, or am I allowed the cancel the ns? This means n must equal 1 BUT it's a later part of the question to calculate n which is why I'm panicing a little! Also I assumed monatomic gas, is there no way to do this for a general case because it's also a later part of the question to determine if its monatomic/diatomic etc. lol. I have all the answers but in the wrong order!
    Thanks a lot anyway, you've been a great help. Do you have any other ideas?
     
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