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    This is the question:

    A mineral has a heat capacity Cp which varies in the temperature range 0-2000K according to Cp = 4.0 + 2x10-3T + 3x105T-2. What is the 3rd Law Entropy at 300K?

    I am genuinely not really sure how to go about doing this. Since Cp(T) = dQ/dT, dQ = Cp(T)dT. dS = dQ/T so dS = Cp(T)/TdT. So dS = 4.0T-1 + 2x10-3 + 3x105T-3dT. I then thought you'd integrate that between 0K (where S = 0) and 300K but that's not impossible... and to be honest this entire process feels like guesswork anyway.

    Could someone give me some pointers? Thanks!
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    Sorry you've not had any responses about this.

    Why not try posting in a specific subject forum- you might have more luck there.

    Here's a link to our subject forum which should help get you more responses.

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    (Original post by somemightsay888)
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    (Original post by Plagioclase)
    This is the question:

    A mineral has a heat capacity Cp which varies in the temperature range 0-2000K according to Cp = 4.0 + 2x10-3T + 3x105T-2. What is the 3rd Law Entropy at 300K?

    I am genuinely not really sure how to go about doing this. Since Cp(T) = dQ/dT, dQ = Cp(T)dT. dS = dQ/T so dS = Cp(T)/TdT. So dS = 4.0T-1 + 2x10-3 + 3x105T-3dT. I then thought you'd integrate that between 0K (where S = 0) and 300K but that's not impossible... and to be honest this entire process feels like guesswork anyway.

    Could someone give me some pointers? Thanks!
    What you've written looks fine to me. I dont understand why you say "that's not impossible" though. No, it's not impossible - it's quite possible, and correct AFAICS.
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    (Original post by atsruser)
    What you've written looks fine to me. I dont understand why you say "that's not impossible" though. No, it's not impossible - it's quite possible, and correct AFAICS.
    Unless I've forgotten how to integrate, aren't you going to end up dividing by zero?
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    (Original post by Plagioclase)
    Unless I've forgotten how to integrate, aren't you going to end up dividing by zero?
    Sorry. You're right - I didn't think about the limits. Yes, that looks a bit odd. However, your approach looks fine for calculating the entropy *change* from one temperature to another, and I assume that this question wants the absolute entropy, which is usually defined as the entropy change from 0 K, I think.

    Have you been given the correct C_p? The constant term is going to cause problems here, as it will always give you a log term. Or maybe there's some engineering hack that I'm unaware of, to allow us to ignore the low temperatures, where the entropy contribution will be small anyway.

    [edit: this page talks about this problem, but I don't think that it helps you, due to the form of C_p:

    http://cbc.arizona.edu/~salzmanr/480...aw/3rdlaw.html ]
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    (Original post by atsruser)
    Sorry. You're right - I didn't think about the limits. Yes, that looks a bit odd. However, your approach looks fine for calculating the entropy *change* from one temperature to another, and I assume that this question wants the absolute entropy, which is usually defined as the entropy change from 0 K, I think.

    Have you been given the correct C_p? The constant term is going to cause problems here, as it will always give you a log term. Or maybe there's some engineering hack that I'm unaware of, to allow us to ignore the low temperatures, where the entropy contribution will be small anyway.

    [edit: this page talks about this problem, but I don't think that it helps you, due to the form of C_p:

    http://cbc.arizona.edu/~salzmanr/480...aw/3rdlaw.html ]
    I'm pretty sure that I've read the question correctly, maybe there was a typo since according to the question Cp approaches infinity as T approaches zero whereas according to the link you've sent, it should approach zero. Thanks anyway, I appreciate the help
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    (Original post by Plagioclase)
    I'm pretty sure that I've read the question correctly, maybe there was a typo since according to the question Cp approaches infinity as T approaches zero whereas according to the link you've sent, it should approach zero. Thanks anyway, I appreciate the help
    Experimentally C_p \propto T^3 at low temperatures for most materials (and theoretically, too using Debye's model) so T \to 0 \Rightarrow C_p \to 0 as you would require to do the integral.

    So my feeling is that your lecturer has screwed up, assuming that s/he wanted you to calculate the absolute entropy - the quoted formula is very unphysical, and the calculation won't work with what you've been given.
 
 
 
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