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Heat required to increase the temperature

  1. Jul 27, 2017 #1

    gj2

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    1. The problem statement, all variables and given/known data

    At high temperatures the nitrogen molecule behaves like a one-dimensional harmonic oscillator. In this situation, estimate how much heat must be added to the system in order to increase the temperature of 1 mole of nitrogen gas by 10 degrees Celsius (for constant volume and constant pressure respectively). Take into account all degrees of freedom: translational, rotational and vibrational.

    2. Relevant equations
    Average energy per degree of freedom: ##kT/2##

    3. The attempt at a solution
    A one-dimensional harmonic oscillator has two degrees of freedom, therefore according to the equipartition theorem the average energy of a nitrogen molecule must be ##kT##. One mole of nitrogen has ##N_A## molecules and so the total internal energy of the gas is ##U=N_A k T##. Therefore if the process is isochoric the amount of heat we need to add to the system in order to increase the temperature by 10 degrees is
    $$dQ=N_A k dT=N_A k \cdot 10\text{K} \approx 19.8 \text{cal}$$
    However the correct answer is ##70 \text{cal}## and I don't understand why.
    Also, I have no idea how to find the heat for the case of an isobaric process since this is not an ideal gas (the answer for constant pressure case is ##90 \text{cal}##).
     
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  3. Jul 27, 2017 #2

    Orodruin

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    First of all, a nitrogen molecule is not a one dimensional harmonic oscillator. You are missing a lot of degrees of freedom by this assumption.
     
  4. Jul 27, 2017 #3

    gj2

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    But that's how the question was formulated. It's not my assumption.
     
  5. Jul 27, 2017 #4

    Orodruin

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    It is an oscillator that in addition can translate and rotate. Only the vibrational spectrum is described by an actual harmonic oscillator.
     
  6. Jul 27, 2017 #5

    gj2

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    Oh I see. So the translational motion contributes 3 degrees of freedom whereas the rotational contributes 2. Therefore there are 7 degrees of freedom and the heat required is approximately 70cal, as expected.
    But what about the constant pressure case? How can I approach this problem?
    Thank you.
    Edit: nevermind. I think the reason why it's ##90 \text{cal}## for constant pressure case is because ##C_p-C_v=2 \frac{\text{cal}}{\text{mol}\,\text{K}}## for nitrogen (apparently it does behave approximately as an ideal gas). So it requires additional $$\left(2 \,\frac{\text{cal}}{\text{mol}\,\text{K}} \right )\cdot (10 \,\text{K})=20 \,\frac{\text{cal}}{\text{mol}}$$ heat for constant pressure case.
     
    Last edited: Jul 27, 2017
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