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U(internal energy)=nCvdT. why

  1. Dec 20, 2011 #1
    Internal of a gas is nCvdT. why it is nCvdT.do you know any simple proof or derivation for this.
    Last edited: Dec 20, 2011
  2. jcsd
  3. Dec 21, 2011 #2
    The heat capacity at constant volume is defined as the rate at which the heat changes with respect to temperature per mole. So for an infinitesimal change we can write.
    ##C_{v}=\frac{dQ}{dT} ##

    In an isovolumetric process no work is done so dU=δQ as per the 1st law of Thermodynamics
    so we can write the equation as

    ##U=nC_{v}dT =n \frac{dQ}{dT}_{v}dT =n \frac{dU}{dT}_{v}dT ##
    Last edited: Dec 21, 2011
  4. Dec 21, 2011 #3
    Cv is the molar heat capacity of a gas at constant volume and is defined as 'The heat energy required to warm 1 mole of a gas through one degree when its volume is kept constant'
    Gases have 2 principal heat capacities. If the gas is kept at constant pressure then Cp is the molar heat capacity for gas at constant pressure.
    When heat is supplied to a gas at constant volume no external work is done therefore all of the heat energy shows as a temperature change.
    When heat is supplied to a gas at constant pressure some external work is done [P(V2 - V1)]
    So for a temperature rise of 1 degree extra heat energy is required to provide the external work. This essentially means that Cp is greater than Cv and it can be shown that
    Cp - Cv = R (the gas constant)
  5. Dec 22, 2011 #4
    great, but there is dent in this, that is if process is not isobaric(isovolumetric) then?????????
    friend you seem to tell me that Cp-Cv=R. that is not what am i asking.

    after all thanks to both guys,

    I think i a just a beginner in thermodynamics. so proof of all these formula are out of my scope, I hope i will learn this formula in future.
  6. Dec 22, 2011 #5
    1st thermodynamic law:

    δQ= dU + δW.
    Supposing we have the general form of U=U(T,V)
    then its differential:
    dU=[itex]\frac{\partial U(T,V)}{\partial T}[/itex] dT + [itex]\frac{\partial U(T,V)}{\partial V}[/itex] dV

    and the work is δW=pdV

    we go to the 1st law and replace δW and dU by the quantities we have above. We get:

    δQ=[itex]\frac{\partial U(T,V)}{\partial T}[/itex] dT + [[itex]\frac{\partial U(T,V)}{\partial V}[/itex] + p ] dV

    in case of dV=0 (V:const) you get
    [itex]\frac{\partial U(T,V)}{\partial T}[/itex] =δQ/dT [itex]\equiv[/itex] Cv

    From that you totally see that:

    U= n Cv dT (for n moles now)

    of course that is for Cv constant, which of course is true for ideal gases.
    Last edited: Dec 22, 2011
  7. Dec 22, 2011 #6
    Your original question was :
    'Internal of a gas is nCvdT. why it is nCvdT.do you know any simple proof or derivation for this.'
    The answer is :At constant volume no external work is done by (or on) the gas. Therefore the heat supplied = increase in internal energy.
    To calculate the effect of heat supplied you nedd an 'SHC' equation
    In general Heat energy = mass x SHC x temp change.
    For a gas H = n x Cv x ΔT (n = number of moles rather than mass and C = molar heat capacity rather than specific heat capacity.... specific means 'per kg')

    If you need to know the equation for when the pressure is kept constant you need a different C..... Cp. If you need something in between then you need something other than the principal Cv and Cp
  8. Dec 22, 2011 #7

    I like Serena

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    Homework Helper

    The general equation for dU in terms of temperature and volume is:
    [tex]dU=n C_V dT + n \left[ T \left({\partial P \over \partial T}\right)_V - P \right] dV[/tex]
    (See wikipedia)
    This result can be derived from the general formula dU=TdS-PdV, which is in terms of entropy and volume.
    I'll leave the proof for that out (for now).

    If you substitute the ideal gas law [itex]P={nRT \over V}[/itex], the requested result follows.
    Last edited: Dec 22, 2011
  9. Dec 22, 2011 #8
    In words:
    Heat energy supplied = heat energy to raise temperature + heat energy converted to external work.(basically P x ΔV)
    If there is no external work (constant vol) then
    Heat energy = heat energy to raise temp
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