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I would like to know how to calculate the "internal energy" of an object. What I mean by internal energy is something like this:

  • 2kg of carbon has a higher "internal energy" than 1kg of carbon.
  • 1kg of carbon at 600 kelvin has a higher "internal energy" than 1kg of carbon at 599 kelvin.
  • 1kg of carbon at 600 kelvin has a higher "internal energy" than 1kg of hydrogen at 600 kelvin (because the energy stored in a monoatomic gas would have less degrees of freedom and therefore will be stored just as kinetic energy rather than other forms).

Is what I'm looking for just the Heat capacity?

Is there any way to calculate (not measure empirically) this property?

Edit:

Maybe it's not heat capacity because it turns out it can be negative!

But I think that has more to do with how heat capacity is defined, rather than what I mean by "internal energy", hope is not too confusing.

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  • $\begingroup$ Not in the sense of obtaining an absolute value. This isn't possible for energy, which can be defined only relative to a reference value. Yes, the (constant-volume) heat capacity mediates the relationship between internal energy and temperature: $\partial U/\partial T=C_V$. Once you choose your reference state (e.g., reference temperature), you can obtain $U$ by integrating: $U=U_0+\int C_V\,dT$. $\endgroup$
    – Chemomechanics
    Aug 6 at 16:54
  • $\begingroup$ I assumed that heat capacity increased with increasing temperature. Is this notion correct? In your equation you would still need to obtain the heat capacity, and as far as I'm aware, the only way to do that is to measure it directly. Is it not enough to know the temperature, volume, mass and composition of the object? $\endgroup$
    – gabriel
    Aug 6 at 17:12
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    $\begingroup$ The heat capacity (and its temperature dependence) can be theoretically estimated using models; see the Debye model, for example. $\endgroup$
    – Chemomechanics
    Aug 6 at 17:19

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What you want to calculate is precisely the internal energy of the object. It can be done as soon as you have a reasonable model/theory for the interaction between the relevant microscopic degrees of freedom.

It is routinely done with numerical computer simulations (Monte Carlo or Molecular Dynamics) methods. The internal energy corresponds to the average value of the Hamiltonian in the appropriate ensemble.

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