Heat capacities and negative temperature

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Discussion Overview

The discussion revolves around the relationship between heat capacities \(C_P\) and \(C_V\) in thermodynamic systems, particularly in the context of negative temperatures. Participants explore the implications of negative temperature on the established inequality \(C_P > C_V\) and the conditions under which this holds true.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant states that the mechanical stability of a system implies \(C_P > C_V\) under normal conditions, questioning if this holds true when temperature is negative.
  • Another participant asserts that absolute temperature cannot be negative and that \(C_P\) is always greater than \(C_V\).
  • Contrarily, some participants argue that quantum systems can exhibit negative temperatures, citing examples such as systems with two energy levels.
  • A participant mentions that the definition of temperature in statistical mechanics allows for the possibility of negative temperatures.
  • One participant provides a link to a description of negative temperature, noting that lasing systems can have negative temperatures, but they are not in thermal equilibrium with gases.
  • Another participant challenges the applicability of negative temperature concepts to gases, suggesting that such systems may be unphysical.

Areas of Agreement / Disagreement

Participants express disagreement regarding the existence and implications of negative temperatures, with some asserting that they are not possible while others provide examples and argue for their validity. The discussion remains unresolved on the relationship between heat capacities in the context of negative temperatures.

Contextual Notes

There are varying definitions of temperature that influence the discussion, particularly in statistical mechanics. The implications of negative temperature on thermodynamic properties are not fully explored, and the conditions under which these concepts apply remain unclear.

Einj
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Hi everybody,
I have the following doubt. We know that for a thermodynamic system the following equality holds:
$$
C_P-C_V=-T\frac{\left[\left(\frac{\partial P}{\partial T}\right)_V\right]^2}{\left(\frac{\partial P}{\partial V}\right)_T}
$$

Now, the mechanical stability of the system requires that the volume decreases with increasing pressure, i.e. (\frac{\partial V}{\partial P})_T<0. So this seems to lead to C_P>C_V. Is that always true? What happen if the temperature is negative, T<0?

Thanks
 
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The absolute temperature can't be negative, and Cp is always greater than Cv.

chet
 
I am not really sure about that. There are quantum systems with negative temperature. For example a system with just two energy levels has negative temperature.
 
Einj said:
I am not really sure about that. There are quantum systems with negative temperature. For example a system with just two energy levels has negative temperature.

I never heard of that, but I don't know much about qm. When I studied qm, I did not encountered the concept of negative absolute temperature.
 
It depends on the definition. When you work with statistical mechanics the temperature is defined through its relations with entropy, free energy and so on. And it turns out that from this relations it can also be negative.
 
Yes, but systems with negative temperature are not ever in thermal equilibrium with gasses, and since you're talking about gasses, you are describing an unphysical system.
 

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