# Variation of specific heat with temperature

1. Apr 29, 2010

### chhitiz

how does specific heat of gases vary with temperature? is there an equation to describe it?

2. Apr 29, 2010

### mgb_phys

3. Apr 29, 2010

### chhitiz

To quote wiki "As the temperature approaches absolute zero, the specific heat capacity of a system also approaches zero."
so it does vary with temperature, right?

4. Apr 29, 2010

### mgb_phys

The laws of thermodynamics don't work very well at absolute zero.
There is a probably a theory of the specific heat capacity of a bose-einstein condensate at superfluid temperatures, but it's unlikely to be simple

5. Apr 30, 2010

### stewartcs

In Thermodynamics specific heats are defined as:

$$C_v = \Bigg(\frac{\partial{u}}{\partial{T}}\Bigg)_v$$

$$C_p = \Bigg(\frac{\partial{h}}{\partial{T}}\Bigg)_p$$

So yes, they are temperature dependent.

CS

6. Apr 30, 2010

### Mapes

(I must point out that the second statement doesn't follow from the definitions; if $U\propto T$, for example, then $c_V$ would be constant. And this is approximately the case for common gases at common temperatures, where $c_V$ is an essentially constant 3R/2 or 5R/2 for monatomic and diatomic gases, respectively.)

EDIT: Whoops, meant to say "common temperatures."

Last edited: Apr 30, 2010
7. Apr 30, 2010

### Topher925

This doesn't really state that heat capacity is dependent upon temperature though. It actually states that internal energy or enthalpy is dependent upon temperature, not heat capacity.

Theoretically, the heat capacity of a gas should not change with temperature and should only depend on its molecular composition. In reality, this isn't the case and heat capacities must be determined experimentally at different temperatures. You won't find a single use-all equation for heat capacity of different gases. Instead, empirical correlations are used which are determined using experimental data and curve fits.

For example, the heat capacity of air with change in temperature: http://www.engineeringtoolbox.com/air-properties-d_156.html

For simple analysis at low temperatures, deviations in heat capacity won't make a big difference. But in applications such as combustion, changes in heat capacity can be very significant.

8. Apr 30, 2010

### stewartcs

If $U\propto T$ then $c_V$ isn't necessarily constant (although by those Classical Thermodynamic equations alone it might appear that way). It still depends on the amount of kinetic energy added (and thus the temperature). For a diatomic gas at a certain range of temperatures the only mode excited is the translational mode. As the temperature increases and more KE is added, the rotational mode is excited (and the KE will now be stored there as well). If the temperature is increase yet again and even more KE is added to the gas, the vibrational mode will be excited and the KE stored there as well.

The result is that the commonly quoted value of 5R/2 for a diatomic molecule is only for the translational mode. The other modes are essentially dormant or frozen out until enough KE has been added to excite them.

The value becomes 7R/2 when the temperature increases enough to excite the rotational mode, and then 9R/2 when the temperature increases enough to excite the vibrational mode.

See the attached for a better visual.

CS

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• ###### Specific heats.JPG
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9. Apr 30, 2010

### stewartcs

I'm not sure what you mean. A simple substance's internal energy is a function of temperature and specific volume u = (T,v)

The total differential of u is then:

$$du = \Bigg(\frac{\partial{u}}{\partial{T}}\Bigg)_v dT + \Bigg(\frac{\partial{u}}{\partial{v}}\Bigg)_T dv$$

or using the previous definition of Cv we can right it like this

$$du = C_v dT + \Bigg(\frac{\partial{u}}{\partial{v}}\Bigg)_T dv$$

Which clearly shows that the internal energy is in fact a function of the specific heat capacity.

Depends on what theory you are referring to. In Classical Thermodynamics this would be true. However, when we start talking about Statistical Thermodynamics it can be shown that this is not true. Hence my comments above to Mapes about the different modes.

Of course in reality it is obvious from experimentation that the specific heat capacities are indeed temperature dependent. So regardless of any theory that can explain why this happens, it still happens and should be accounted for during some analyses.

CS

10. Apr 30, 2010

### mgb_phys

I thought for an ideal gas the heat capacity was just 3R/2 (for monotonic) or 7R/2 for (diatomic) ?

11. Apr 30, 2010

### Mapes

Agreed, but please note what temperature these transitions typically occur: for nitrogen between ~100-2000K, for example, the translational and rotational modes (but not the vibrational mode) are excited, the energy $U\propto T$ (specifically, $U\approx 5NRT/2$), and the constant-volume specific heat is an approximately constant 5R/2. That's what I meant when I wrote "common gases at common [temperatures]."

12. Apr 30, 2010

### stewartcs

Very true. So for most practical purposes (at least the ones I deal with) it is certainly safe in most cases to assume it's constant.

CS