Variation of specific heat with temperature

[chhitiz]

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

 

[mgb_phys]

It doesn't - the heat capacity just depends on the degrees of freedom of bonds/atoms of the gas.
http://en.wikipedia.org/wiki/Specific_heat_capacity

(not necessarily true in very extreme cases like the center of gas giant or near absolute zero)

 

[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?

 

[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

 

[stewartcs]

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

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

 

[Mapes]

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

(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."

 

[Topher925]

In Thermodynamics specific heats are defined as:

So yes, they are temperature dependent.

CS

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.

 

[stewartcs]

(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 pressures, where $c_V$ is an essentially constant 3R/2 or 5R/2 for monatomic and diatomic gases, respectively.)

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

 

[stewartcs]

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.

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.

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.

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

 

[mgb_phys]

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

 

[Mapes]

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.

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]."

 

[stewartcs]

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]."

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