## Physics Friday 73

For an ideal gas of N molecules, each with f total available degrees of freedom for each molecule (f≥3), we have equation of state  and internal energy , as discussed here and here. Now, we can use this to find the heat capacity for a given quantity of an ideal gas. The heat capacity is defined as the ratio of a small amount of heat (energy) added to the body to the resulting small temperature increase, with the system under some constraint (to define a particular path through the phase space of the system):

Now, recalling that we can define temperature via , we can define heat capacity as
.
For gases, the two conditions we usually consider are constant volume, CV, and constant pressure, CP.

For a closed system, the internal energy has increment  (see here). Thus, we see that for constant volume (dV=0), , and so .
Now, as our internal energy is , we see that 
With enthalpy defined as H=U+PV, it has increment , which can combine with the internal energy increment equation above to get , and so our heat capacity at constant pressure is .
Using our ideal gas law, , so , and thus .

The specific heat capacity is the heat capacity for a fixed quantity of substance, usually either a unit mass or unit mole. Recalling that for n moles of gas, Nk=nR, we have , so for the molar specific heats, we divide the above by n to get .

Next, we can also note that the heat capacity has the same units as Boltzmann’s constant k, and as it scales with quantity, it scales with the number of molecules. Thus, we can define a dimensionless specific heat capacity:
. The above difference relation between heat capacities for constant pressure and constant volume then becomes ĉPĉV=1, and in terms of f, we see  and .

Recalling the previous discussion of the internal degrees of freedom, we note that for a real gas, f is dependent on temperature: as T increases past various threshold temperatures, more degrees of freedom “unfreeze” and f increases. Thus actual specific heat capacities also depend on temperature, usually increasing with rising temperatures.

### 6 Responses to “Physics Friday 73”

1. Cederash Says:

Оценка 5, базару ноль

2. Ferinannnd Says:

Кстати, по радио программа была об этом. Не помню, правда на какой волне…

3. Physics Friday 74 « Twisted One 151’s Weblog Says:

[…] Friday 74 By twistedone151 Previously, I discussed the heat capacity of an ideal gas, and the relationship to the molecular degrees of […]

4. Physics Friday 80 « Twisted One 151’s Weblog Says:

[…] both possibly functions of particle number N). For the first term on the right, we recall our discussion of heat capacity, and that . This means our entropy expression becomes . For the latter, we use one of the Maxwell […]

5. Physics Friday 81 « Twisted One 151’s Weblog Says:

[…] constant volume, and f is the number of available degrees of freedom for a molecule of the gas (see here). We previously found that (for high enough temperatures) the entropy of an ideal gas can be […]

6. Physics Friday 93 « Twisted One 151’s Weblog Says:

[…] the energy. For T>Tc, we have as before, . For T<Tc, we have . From this, we find the heat capacity at constant volume by . Below the condensation temperature, we take the derivative of the above, to […]