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, *C*_{V}, and constant pressure, *C*_{P}.

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 *ĉ*_{P}–*ĉ*_{V}=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.

### Like this:

Like Loading...

*Related*

Tags: Enthalpy, Entropy, Friday Physics, Heat Capacity, Ideal Gas, Ideal Gas Law, Internal Energy, physics, Specific Heat Capacity, Statistical Mechanics, Thermodynamics

This entry was posted on May 22, 2009 at 12:17 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

May 23, 2009 at 1:40 pm |

Оценка 5, базару ноль

May 24, 2009 at 10:46 am |

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

May 29, 2009 at 12:21 am |

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

July 10, 2009 at 12:06 am |

[…] 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 […]

July 17, 2009 at 12:42 am |

[…] 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 […]

October 16, 2009 at 12:14 am |

[…] 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 […]