Editing Heat capacity (section)

===Heat capacities of a homogeneous system undergoing different thermodynamic processes===

==== At constant pressure, ''δQ'' = ''dU'' + ''pdV'' ([[isobaric process]]) ====
At constant pressure, heat supplied to the system contributes to both the [[Work (thermodynamics)|work]] done and the change in [[internal energy]], according to the [[first law of thermodynamics]]. The heat capacity is called <math>C_p</math> and defined as:

<math display="block">C_p = \left.\frac{\delta Q}{dT}\right|_{p = \text{const}}</math>

From the [[first law of thermodynamics]] follows <math>\delta Q = dU + p\,dV </math> and the inner energy as a function of <math>p</math> and <math>T</math> is:

<math display="block">\delta Q = \left(\frac{\partial U}{\partial T}\right)_p dT + \left(\frac{\partial U}{\partial p}\right)_T dp + p\left[ \left(\frac{\partial V}{\partial T}\right)_p dT + \left(\frac{\partial V}{\partial p}\right)_T dp  \right]</math>

For constant pressure <math>(dp = 0)</math> the equation simplifies to:

<math display="block">C_p = \left.\frac{\delta Q}{dT}\right|_{p = \text{const}}
= \left(\frac{\partial U}{\partial T}\right)_p + p\left(\frac{\partial V}{\partial T}\right)_p
= \left(\frac{\partial H}{\partial T}\right)_p</math>

where the final equality follows from the appropriate [[Maxwell relations]], and is commonly used as the definition of the isobaric heat capacity.

==== At constant volume, ''dV'' = 0, ''δQ'' = ''dU'' ([[isochoric process]]) ====
A system undergoing a process at constant volume implies that no expansion work is done, so the heat supplied contributes only to the change in internal energy. The heat capacity obtained this way is denoted <math>C_V.</math> The value of <math>C_V</math> is always less than the value of <math>C_p</math>. (<math>C_V < C_p</math>.)

Expressing the inner energy as a function of the variables <math>T</math> and <math>V</math> gives:

<math display="block">\delta Q = \left(\frac{\partial U}{\partial T}\right)_V dT + \left(\frac{\partial U}{\partial V}\right)_T dV + pdV</math>

For a constant volume (<math>dV = 0</math>) the heat capacity reads:

<math display="block">C_V = \left.\frac{\delta Q}{dT}\right|_{V = \text{const}} = \left(\frac{\partial U}{\partial T}\right)_V</math>

The relation between <math>C_V</math> and <math>C_p</math> is then:

<math display="block">C_p = C_V + \left(\left(\frac{\partial U}{\partial V}\right)_T + p\right)\left(\frac{\partial V}{\partial T}\right)_p</math>

==== Calculating ''C<sub>p</sub>'' and ''C<sub>V</sub>'' for an ideal gas ====

[[Mayer's relation]]:

<math display="block">C_p - C_V = nR.</math>
<math display="block">C_p/C_V = \gamma,</math>

where:

* <math>n</math> is the number of moles of the gas,
* <math>R</math> is the [[Gas constant|universal gas constant]],
* <math>\gamma</math> is the [[heat capacity ratio]] (which can be calculated by knowing the number of [[Degrees of freedom (physics and chemistry)|degrees of freedom]] of the gas molecule).

Using the above two relations, the specific heats can be deduced as follows:

<math display="block">C_V = \frac{nR}{\gamma - 1},</math>
<math display="block">C_p = \gamma \frac{nR}{\gamma - 1}.</math>
Following from the [[equipartition of energy]], it is deduced that an ideal gas has the isochoric heat capacity

<math display="block">C_V = n R \frac{N_f}{2} = n R \frac{3 + N_i}{2}</math>

where <math>N_f</math> is the number of [[Degrees of freedom (physics and chemistry)|degrees of freedom]] of each individual particle in the gas, and <math>N_i = N_f - 3</math> is the number of [[Degrees of freedom (physics and chemistry)#Thermodynamic degrees of freedom for gases|internal degrees of freedom]], where the number 3 comes from the three translational degrees of freedom (for a gas in 3D space). This means that a [[Monatomic gas|monoatomic ideal gas]] (with zero internal degrees of freedom) will have isochoric heat capacity <math>C_v = \frac{3nR}{2}</math>.

==== At constant temperature ([[Isothermal process]]) ====
No change in internal energy (as the temperature of the system is constant throughout the process) leads to only work done by the total supplied heat, and thus an [[Infinity|infinite]] amount of heat is required to increase the temperature of the system by a unit temperature, leading to infinite or undefined heat capacity of the system.

==== At the time of phase change ([[Phase transition]]) ====
Heat capacity of a system undergoing phase transition is [[infinity (mathematics)|infinite]], because the heat is utilized in changing the state of the material rather than raising the overall temperature.