Editing Heat capacity (section)

== Definition ==
=== Basic definition ===
The heat capacity of an object, denoted by <math>C</math>, is the limit
<math display="block"> C = \lim_{\Delta T\to 0}\frac{\Delta Q}{\Delta T}, </math>
where <math>\Delta Q</math> is the amount of heat that must be added to the object (of mass ''M'') in order to raise its temperature by <math>\Delta T</math>.

The value of this parameter usually varies considerably depending on the starting temperature <math>T</math> of the object and the pressure <math>p</math> applied to it. In particular, it typically varies dramatically with [[phase transition]]s such as melting or vaporization (see [[enthalpy of fusion]] and [[enthalpy of vaporization]]). Therefore, it should be considered a function <math>C(p,T)</math> of those two variables.

=== Variation with temperature ===
[[File:Heat capacity of water 2.jpg|thumb|403x403px|Specific heat capacity of water<ref>{{Cite web |title=Heat capacity of water online |url=https://www.desmos.com/calculator/wicmrvrznj?lang=ru |access-date=2022-06-03 |website=Desmos |language=ru}}</ref>]]The variation can be ignored in contexts when working with objects in narrow ranges of temperature and pressure. For example, the heat capacity of a block of [[iron]] weighing one [[pound (mass)|pound]] is about 204&nbsp;J/K when measured from a starting temperature ''T''&nbsp;=&nbsp;25&nbsp;°C and ''P''&nbsp;=&nbsp;1&nbsp;atm of pressure. That approximate value is adequate for temperatures between 15&nbsp;°C and 35&nbsp;°C, and surrounding pressures from 0 to 10 atmospheres, because the exact value varies very little in those ranges. One can trust that the same heat input of 204&nbsp;J will raise the temperature of the block from 15&nbsp;°C to 16&nbsp;°C, or from 34&nbsp;°C to 35&nbsp;°C, with negligible error.

===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.

===Heterogeneous objects===
The heat capacity may be well-defined even for heterogeneous objects, with separate parts made of different materials; such as an [[electric motor]], a [[crucible]] with some metal, or a whole building.  In many cases, the (isobaric) heat capacity of such objects can be computed by simply adding together the (isobaric) heat capacities of the individual parts.

However, this computation is valid only when all parts of the object are at the same external pressure before and after the measurement. That may not be possible in some cases.  For example, when heating an amount of gas in an elastic container, its volume ''and pressure'' will both increase, even if the atmospheric pressure outside the container is kept constant. Therefore, the effective heat capacity of the gas, in that situation, will have a value intermediate between its isobaric and isochoric capacities <math>C_p</math> and <math>C_V</math>.

For complex [[thermodynamic system]]s with several interacting parts and [[state variables]], or for measurement conditions that are neither constant pressure nor constant volume, or for situations where the temperature is significantly non-uniform, the simple definitions of heat capacity above are not useful or even meaningful. The heat energy that is supplied may end up as [[kinetic energy]] (energy of motion) and [[potential energy]] (energy stored in force fields), both at macroscopic and atomic scales.  Then the change in temperature will depend on the particular path that the system followed through its [[phase space]] between the initial and final states.  Namely, one must somehow specify how the positions, velocities, pressures, volumes, etc. changed between the initial and final states; and use the general tools of [[thermodynamics]] to predict the system's reaction to a small energy input. The "constant volume" and "constant pressure" heating modes are just two among infinitely many paths that a simple homogeneous system can follow.