Heat capacity and Wikipedia:Peer review: Difference between pages

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'''Heat capacity''' is a measure of the ability of a body to store [[heat]].

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The heat capacity of a body at a certain temperature is the ratio of a small amount of heat energy added to the body to the corresponding small increase in temperature of the body.

This page is for nearly Featured-standard articles that need the final checking by peers before being nominated as [[Wikipedia:Featured article candidates|Featured article candidates]]. '''Requests for peer review''' are listed here to expose articles to closer scrutiny than they might otherwise receive. See [[:Category:Wikipedia style guidelines]] and [[:Category:Wikipedia how-to]] for advice on writing [[Wikipedia:How to write a great article|great articles]]. Or look at the discussion of [[Wikipedia:The perfect article|the perfect article]] and try to reach as many of those ideals as possible. If an article needs extensive work, please list it on [[Wikipedia:Pages needing attention|Pages needing attention]], [[Wikipedia:Requests for expansion|Requests for expansion]] or [[Wikipedia:Cleanup|Cleanup]]. Please list article content disputes on [[Wikipedia:Requests for comment|Requests for comment]] rather than here.

Dimension: energy/temperature.

'''Note:''' [[Peer review]] is the process of review by peers and usually implies a group of authoritative reviewers who are equally familiar and expert in the subject. The process represented by this page is not formal peer review in that sense and articles that undergo this process cannot be assumed to have greater authority than any other.

[[SI]] unit: J&middot;K^&minus;1 ([[joule]] per [[kelvin]]).

{{FAPath}}

Abbreviation: ''C''.

{{PR-instructions}}

''See also [[Wikipedia:WikiProject Computer and video games/Peer review|Peer reviews]] for [[Wikipedia:WikiProject Computer and video games]]''

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==Heat capacity of compressible bodies==

The state of a compressible body is described by a thermodynamic parameter such as its temperature, ''T'', or entropy ''S'', ''and'' by a mechanical parameter such as the pressure, ''p'', or the volume, ''V''. Therefore one must distinguish between ''heat capacity at constant volume'', <math>C_V</math>, and ''heat capacity at constant pressure'', <math>C_p</math>.

==Requests==

Mathematically,

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==Archives==

:<math>C_V=\left(\frac{\delta Q}{dT}\right)_V</math>

*'''[[Wikipedia:Peer review/November 2005]] (current)'''

:<math>C_p=\left(\frac{\delta Q}{dT}\right)_p</math>

*[[Wikipedia:Peer review/October 2005]]

*[[Wikipedia:Peer review/September 2005]]

*[[Wikipedia:Peer review/August 2005]]

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[[Category:Editorial validation]]

where <math>\delta Q</math> is the infinitesimal amount of heat added, and <math>dT</math> is the subsequent rise in temperature.

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From the definition of [[Entropy#Thermodynamic_definition_of_entropy|entropy]], for a [[reversible process|reversible]] transformation, we have

[[fr:Wikipédia:Comité de lecture]]

[[simple:Wikipedia:Peer review]]

:<math>\delta Q=T\,dS.</math>

[[fi:Wikipedia:Vertaisarviointi]]

[[zh:Wikipedia:同行评审]]

The increment of [[internal energy]] is the heat added and the work added:

:<math>dU=T\,dS-p\,dV</math>

So the heat capacity at constant volume is

:<math>C_V=\left(\frac{\partial U}{\partial T}\right)_V

=T\left(\frac{\partial S}{\partial T}\right)_V. </math>

The [[enthalpy]] is defined by <math>H=U+pV</math>. The increment of enthalpy is

:<math>dH=T\,dS+V\,dp.</math>

So the heat capacity at constant pressure is

:<math>C_p=\left(\frac{\partial H}{\partial T}\right)_p

=T\left(\frac{\partial S}{\partial T}\right)_p. </math>

==Heat capacity at absolute zero ==

Using the definition of entropy

:<math>TdS=\delta Q\,</math>

we can calculate the absolute entropy by integrating:

:<math>S(T_f,x)=\int_{T=0}^{T_f} \frac{\delta Q}{T}

=\int_0^{T_f} \left(\frac{\delta Q}{dT}\right)_x\frac{dT}{T}

=\int_0^{T_f} C_x(T,x)\,\frac{dT}{T}</math>

Where <math>C_x</math>&nbsp; is the heat capacity measured while holding some thermodynamic parameter ''x'' constant. Any such heat capacity must be zero at zero temperature in order for the above integral not to diverge and yield an infinite absolute entropy violating the [[third law of thermodynamics]].

==Specific heat capacity ==

The [[specific heat capacity]] of a material (measured in J&middot;K^&minus;1&middot;kg^&minus;1 ) is

:<math>{C \over m} = {C \over {\rho V}}</math>

where

''C'' is the heat capacity of a body made of the material in question (measured in J&middot;K^&minus;1)

''m'' is the mass of the body (measured in kg)

''V'' is the volume of the body (measured in m³)

&rho; = ''mV''^&minus;1 is the density of the material (measured in kg&middot;m^&minus;3)

''CV''^&minus;1 is known as [[volumetric heat capacity]], and has units of J&middot;m^&minus;3K^&minus;1.

==Dimensionless heat capacity ==

The ''[[dimensionless number|dimensionless]] heat capacity'' of a material is

:<math>{C \over nR} = {C \over {Nk}}</math>

where

''C'' is the heat capacity of a body made of the material in question (measured in J&middot;K^&minus;1)

''n'' is the amount of matter in the body (measured in [[mole (unit)|mol]])

''R'' is the [[gas constant]] (measured in J&middot;K^&minus;1&middot;mol^&minus;1)

''nR=Nk'' is the amount of matter in the body (measured in J&middot;K^&minus;1)

''N'' is the number of molecules in the body.

''k'' is [[Boltzmann's constant]] (measured in J&middot;K^&minus;1&middot;molecule^&minus;1)

==Gas phase heat capacities==

According to the [[equipartition theorem]] from classical [[statistical mechanics]], for a system made up of [[statistical independence|independent]] and quadratic [[degrees of freedom (physics and chemistry)|degrees of freedom]], any input of energy into a closed system composed of N molecules is evenly divided among the degrees of freedom available to each molecule. It can be shown that, in the classical limit of [[statistical mechanics]], for each [[statistical independence|independent]] and quadratic [[degrees of freedom (physics and chemistry)|degree of freedom]], that

:<math>E_i=\frac{k_B T}{2}</math>

where

<math>E_i</math> is the [[mean]] energy (measured in [[joule]]s) associated with degree of freedom ''i''.

''T'' is the temperature (measured in [[kelvin]]s)

<math>k_B</math> is [[Boltzman's constant]], (1.380 6505(24) × 10^&minus;23 J K^&minus;1)

In the case of a monatomic gas such as [[helium]] under constant volume, if it assumed that no electronic or nuclear quantum excitations occur, each atom in the gas has only 6 [[degrees of freedom (physics and chemistry)|degrees of freedom]], all of a translational type. No energy is attached to the [[degrees of freedom (physics and chemistry)|degrees of freedom]] attached to the position of atoms, while [[degrees of freedom (physics and chemistry)|degrees of freedom]] corresponding to the [[momentum]]s of atoms are quadratic degrees of freedom. ''N'' atoms thus correspond to ''3N'' [[degrees of freedom (physics and chemistry)|degrees of freedom]], leading to the equation

:<math>C_v=\frac{\partial E}{\partial T}=\frac{3}{2}N\,k_B =\frac{3}{2}n\,R</math>

:<math>c_{v,m}=\frac{C_v}{n}=\frac{3}{2}R</math>

where

<math>C_v</math> is the ''heat capacity'' at constant volume of the gas

<math>c_{v,m}</math> is the ''molar heat capacity'' at constant volume of the gas

''N'' is the total number of atoms present in the container

''n'' is the number of [[Mole (unit)|moles]] of atoms present in the container (''n'' is the ratio of ''N'' and [[Avogadro's number]])

''R'' is the [[ideal gas constant]], (8.314570[70] J K^&minus;1mol^&minus;1). ''R'' is equal to the product of [[Boltzman's constant]] <math>k_B</math> and [[Avogadro's number]]

The following table shows experimental molar constant volume heat capacity measurements taken for each noble monatomic gas (at 1 atm and 25 &deg;C):

{| border="1" cellpadding="2"

|Monatomic gas||''C''_v,m (J K^&minus;1 mol^&minus;1),||''C''_v,m/''R''

|-

|He||12.5||1.50

|-

|Ne||12.5||1.50

|-

|Ar||12.5||1.50

|-

|Kr||12.5||1.50

|-

|Xe||12.5||1.50

|}

It is apparent from the table that the experimental heat capacities of the monatomic noble gases agrees with this simple application of statistical mechanics to a very high degree. In the somewhat more complex case of an ideal gas of diatomic molecules, the presence of internal degrees of freedom are apparent. In addition to the three translational degrees of freedom, there are rotational and vibrational degrees of freedom. In general, there are three degrees of freedom ''f'' per atom in the molecule ''n_a''

:<math>f=3n_a \,</math>

Mathematically, there are a total of three rotational degrees of freedom, one corresponding to rotation about each of the axes of three dimensional space. However, in practice we shall only consider the existence of two degrees of rotational freedom for linear molecules. This approximation is valid because the moment of inertia about the internuclear axis is essentially zero. Quantum mechanically, it can be shown that the interval between successive rotational energy [[eigenstates]] is inversely proportional to the moment of inertia about that axis. Because the moment of inertia about the internuclear axis is vanishingly small relative to the other two rotational axes, the energy spacing can be considered so high that no excitations of the rotational state can possibly occur unless the temperature is extremely high.

We can easily calculate the expected number of vibrational degrees of freedom (or [[vibrational modes]]). There are three degrees of translational freedom, and two degrees of rotational freedom, therefore

:<math>f_\mathrm{vib}=f-f_\mathrm{trans}-f_\mathrm{rot}=6-3-2=1 \,</math>

Each rotational and translational degree of freedom will contribute ''R''/2 in the total molar heat capacity of the gas. Each vibrational mode will contribute <math>R</math> in the total molar heat capacity, however. This is because for each vibrational mode, there is a potential and kinetic energy component. Both the potential and kinetic components will contribute ''R''/2 to the total molar heat capacity of the gas. Therefore, we expect that a diatomic molecule would have a constant volume heat capacity of

:<math>C_v=\frac{3R}{2}+R+R=\frac{7R}{2}</math>

where the terms originate from the translational, rotational, and vibrational degrees of freedom, respectively. The following is a table of some constant volume heat capacities of various diatomics

{| border="1" cellpadding="2"

|Diatomic gas||''C''_v,m (J K^&minus;1 mol^&minus;1),||''C''_v,m/''R''

|-

|H₂||20.18||2.427

|-

|CO||20.2||2.43

|-

|N₂||19.9||2.39

|-

|Cl₂||24.1||2.90

|-

|Br₂||32.0||3.84

|}

From the above table, clearly there is a problem with the above theory. All of the diatomics examined have heat capacities that are lower than those predicted by the Equipartition theorem, except <math>Br_2</math>. However, as the atoms composing the molecules become heavier, the heat capacities move closer to their expected values. One of the reasons for this phenomenon is the quantization of vibrational, and to a lesser extent, rotational states. In fact, if it is assumed that the molecules remain in their lowest energy vibrational state because the interlevel energy spacings are large, the predicted constant volume heat capacity for a diatomic molecule becomes

:<math>C_v=\frac{3R}{2}+R=\frac{5R}{2}</math>

which is a fairly close approximation of the heat capacities of the lighter molecules in the above table. If the quantum [[harmonic oscillator]] approximation is made, it turns out that the quantum vibrational energy level spacings are actually inversely proportional to the square root of the [[reduced mass]] of the atoms composing the diatomic molecule. Therefore, in the case of the heavier diatomic molecules, the quantum vibrational energy level spacings become finer, which allows more excitations into higher vibrational levels at a fixed temperature.

==Solid phase heat capacities==

[[Image:DebyeVSEinstein.jpg|thumb|300px|The dimensionless heat capacity divided by three, as a function of temperature as predicted by the [[Debye model]] and by Einstein's earlier model. The horizontal axis is the temperature divided by the Debye temperature. Note that, as expected, the dimensionless heat capacity is zero at absolute zero, and rises to a value of three as the temperature becomes much larger than the Debye temperature.]]

For matter in a crystalline solid phase, the [[Dulong-Petit law]] states that the dimensionless specific heat capacity assumes the value 3. For more modern and precise analysis it is useful to use the idea of [[phonons]]. See [[Debye model]].

== See also ==

* [[Quantum statistical mechanics]]

* [[Specific heat capacity]]

* [[Statistical mechanics]]

* [[Volumetric heat capacity]]

* [[thermodynamics equations]]

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