Specific heat capacity

The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat, even for the same starting pressure ${\displaystyle p}$ and starting temperature ${\displaystyle T}$. Two particular choices are widely used:

• If the pressure is kept constant (for instance, at the ambient atmospheric pressure), and the sample is allowed to expand, the expansion generates work as the force from the pressure displaces the enclosure or the surrounding fluid. That work must come from the heat energy provided. The specific heat thus obtained is said to be measured at constant pressure (or isobaric), and is often denoted ${\displaystyle c_{p}}$, ${\displaystyle c_{\mathrm {p} }}$, etc.
• On the other hand, if the expansion is prevented — for example by a sufficiently rigid enclosure, or by increasing the external pressure to counteract the internal one — no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the sample, including raising its temperature by an extra amount. The specific heat obtained this way is said to be measured at constant volume (or isochoric) and denoted ${\displaystyle c_{V}}$, ${\displaystyle c_{v}}$ ${\displaystyle c_{\mathrm {v} }}$, etc.

The value of ${\displaystyle c_{V}}$ is usually less than the value of ${\displaystyle c_{p}}$. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the heat capacity ratio of gases is typically between 1.3 and 1.67.[10]

The specific heat can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale.

The specific heat can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops.

The specific heat is not meaningful if the substance undergoes irreversible chemical changes, or if there is a phase change, such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement.

The SI unit for specific heat is joule per kelvin per kilogram (J/K/kg, J/(kg K), J K⁻¹ kg⁻¹, etc.). Since an increment of temperature of one degree Celsius is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram (J/°C/kg). Sometimes the gram is used instead of kilogram for the unit of mass: 1 J/K/kg = 0.001 J/K/g.

The specific heat of a substance (per unit of mass) has dimension L²·Θ⁻¹·T⁻², or (L/T)²/Θ. Therefore, the SI unit J/K/kg is equivalent to metre squared per second squared per kelvin (m² K⁻¹ s⁻²).

Professionals in construction, civil engineering, chemical engineering, and other technical disciplines, especially in the United States, may use English Engineering units including the Imperial pound (lb = 0.45359237 kg) as the unit of mass, the degree Fahrenheit or Rankine (°F = 5/9 K, about 0.555556 K) as the unit of temperature increment, and the British thermal unit (BTU ≈ 1055.06 J),[13][14] as the unit of heat.

In those contexts, the unit of specific heat is BTU/°F/lb = 4177.6 J/K/kg. The BTU was originally defined so that the average specific heat of water would be 1 BTU/°F/lb.

In chemistry, heat amounts were often measured in calories. Confusingly, two units with that name, denoted "cal" or "Cal", have been commonly used to measure amounts of heat:

• the "small calorie" (or "gram-calorie", "cal") is 4.184 J, exactly. It was originally defined so that the specific heat of liquid water would be 1 cal/C°/g.
• The "grand calorie" (also "kilocalorie", "kilogram-calorie", or "food calorie"; "kcal" or "Cal") is 1000 small calories, that is, 4184 J, exactly. It was originally defined so that the specific heat of water would be 1 Cal/C°/kg.

While these units are still used in some contexts (such as kilogram calorie in nutrition), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat is usually

1 cal/°C/g ("small calorie") = 1 Cal/°C/kg = 1 kcal/°C/kg ("large calorie") = 4184 J/K/kg.

In either unit, the specific heat of water is approximately 1. The combinations cal/°C/kg = 4.184 J/K/kg and kcal/°C/g = 4184,000 J/K/kg do not seem to be widely used.

Quantum mechanics predicts that, at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy. Thus, heat capacity per mole is the same for all monoatomic gases (such as the noble gases). More precisely, ${\displaystyle c_{V,\mathrm {m} }=3R/2\approx {}}$12.5 J/K/mol and ${\displaystyle c_{P,\mathrm {m} }=5R/2\approx {}}$21 J/K/mol, where ${\displaystyle R\approx {}}$8.31446 J/K/mol is the ideal gas unit (which is the product of Boltzmann conversion constant from kelvin microscopic energy unit to the macroscopic energy unit joule, and Avogadro's number).

Therefore, the specific heat (per unit of mass, not per mole) of a monoatomic gas will be inversely proportional to its (adimensional) atomic weight ${\displaystyle A}$. That is, approximately,

${\displaystyle c_{V}\approx {}}$12470 J/K/kg${\displaystyle /A\quad \quad \quad c_{p}\approx {}}$20785 J/K/kg${\displaystyle /A}$

For the noble gases, from helium to xenon, these computed values are

Gas	He	Ne	Ar	Kr	Xe
${\displaystyle A}$	4.00	20.17	39.95	83.80	131.29
${\displaystyle c_{V}}$ (J/K/m3)	3118	618.3	312.2	148.8	94.99
${\displaystyle c_{p}}$ (J/K/kg)	5197	1031	520.3	248.0	158.3

On the other hand, a polyatomic gas molecule (consisting of two or more atoms bound together) can store heat energy in other forms besides its kinetic energy. These forms include rotation of the molecule, and vibration of the atoms relative to its center of mass.

These extra degrees of freedom or "modes" contribute to the specific heat of the substance. Namely, when heat energy is injected into a gas with polyatomic molecules, only part of it will go into increasing their kinetic energy, and hence the temperature; the rest will go to into those other degrees of freedom. In order to achieve the same increase in temperature, more heat energy will have to be provided to a mol of that substance than to a mol of a monoatomic gas. Therefore, the specific heat of a polyatomic gas depends not only on its molecular mass, but also on the number of degrees of freedom that the molecules have.[15][16] [17]

Quantum mechanics further says that each rotational or vibrational mode can only take or lose energy in certain discrete amount (quanta). Depending on the temperature, the average heat energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be "frozen out". In that case, the specific heat of the substance is going to increase with temperature, sometimes in a step-like fashion, as more modes become unfrozen and start absorbing part of the input heat energy.

For example, the molar heat capacity of nitrogen N
₂ at constant volume is ${\displaystyle c_{V,\mathrm {m} }={}}$ 20.6 J/K/mol (at 15 °C, 1 atm), which is 2.49${\displaystyle R}$.[18] That is the value expected from theory if each molecule had 5 degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. Because of those two extra degrees of freedom, the specific heat ${\displaystyle c_{V}}$ of N
₂ (736 J/K/kg) is greater than that of an hypothetical monoatomic gas with the same molecular mass 28 (445 J/K/kg), by a factor of 5/3.

This value for the specific heat of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still "frozen out". At about that temperature, those modes begin to "un-freeze", and as a result ${\displaystyle c_{V}}$ starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J/K/mol at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.[19] The last value corresponds almost exactly to the predicted value for 7 degrees of freedom per molecule.

To apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat can be defined; in particular, that it has homogeneous composition and fixed mass ${\displaystyle M}$. Assume that the evolution of the system is always slow enough for the internal pressure ${\displaystyle P}$ and temperature ${\displaystyle T}$ be considered uniform throughout. The pressure ${\displaystyle P}$ would be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air.

The state of the material can then be specified by three parameters: its temperature ${\displaystyle T}$, the pressure ${\displaystyle P}$, and its specific volume ${\displaystyle V={\boldsymbol {\mathrm {V} }}/M}$, where ${\displaystyle {\boldsymbol {\mathrm {V} }}}$ is the volume of the sample. (This quantity is the reciprocal ${\displaystyle 1/\rho }$ of the material's density ${\displaystyle \rho =M/{\boldsymbol {\mathrm {V} }}}$.) Like ${\displaystyle T}$ and ${\displaystyle P}$, the specific volume ${\displaystyle V}$ is an intensive property of the material and its state, that does not depend on the amount of substance in the sample.

Those variables are not independent. The allowed states are defined by an equation of state relating those three variables: ${\displaystyle F(T,P,V)=0.}$ The function ${\displaystyle F}$ depends on the material under consideration. The specific internal energy stored internally in the sample, per unit of mass, will then be another function ${\displaystyle U(T,P,V)}$ of these state variables, that is also specific of the material. The total internal energy in the sample then will be ${\displaystyle MU(T,P,V)}$.

For some simple materials, like an ideal gas, one can derive from basic theory the equation of state ${\displaystyle F=0}$ and even the specific internal energy ${\displaystyle U}$ In general, these functions must be determined experimentally for each substance.

The absolute value of this quantity is undefined, and (for the purposes of thermodynamics) the state of "zero internal energy" can be chosen arbitrarily. However, by the law of conservation of energy, any infinitesimal increase ${\displaystyle M\mathrm {d} U}$ in the total internal energy ${\displaystyle MU}$ must be matched by the net flow of heat energy ${\displaystyle \mathrm {d} Q}$ into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is ${\displaystyle -P\mathrm {d} {\boldsymbol {\mathrm {V} }}}$, where ${\displaystyle \mathrm {d} {\boldsymbol {\mathrm {V} }}}$ is the change in the sample's volume in that infinitesimal step.[20] Therefore

${\displaystyle \mathrm {d} Q-P\mathrm {d} {\boldsymbol {\mathrm {V} }}=M\mathrm {d} U}$

hence

${\displaystyle {\frac {\mathrm {d} Q}{M}}-P\mathrm {d} V=\mathrm {d} U}$

If the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount ${\displaystyle \mathrm {d} Q}$, then the term ${\displaystyle P\mathrm {d} V}$ is zero (no mechanical work is done). Then, dividing by ${\displaystyle \mathrm {d} T}$,

${\displaystyle {\frac {\mathrm {d} Q}{M\mathrm {d} T}}={\frac {\mathrm {d} U}{\mathrm {d} T}}}$

where ${\displaystyle \mathrm {d} T}$ is the change in temperature that resulted from the heat input. The left-hand side is the specific heat at constant volume ${\displaystyle c_{V}}$ of the material.

For the heat capacity at constant pressure, it is useful to define the specific enthalpy of the system as the sum ${\displaystyle H(T,P,V)=U(T,P,V)+PV}$. An infinitesimal change in the specific enthalpy will then be

${\displaystyle \mathrm {d} H=\mathrm {d} U+V\mathrm {d} P+P\mathrm {d} V}$

therefore

${\displaystyle {\frac {\mathrm {d} Q}{M}}+V\mathrm {d} P=\mathrm {d} H}$

If the pressure is kept constant, the second term on the left-hand side is zero, and

${\displaystyle {\frac {\mathrm {d} Q}{M\mathrm {d} T}}={\frac {\mathrm {d} H}{\mathrm {d} T}}}$

The left-hand side is the specific heat at constant pressure ${\displaystyle c_{P}}$ of the material.

In general, the infinitesimal quantities ${\displaystyle \mathrm {d} T,\mathrm {d} P,\mathrm {d} V,\mathrm {d} U}$ are constrained by the equation of state and the specific internal energy function. Namely,

${\displaystyle \left\{{\begin{array}{lcl}\displaystyle \mathrm {d} T{\frac {\partial F}{\partial T}}(T,P,V)+\mathrm {d} P{\frac {\partial F}{\partial P}}(T,P,V)+\mathrm {d} V{\frac {\partial F}{\partial V}}(T,P,V)&=&0\\[2ex]\displaystyle \mathrm {d} T{\frac {\partial U}{\partial T}}(T,P,V)+\mathrm {d} P{\frac {\partial U}{\partial P}}(T,P,V)+\mathrm {d} V{\frac {\partial U}{\partial V}}(T,P,V)&=&\mathrm {d} U\end{array}}\right.}$

Here ${\displaystyle (\partial F/\partial T)(T,P,V)}$ denotes the (partial) derivative of the state equation ${\displaystyle F}$ with respect to its ${\displaystyle T}$ argument, keeping the other two arguments fixed, evaluated at the state ${\displaystyle (T,P,V)}$ in question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state. The constant-volume and constant-pressure changes are only two particular directions in this space.

This analysis also holds no matter how the energy increment ${\displaystyle \mathrm {d} Q}$ is injected into the sample, namely by heat conduction, irradiation, electromagnetic induction, radioactive decay, etc.

For any specific volume ${\displaystyle V}$, denote ${\displaystyle p_{V}(T)}$ the function that describes how the pressure varies with the temperature ${\displaystyle T}$, as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at ${\displaystyle V}$. Analogously, for any pressure ${\displaystyle P}$, let ${\displaystyle v_{P}(T)}$ be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at ${\displaystyle P}$. Namely, those functions are such that

${\displaystyle F(T,p_{V}(T),V)=0\quad \quad {}}$and${\displaystyle {}\quad \quad F(T,P,v_{P}(T))=0}$

for any values of ${\displaystyle T,P,V}$. In other words, the graphs of ${\displaystyle p_{V}(T)}$ and ${\displaystyle v_{P}(T)}$ are slices of the surface defined by the state equation, cut by planes of constant ${\displaystyle V}$ and constant ${\displaystyle P}$, respectively.

Then, from the fundamental thermodynamic relation it follows that

${\displaystyle c_{P}(T,P,V)-c_{V}(T,P,V)=T\left[{\frac {\mathrm {d} p_{V}}{\mathrm {d} T}}(T)\right]\left[{\frac {\mathrm {d} v_{P}}{\mathrm {d} T}}(T)\right]}$

This equation can be rewritten as

${\displaystyle c_{P}(T,P,V)-c_{V}(T,P,V)=VT{\frac {\alpha ^{2}}{\beta _{T}}},}$

where

${\displaystyle \alpha }$ is the coefficient of thermal expansion,

${\displaystyle \beta _{T}}$ is the isothermal compressibility,

both depending on the state ${\displaystyle (T,P,V)}$.

The heat capacity ratio, or adiabatic index, is the ratio ${\displaystyle c_{P}/c_{V}}$ of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.

The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below.

For an ideal gas, evaluating the partial derivatives above according to the equation of state, where R is the gas constant, for an ideal gas[21]

${\displaystyle PV=nRT,}$

${\displaystyle C_{P}-C_{V}=T\left({\frac {\partial P}{\partial T}}\right)_{V,n}\left({\frac {\partial V}{\partial T}}\right)_{P,n},}$

${\displaystyle P={\frac {nRT}{V}}\Rightarrow \left({\frac {\partial P}{\partial T}}\right)_{V,n}={\frac {nR}{V}},}$

${\displaystyle V={\frac {nRT}{P}}\Rightarrow \left({\frac {\partial V}{\partial T}}\right)_{P,n}={\frac {nR}{P}}.}$

Substituting

${\displaystyle T\left({\frac {\partial P}{\partial T}}\right)_{V,n}\left({\frac {\partial V}{\partial T}}\right)_{P,n}=T{\frac {nR}{V}}{\frac {nR}{P}}={\frac {nRT}{V}}{\frac {nR}{P}}=P{\frac {nR}{P}}=nR,}$

this equation reduces simply to Mayer's relation:

${\displaystyle C_{P,m}-C_{V,m}=R.}$

The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. -->