# Specific heat capacity

Physical size
Surname specific heat capacity
Formula symbol ${\ displaystyle c}$
Size and
unit system
unit dimension
SI J / ( kg · K ) L 2 · T −2 · Θ −1

The specific heat capacity , also specific heat or shortened heat capacity , is a material property of thermodynamics . It measures the ability of a substance to store thermal energy .

## definition

### Definition of the specific heat capacity

The specific heat capacity of a substance in a certain state is the heat that is added to or withdrawn from a quantity of the substance, divided by the associated temperature increase or decrease and the mass of the substance:

${\ displaystyle c = {\ frac {\ Delta Q} {m \ cdot \ Delta T}}}$

It is

• ${\ displaystyle \ Delta Q}$ the heat that is added to or withdrawn from the material,
• ${\ displaystyle m}$ the mass of the substance,
• ${\ displaystyle \ Delta T = T_ {2} -T_ {1}}$ the difference between the final and initial temperature.

The unit of specific heat capacity in the International System of Units (SI) is:

${\ displaystyle [c] = \ mathrm {\ frac {J} {kg \ cdot K}}}$

For example, the specific heat capacity of liquid water is approximately . This means that one kilogram of water has to be supplied with an energy of 4.2  kilojoules in order to heat it up by 1  Kelvin . ${\ displaystyle c = 4 {,} 2 \, \ mathrm {\ tfrac {kJ} {kg \ cdot K}}}$

In general, the specific heat capacity of state variables depends in particular on the temperature. Therefore values ​​for the specific heat capacity only apply to a certain temperature, often to 25 ° C. Measurements of the temperature dependency take place e.g. B. by dynamic power (difference) calorimetry . Historically, measurements of this kind, especially at low temperatures, have significantly advanced solid-state physics . ${\ displaystyle c (T)}$

In the case of a phase transition of the first order , the heat capacity is not defined, measured values ​​diverge there. A jump in , on the other hand, indicates a second order phase transition in which the number of degrees of freedom in the material changes. ${\ displaystyle c (T)}$

In addition, the specific heat capacity depends on the process management of the heating or cooling, especially in the case of gases . In particular, a distinction is made between the specific heat at constant volume and that at constant pressure . With a constant volume, the entire heat input benefits the temperature increase. If the gas can expand, however, then part of the heat is used to do the work of expansion and is therefore missing for the temperature increase. ${\ displaystyle c_ {V}}$ ${\ displaystyle c_ {p}}$

### Mean and true specific heat capacity

The formula in the introduction indicates the mean specific heat capacity for the temperature interval . This can ${\ displaystyle c \ vert _ {T_ {1}} ^ {T_ {2}}}$${\ displaystyle [T_ {1}, T_ {2}]}$

• for temperature ranges between and any temperature , read from tables or, if it is not listed there, approximate from them by interpolation ;${\ displaystyle T_ {0} = 0 \; ^ {\ circ} \ mathrm {C}}$${\ displaystyle T_ {x}}$
• for temperature ranges that do not start at, calculate as follows:${\ displaystyle T_ {0} = 0 \; ^ {\ circ} \ mathrm {C}}$
${\ displaystyle c \ vert _ {T_ {1}} ^ {T_ {2}} = {\ frac {c \ vert _ {T_ {0}} ^ {T_ {2}} \ cdot (T_ {2} - T_ {0}) - c \ vert _ {T_ {0}} ^ {T_ {1}} \ cdot (T_ {1} -T_ {0})} {T_ {2} -T_ {1}}}}$

For more detailed considerations, go over to the true specific heat capacity at temperature , i. H. for the borderline case of arbitrarily small temperature changes: ${\ displaystyle T_ {1}}$

${\ displaystyle c \ vert _ {T_ {1}} = \ lim _ {T_ {2} \ rightarrow T_ {1}} {\ frac {\ Delta Q} {m \ cdot \ Delta T}}}$

## Values ​​for selected materials

material material c in kJ / (kg K) c in kJ / (kg K) Ice (0 ° C) 02.060 hydrogen 14.32 Ice (−10 ° C) 02.22 helium 05.193 sodium 01,234 methane 02.158 magnesium 01.046 Water vapor (100 ° C) 02.080 aluminum 00.896 butane 01.658 iron 00.452 Air (20 ° C dry) 01.005 copper 00.382 Carbon dioxide 00.846 silver 00.235 argon 00.523 lead 00.129 Wood fiber insulation , cellulose flakes 02.1 Water (20 ° C) 04.184 Polystyrene 01.4 Ethanol 02.43 Plaster of paris , chamotte ≈1 petroleum 02.14 concrete 00.88 mercury 00.139 Mineral fiber insulation 00.8

## Relationships to heat capacity and molar heat capacity

If the temperature of a body changes by the temperature difference , the heat will be increased ${\ displaystyle \ Delta T}$

${\ displaystyle \ Delta Q = C \, \ Delta T}$

passed, provided that the heat capacity of the body is at least approximately independent of temperature in this temperature interval. In contrast to the volume or mass-related heat capacity, the (absolute) heat capacity is not a material property, but the property of a body. ${\ displaystyle C}$

If it is a homogeneous body, the mass-specific heat capacity can also be given:

${\ displaystyle \ Delta Q = c \, m \, \ Delta T}$

If the heat capacity is not related to the mass of the substance, but to its amount of substance , the above equation using the molar heat capacity (outdated also called molar heat ) reads : ${\ displaystyle n}$ ${\ displaystyle C _ {\ mathrm {m}}}$

${\ displaystyle \ Delta Q = C _ {\ mathrm {m}} \, n \, \ Delta T}$

There is a relationship between the heat capacity , the specific heat capacity and the molar heat capacity${\ displaystyle C}$${\ displaystyle c}$${\ displaystyle C _ {\ mathrm {m}}}$

${\ displaystyle C = c \, m = C _ {\ mathrm {m}} \, n}$.

After dividing by the amount of substance it becomes ${\ displaystyle n}$

${\ displaystyle {\ frac {C} {n}} = c \, M = C _ {\ mathrm {m}}}$

with the molar mass . ${\ displaystyle M = {\ tfrac {m} {n}}}$

Using the example of copper :${\ displaystyle c = {0 {,} 38 \, \ mathrm {\ tfrac {J} {g \ cdot K}}, \, M = 63 \, \ mathrm {\ tfrac {g} {mol}} \, \ Rightarrow C _ {\ mathrm {m}} = c \ cdot M = 24 \, \ mathrm {\ tfrac {J} {mol \ cdot K}}}}$

## Heat capacity of ideal gases

From the thermodynamic equations of state of the ideal gas

thermal :${\ displaystyle p \, V = n \, R \, T}$
caloric :${\ displaystyle U = n \, C _ {\ mathrm {m}, V} \, T}$

and the definition of enthalpy :

${\ displaystyle H = n \, C _ {\ mathrm {m}, p} \, T = U + p \, V}$

it follows for the molar heat capacities at constant volume ( isochoric ) and at constant pressure ( isobaric ): ${\ displaystyle C _ {\ mathrm {m}, V}}$${\ displaystyle C _ {\ mathrm {m}, p}}$

${\ displaystyle C _ {\ mathrm {m}, p} = C _ {\ mathrm {m}, V} + R}$

with the universal gas constant . ${\ displaystyle R = 8 {,} 314 \; \ mathrm {\ tfrac {J} {mol \; K}}}$

The individual symbols stand for the following quantities :

• ${\ displaystyle p}$- pressure
• ${\ displaystyle V}$- volume
• ${\ displaystyle n}$- Amount of substance
• ${\ displaystyle T}$- absolute temperature
• ${\ displaystyle U}$- Inner energy
• ${\ displaystyle H}$- enthalpy

Compared to the molar heat capacity at constant volume that falls at a constant pressure of greater because the gas expands in this case, upon heating and thus against the external pressure working guaranteed. The corresponding proportion of the supplied heat does not benefit the internal energy of the gas and thus also does not benefit the temperature increase. Therefore, more heat has to be supplied for a certain temperature increase, the quotient and thus the molar heat capacity increase.

The isentropic exponent is defined as:

${\ displaystyle \ kappa = {\ frac {C _ {\ mathrm {m}, p}} {C _ {\ mathrm {m}, V}}} = {\ frac {c_ {m, p}} {c_ {m , V}}} = {\ frac {C_ {p}} {C_ {V}}} = {\ frac {c_ {p}} {c_ {V}}}}$

### General case

As a good approximation:

with and follows .${\ displaystyle \; C _ {\ mathrm {m}, V} = {\ frac {f} {2}} \, R \;}$${\ displaystyle \; C _ {\ mathrm {m}, p} = {\ frac {f + 2} {2}} \, R \;}$${\ displaystyle \; \ kappa = {\ frac {f + 2} {f}} = 1 + {\ frac {2} {f}} \;}$

with the total number of energetic degrees of freedom of a molecule with the proportions ${\ displaystyle f = f _ {\ mathrm {trans}} + f _ {\ mathrm {red}} + f _ {\ mathrm {vib}}}$

• ${\ displaystyle f _ {\ mathrm {trans}} = 3}$for the translational kinetic energy of the center of gravity
• ${\ displaystyle f _ {\ mathrm {red}} \ in \ {0,2,3 \}}$for the rotational energy (explanation see below)
• ${\ displaystyle f _ {\ mathrm {vib}} = 2 \, l}$for the internal energy of the normal oscillations of the atomic nuclei against each other (each oscillation brings an additional degree of freedom for the kinetic energy and one for the potential energy ).${\ displaystyle l}$

### 1 atomic gas

The simplest model system regards the atoms as mass points : from them ( number of particles ) fly around freely in a box with volume and exert a pressure through impacts against the wall . According to the kinetic gas theory, the time mean for the pressure on the wall is: ${\ displaystyle N \ gg 1}$${\ displaystyle V}$${\ displaystyle p}$

${\ displaystyle p \, V = {\ frac {2} {3}} \, N \, \ langle E _ {\ mathrm {kin}} \ rangle}$

Therein is the average kinetic energy of a particle . ${\ displaystyle \ langle E _ {\ mathrm {kin}} \ rangle}$

For the total kinetic energy of all particles, a comparison with the equation of state of the ideal gas results : ${\ displaystyle N \, \ langle E _ {\ mathrm {kin}} \ rangle}$${\ displaystyle p \, V = n \, R \, T}$

${\ displaystyle \ Rightarrow N \, \ langle E _ {\ mathrm {kin}} \ rangle = {\ frac {3} {2}} \, n \, R \, T}$

This result also follows from the uniform distribution theorem of statistical mechanics , according to which every particle has on average the energy in each of its degrees of freedom of motion ; with the three degrees of freedom of the monatomic gas we get: ${\ displaystyle k _ {\ mathrm {B}} T / 2}$

${\ displaystyle \ langle E _ {\ mathrm {kin}} \ rangle = {\ frac {3} {2}} \, k _ {\ mathrm {B}} \, T}$

With

• the Boltzmann constant ${\ displaystyle k _ {\ mathrm {B}} = {\ tfrac {R} {N _ {\ mathrm {A}}}}}$
• the Avogadro constant .${\ displaystyle N _ {\ mathrm {A}} = {\ tfrac {N} {n}}}$

The mass point has degrees of freedom, corresponding to the three spatial dimensions . It is true that a single atom can also rotate in the sense that it has a higher angular momentum in its excited states than in its ground state . These states correspond to electronic excitations and have excitation energies which, due to the smallness of the mass moment of inertia, are at least a few eV due to the angular momentum quantization , i.e. far higher than the typical thermal energy , so that no excitation can take place in thermal equilibrium . ${\ displaystyle f = f _ {\ text {trans}} = 3}$ ${\ displaystyle k _ {\ mathrm {B}} \, T}$${\ displaystyle \ left (f _ {\ text {red}} = 0 \ right)}$

If the thermodynamic internal energy is identified with the total kinetic energy, then the caloric equation of state of the monatomic ideal gas follows: ${\ displaystyle U}$

${\ displaystyle U = {\ frac {3} {2}} \, n \, R \, T}$

Hence is

${\ displaystyle C _ {\ mathrm {m}, V} = {\ frac {3} {2}} \, R \ \ Rightarrow \ C _ {\ mathrm {m}, p} = {\ frac {5} {2 }} \, R \ \ Rightarrow \ \ kappa = {\ frac {5} {3}} = 1 {,} 666 \ ldots}$

#### Larger temperature range

These values ​​agree perfectly with measurements on noble gases and mercury vapor if the temperature or pressure is sufficiently far above the liquefaction point . The first measurement was made in 1876 on thin mercury vapor at around 300 ° C. The isentropic exponent determined by the speed of sound confirmed for the first time that free atoms behave like mass points over a large temperature range. ${\ displaystyle \ kappa \ approx 1 {,} 66}$

### 2 atomic gas

The simplest model for a diatomic gas is a rigid dumbbell . It has degrees of freedom for translational movements of the center of gravity and degrees of freedom for rotations around the two axes perpendicular to the dumbbell axis; the possibility of rotation around the dumbbell axis (given in the macroscopic mechanical model) is not counted, since both atomic nuclei lie on the rotation axis. Therefore - as with monatomic gas - they have no mass moment of inertia around this axis and therefore no rotational energy either . ${\ displaystyle \ left (l = 0 \ Rightarrow f _ {\ text {vib}} = 0 \ right)}$${\ displaystyle f _ {\ text {trans}} = 3}$${\ displaystyle f _ {\ text {red}} = 2}$

With the above degrees of freedom it follows from the uniform distribution theorem: ${\ displaystyle f = 3 + 2 = 5}$

${\ displaystyle U = {\ frac {5} {2}} \, n \, R \, T}$

Hence is

${\ displaystyle C _ {\ mathrm {m}, V} = {\ frac {5} {2}} \, R \ \ Rightarrow \ C _ {\ mathrm {m}, p} = {\ frac {7} {2 }} \, R \ \ Rightarrow \ \ kappa = {\ frac {7} {5}} = 1 {,} 4}$

Measured values ​​for oxygen , nitrogen , hydrogen etc. fit this perfectly under normal conditions .

#### At very low temperatures

In the case of very cold hydrogen , a decrease in the molar heat is observed up to , which corresponds to the behavior of a single mass point. This is explained by the quantum physical effect that the rotational energy can only assume discrete values ​​with certain intervals (energy levels, quantization ). At low temperatures, the order of magnitude of the energies that are typically exchanged between the molecules during collisions (given approximately by the thermal energy ) can drop below the lowest level of the rotational energy: ${\ displaystyle \ left (T <200 {\ text {K}} \ right)}$${\ displaystyle C _ {\ mathrm {m}, V} = {\ frac {3} {2}} \, R}$${\ displaystyle E _ {\ text {th}} = k _ {\ mathrm {B}} \, T}$

${\ displaystyle E _ {\ text {th}}

In this case, no more rotations can be excited by the thermal collisions, the degrees of freedom of rotation "freeze", which is why diatomic gases at low temperatures can be treated like monatomic gases at low temperatures:

${\ displaystyle f _ {\ text {red}} = 0 \ Rightarrow f = f _ {\ text {trans}} = 3}$

This effect is most clearly pronounced with hydrogen, which remains gaseous down to very low temperatures and whose molecules have the smallest moment of inertia and thus also the lowest rotational energy ( ). ${\ displaystyle \ kappa = {\ frac {5} {3}} \ approx 1 {,} 67}$

#### At very high temperatures

At higher or very high temperatures , the molar heats can increase up to:

${\ displaystyle C _ {\ mathrm {m}, V} = {\ frac {7} {2}} \, R \ Rightarrow C _ {\ mathrm {m}, p} = {\ frac {9} {2}} \, R \ Rightarrow \ kappa = {\ frac {9} {7}} \ approx 1 {,} 29}$

This is explained by the gradual "thawing" of the degrees of freedom for the oscillation of the two atoms against each other, i. That is, the rigid dumbbell model no longer applies at high temperatures:

${\ displaystyle l = 1 \ Rightarrow f _ {\ text {vib}} = 2 \ Rightarrow f = 3 + 2 + 2 = 7}$

### 3- and polyatomic gas

Translational and rotational movements each bring three degrees of freedom:

${\ displaystyle f _ {\ text {trans}} = f _ {\ text {red}} = 3,}$

if not all nuclei lie on one line (then there are only two degrees of freedom of rotation, explanation see above for diatomic gas).

With larger molecules, parts of the vibrational degrees of freedom are already excited under normal conditions:

${\ displaystyle l \ geq 2 \ Rightarrow f _ {\ text {vib}} \ geq 4 \ Rightarrow f \ geq 10}$

As a result, the molar heats rise higher than with 2-atom gases:

${\ displaystyle C _ {\ mathrm {m}, V} \ geq 5 \, R \ Rightarrow C _ {\ mathrm {m}, p} \ geq 6 \, R,}$

why the isentropic exponent continues to fall: ${\ displaystyle \ kappa}$

${\ displaystyle \ kappa \ leq {\ frac {6} {5}} = 1 {,} 2}$

## Heat capacity of solids

### Observations

Temperature profile of the heat capacity of iron (with a peak at the Curie temperature )

According to the empirically found Dulong-Petit law, the molar heat of solids reaches approximately the same value at sufficiently high temperatures:

${\ displaystyle C _ {\ mathrm {m}} = 3 \ cdot R \ approx 25 \; \ mathrm {\ tfrac {J} {mol \ cdot K}}}$

The specific heat decreases towards low temperatures, the form of this dependence being very similar for all solids if the temperature is scaled appropriately . At very low temperatures, the specific heat approaches the value zero; the curve for non-conductors is similar to the function , for metals to the function . For ferromagnetic materials such as. B. Iron provides the change in magnetization a contribution to the heat capacity. ${\ displaystyle C _ {\ mathrm {m}} = f (T ^ {3})}$${\ displaystyle C _ {\ mathrm {m}} = f (T)}$

### Model system of mass points

The simplest model system of the solid consists of mass points, which are bound to their rest position by elastic forces and can oscillate independently of one another in three directions of space. Since each oscillation contributes two degrees of freedom, the total number of degrees of freedom and the molar heat capacity predicted according to the uniform distribution theorem is${\ displaystyle N \ gg 1}$${\ displaystyle f = 6}$

${\ displaystyle C _ {\ mathrm {m}} = {\ tfrac {6} {2}} R = 3R,}$

which is in accordance with Dulong-Petit's rule.

### Einstein model

The decrease towards lower temperatures shows the freezing of the vibrations. Albert Einstein assumed in 1907 that the vibrations of all particles have the same frequency and that their energy can only change gradually by each ( is Planck's quantum of action ). ${\ displaystyle \ nu}$${\ displaystyle \ Delta E = h \ cdot \ nu}$${\ displaystyle h}$

### Debye model

Debye temperature of selected metals
metal Debye
temperature
iron 0464 K
aluminum 0426 K
magnesium 0406 K
copper 0345 K
tin 0195 K

Peter Debye refined the model in 1912 in such a way that instead of assuming independent, individual vibrations of the individual atoms, he assumed the elastic vibrations of the whole body. At a high temperature they are all excited according to the equation of distribution and give the specific heat in accordance with the value . However, they have different frequencies depending on the wavelength , so that their energy levels are differently far apart and therefore the effect of freezing is distributed over a wider temperature range. According to this Debye model, the molar heat capacity is determined as a function of the temperature: ${\ displaystyle C _ {\ mathrm {m}} = 3R}$

${\ displaystyle c_ {V} (T) = 9R \ cdot \ left ({\ frac {T} {\ Theta _ {\ mathrm {D}}}} \ right) ^ {3} \ cdot \ int _ {0 } ^ {\ frac {\ Theta _ {D}} {T}} {\ frac {x ^ {4} \ cdot \ mathrm {e} ^ {x}} {\ left (\ mathrm {e} ^ {x } -1 \ right) ^ {2}}} \, \ mathrm {d} x}$

The Debye temperature, the only variable that depends on the material, indicates the value with which the temperature must be scaled in order to obtain a curve that is uniform for all substances: at around this temperature , the molar heat has dropped to half its full value. ${\ displaystyle \ Theta _ {\ mathrm {D}}}$${\ displaystyle T = 0 {,} 2 \ cdot \ Theta _ {\ mathrm {D}}}$

The Debye model agrees very well with the measurements on solids at all temperatures. In particular, it also correctly shows the increase in heat capacity in the case of non-conductors in the vicinity of absolute zero , while the Einstein model predicts an increase that is much too weak here. ${\ displaystyle T ^ {3}}$

### Model system electron gas

In order to understand the linear dependence of the heat capacity on the temperature, which the electrical conductors show near absolute zero, one can understand the conduction electrons as a degenerate Fermi gas . With the help of the Fermi distribution and the density of states of a free electron, the temperature dependence of the total energy and consequently also the heat capacity can be calculated for low temperatures.

The result agrees with the measured values ​​and is far lower than if the conduction electrons were viewed as a classic monatomic ideal gas (as above) that is in the solid body in addition to the atomic cores . The elucidation of this discrepancy is considered to be a major advance in solid-state physics in the first half of the 20th century. ${\ displaystyle {\ tfrac {3} {2}} R}$

## literature

• Wolfgang Demtröder: Experimental Physics 3: Atoms, Molecules, Solids. Springer textbook 2005.
• GR Stewart: Measurement of low-temperature specific heat . In: Review of Scientific Instruments . tape 54 , no. 1 , 1983, p. 1-11 , doi : 10.1063 / 1.1137207 .
• Michael Tausch: Chemistry SII, Substance - Formula - Environment. CC Buchners Verlag, Bamberg 1993, ISBN 978-3-7661-6453-7 .
• Gustav Kortüm: Introduction to chemical thermodynamics. Verlag Chemie, Basel 1981, ISBN 3-527-25881-7 (or Vandenhoeck & Ruprecht, Göttingen 1981, ISBN 3-525-42310-1 ).
• Walter J. Moore, Dieter O. Hummel: Physical chemistry. Verlag de Gruyter, Berlin / New York 1986, ISBN 3-11-010979-4 .
• David R. Lide: Handbook of Chemistry and Physics. 59th edition. CRC Press, Boca Raton 1978, ISBN 978-0-8493-0486-6 , pp. D-210, D-211.
• Callen: Thermodynamics and an Introduction to Thermostatistics. Wiley & Sons. ISBN 978-0-471-86256-7 .