# What is the concentrated heat capacity method

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The Heat capacity indicates how much thermal energy \$ \ Delta E_ {th} \$ a body can store in relation to the temperature change \$ \ Delta T \$:

\$ C = \ frac {\ Delta E_ {th}} {\ Delta T} \$

The unit is \$ [C] = \ mathrm {\ tfrac {J} {K}} \$ (Joule per Kelvin).

A distinction must be made between the following terms:

• Specific heat capacity\$ c \$, which is related to the mass \$ [c] = \ mathrm {\ tfrac {J} {kg \ cdot K}} \$
• Molar heat capacity\$ C_ \ mathrm {mol} \$ (also Molar heat), which is related to the amount of substance \$ [C_ \ mathrm {mol}] = \ mathrm {\ tfrac {J} {mol \ cdot K}} \$
• Heat storage number\$ s \$, which is related to the volume \$ [s] = \ mathrm {\ tfrac {J} {m ^ {3} \ cdot K}} \$

### definition

The heat capacity is a term from thermodynamics and describes the ability of a body to store energy in the form of thermal energy, statistically distributed over the degrees of freedom. The heat capacity does not apply across phase boundaries (see heat of fusion and heat of vaporization) and is more or less dependent on the temperature within an aggregate state, which makes a differential consideration useful.

\$ C_X = \ left. \ Frac {\ mathrm {d} E_ {th}} {\ mathrm {d} T} \ right | _X \$

Note that the heat \$ E_ {th} \$ is not a state variable and therefore does not have a total differential. The supplied heat \$ \ delta E_ {th} \$ depends on the temperature \$ T \$ and entropy \$ S \$ via \$ \ delta E_ {th} = \ mathrm {d} U - \ delta W = \ mathrm {d} U + p \ mathrm {d} V \$, so that the following applies to the heat capacity:

\$ C_X = \ left. \ Frac {\ mathrm {d} U + p \ mathrm {d} V} {\ mathrm {d} T} \ right | _X \$

It is necessary that the process of heating takes place quasi-statically (i.e. very slowly) so that irreversible phenomena do not play an essential role during the process. You should therefore be more precise about the Equilibrium heat capacity speak.

In general, the external conditions under which the body is heated (pressure, temperature, etc.) play a role. At constant pressure, for example, work is done at the same time in the form of thermal expansion of the body, which, due to the conservation of energy, leads to greater heat absorption per temperature unit. A distinction is therefore made between the heat capacity for a constant volume, the isochoric heat capacity\$ C_V \$, and for constant pressure that isobaric heat capacity\$ C_p \$.

\$ C_ {V} = T \ left. \ Frac {\ partial S} {\ partial T} \ right | _ {V, N} = \ left. \ Frac {\ partial U} {\ partial T} \ right | _ {V, N} \$
\$ C_ {p} = T \ left. \ Frac {\ partial S} {\ partial T} \ right | _ {p, N} = \ left. \ Frac {\ partial H} {\ partial T} \ right | _ {p, N} \$

\$ U (S, V, N) \$ is the internal energy with \$ \ mathrm {d} U = T \ mathrm {d} Sp \ mathrm {d} V + \ mu \ mathrm {d} N \$ and \$ H ( S, p, N) \$ the enthalpy with \$ \ mathrm {d} H = T \ mathrm {d} S + V \ mathrm {d} p + \ mu \ mathrm {d} N \$.

If the body is physically homogeneous, it makes sense to indicate the heat capacity per unit of mass (or unit of material), which is then the specific heat capacityc (when referring to 1 kg of a substance) or molar heat or molar heat capacity \$ c_ \ text {n} \$ (when referring to 1 mol of a substance).

The physical unit of the heat capacity results from its definition as [J / K], that of the specific heat capacity as [J / (kg · K)] or [J / (mol · K)] depending on the related material unit.

The temperature of the substance also has an influence on the (specific) heat capacity. This can be seen in the following example:

• Water at 20 ° C: \$ c = 4 {,} 190 ~ \ mathrm {kJ / (kg \, K)} \$
• Water at 15 ° C: \$ c = 4 {,} 186 ~ \ mathrm {kJ / (kg \, K)} \$

A mean specific heat capacity can be used for solids in small temperature ranges. At very low temperatures, the values ​​are extremely low and approach zero near absolute zero.

### determination

The heat capacity of a substance is determined with the help of a calorimeter, for example with dynamic differential calorimetry (DSC). It is important for the measurement that the sample to be examined does not have any reaction associated with a heat tone in the temperature range to be examined. A comparison measurement with a sapphire is usually carried out for an exact calculation. If no experimental measured values ​​are available, the heat capacity of many substances can also be estimated using group contribution methods such as the Benson method.

### General relationships

The general equation for heat capacity is:

\$ \ Delta E_ {th} = C \, \ Delta T \$

The equation relating heat, mass, temperature change and specific heat capacity is:

\$ \ Delta E_ {th} = c \, m \, \ Delta T \$

And the equation for molar heat capacity is:

\$ \ Delta E_ {th} = C_ {mol} \, n \, \ Delta T \$

where \$ \ Delta E_ {th} \$ is the heat that is added to or withdrawn from the matter, \$ m \$ is the mass of the substance, \$ n \$ is the amount of substance, \$ c \$ is the specific heat capacity and \$ \ Delta T \$ is the temperature change.

The different heat capacities are related to:

\$ C = c \, m = C_ {mol} \, n \$

About the relationship between mass and amount of substance

\$ m = n \, M \$

with the molar mass \$ M \$ the relationship between molar (\$ C_ \ mathrm {mol} \$) and specific (\$ c \$) heat capacity follows:

\$ C_ \ mathrm {mol} = c \, M \$

Using the example of copper, the result is: \$ C_ \ mathrm {mol} = \ mathrm {0 {,} 38 \, \ mathrm {J / (g \ cdot K)} \ cdot 63 \, \ mathrm {g / mol} = 24 \, \ mathrm {J / (mol \ cdot K)}} \$

### Linking with other thermodynamic quantities

As described below, \$ C_p> C_V \$ is always. The two variables are linked to one another via the following relationships:

\$ C_p - C_V = TV \ frac {\ alpha ^ 2} {\ kappa_T} \$
\$ \ frac {C_p} {C_V} = \ frac {\ kappa_T} {\ kappa_S} = \ kappa \$

Here \$ \ alpha = \ tfrac {1} {V} (\ tfrac {\ partial V} {\ partial T}) _ p \$ is the thermal expansion coefficient, \$ \ kappa_T = - \ tfrac {1} {V} (\ tfrac {\ partial V} {\ partial p}) _ T \$ the isothermal compressibility, \$ \ kappa_S = - \ tfrac {1} {V} (\ tfrac {\ partial V} {\ partial p}) _ S \$ the isentropic resp. adiabatic compressibility, \$ \ kappa \$ the isentropic exponent and \$ T \$ the absolute temperature

### Heat capacity of solids Course of the heat capacity of iron with temperature
material Debye temperature
aluminum 426 K
magnesium 406 K
iron 464 K
copper 345 K
tin 195 K

For solids, the Dulong-Petit law is fulfilled to a good approximation for heavy elements and high temperatures, which has a constant molar heat capacity of \$ C_ \ mathrm {mol} = 3 R \ approx 25 \ mathrm {J / (mol \ cdot K) } \$ predicts for the solid.

This model fails at low temperatures. In this area, the Debye model predicts a \$ T ^ 3 \$ dependence of the heat capacity. According to the Debye model, the molar heat capacity is determined as a function of the temperature by only one substance quantity, the so-called Debye temperature \$ \ Theta_ \ mathrm {D} \$:

\$ c_V (T) = 9R \ cdot \ left (\ frac {T} {\ Theta_ \ mathrm {D}} \ right) ^ 3 \ int_0 ^ {\ frac {\ Theta_D} {T}} \ frac {x ^ 4 \ cdot \ mathrm e ^ x} {\ left (\ mathrm e ^ x-1 \ right) ^ 2} \ mathrm dx \$

The forerunner of the Debye model is the Einstein model, which is too imprecise for practical applications, especially at low temperatures. This article or the following section is not adequately provided with supporting documents (e.g. individual evidence). The data in question are therefore possibly soon removed. Please help Wikipedia by researching the information and adding good evidence. More details may be given on the talk page or in the version history. Finally, please remove this warning mark.

In the case of ladders, there is a linear dependence of the heat capacity on the temperature in the low temperature limit. To understand this, one can understand the thermally conductive conduction electrons as an ideal Fermi gas. With the help of the Fermi distribution and the density of states of a free electron, the temperature dependence of energy and consequently also heat capacity can be calculated for low temperatures. The resulting result is far lower than a first estimate of the heat capacity of the conduction electrons as an ideal gas (see below) would lead one to expect and is considered to be a major advance in solid-state physics in the first half of the 20th century.

### Linking the heat capacity C at constant pressure \$ C_p \$ and constant volume \$ C_V \$

In the case of gases in particular, the heat capacity depends on the external constraints. A distinction is made between the heat capacity at constant pressureC.p and at constant volume C.V.. In the case of isochoric changes in state, the entire amount of heat supplied is used to increase the temperature of the gas (i.e. to increase the kinetic energy of the gas particles). In isobaric processes, on the other hand, volume work has to be done, since the gas has to expand when heated if the pressure is to remain constant. This means that at constant pressure, part of the supplied thermal energy is “consumed” in the form of volume work. Therefore, in the case of isobaric changes in state, more energy has to be supplied in the form of heat to heat a gas by one degree than in the case of isochoric changes in state.

The following applies to ideal gases:

\$ \, C_p = C_V + N \, k_B = C_V + n \, R \$.

Here \$ N \$ is the number of particles, \$ k_B \$ the Boltzmann constant, \$ n \$ the amount of substance and \$ R = N_A \, k_B \$ the universal gas constant. For 1 mol this gives:

\$ \, C_ {mol, p} = C_ {mol, V} + R \$

or.

\$ \, C_ {mol, p} - C_ {mol, V} = R \$.

After introducing the specific gas constant \$ R_s = \ tfrac {R} {M} \$ the result is

\$ c_p - c_V = R_s \$.

example - Hydrogen (\$ H_2 \$) at 20 ° C and 1.013 bar, experimentally measured values:

• with constant pressure: \$ c_p = 14 {,} 24 \, \ mathrm {kJ / (kg \, K)} \$
• with constant volume: \$ c_V = 10 {,} 1 \, \ mathrm {kJ / (kg \, K)} \$

The difference between the experimentally measured quantities \$ c_p - c_V = 4 {,} 14 \, \ mathrm {kJ / (kg \, K)} \$ instead of \$ c_p - c_V = \ tfrac {R} {M} = 4 {,} 124 \, \ mathrm {kJ / (kg \, K)} \$ from theoretical considerations can be explained by the fact that hydrogen is not a one hundred percent ideal gas.

### Ab initio calculation of \$ C_V \$

In order to estimate the heat capacity of ideal gases, it is possible to calculate it using the degrees of freedom. According to the uniform distribution theorem, the total energy \$ E \$ of a gas consisting of \$ N \$ identical particles (atoms, molecules) is \$ E = \ tfrac {f} {2} N k_B T = \ tfrac {f} {2} n R T \$. From \$ C_ {V} = (\ tfrac {\ partial E} {\ partial T}) _ {V, N} \$ follows:

\$ C_V = \ frac {f} {2} n R \$ or \$ C_ {mol, V} = \ frac {f} {2} R \$

where \$ f \ geq 3 \$ indicates the number of energetic degrees of freedom of a molecule. Every atom has three degrees of freedom, namely the translations in the three spatial dimensions. The degrees of freedom of a molecule include:

• \$ f_ \ mathrm {trans} = 3 \$, three degrees of freedom for the translation,
• \$ f_ \ mathrm {rot} \ in \ {0,1,2,3 \} \$, zero to three degrees of freedom for the rotational energy and
• \$ f_ \ mathrm {vib} = 3A - f_ \ mathrm {trans} - f_ \ mathrm {rot} \$ degrees of freedom for the internal vibration energy. A. is the total number of atoms in the molecule.

A diatomic molecule \$ A = 2 \$ has \$ f_ \ mathrm {trans} = 3 \$, \$ f_ \ mathrm {rot} = 2 \$ and \$ f_ \ mathrm {vib} = 3 \ cdot 2 - 3 - 2 = 1 \$ . A triatomic angled molecule \$ A = 3 \$ has \$ f_ \ mathrm {trans} = 3 \$, \$ f_ \ mathrm {rot} = 3 \$ and \$ f_ \ mathrm {vib} = 3 \$. Individual atoms \$ A = 1 \$ have only three translational degrees of freedom \$ f_ \ mathrm {trans} = 3 \$.

The following applies to gases whose components are individual atoms:

\$ C_ {mol, V} = \ frac {3} {2} R \$

For gases whose constituents are \$ A \$ -atomic molecules:

\$ C_ {mol, V} = \ frac {1} {2} R \ left (f_ \ mathrm {trans} + f_ \ mathrm {rot} + 2 \, f_ \ mathrm {vib} \ right) \$ with \$ \ , f_ \ mathrm {trans} + f_ \ mathrm {red} + f_ \ mathrm {vib} = 3A \$

Each translational or rotational degree of freedom thus contributes with \$ \ tfrac {1} {2} R \$ to the molar heat capacity, each vibration degree of freedom with \$ R \$. It should be noted that all possible rotations are usually excited at room temperature (IR spectrum), while vibrations (i.e. oscillations), on the other hand, often only to a small extent (UV spectrum). In the borderline case of complete neglect of the vibrations one obtains:

\$ C_ {mol, V} = \ frac {1} {2} R \ left (f_ \ mathrm {trans} + f_ \ mathrm {red} \ right) \$
\$ C_V \$\$ C_p \$\$ \ kappa = \ frac {C_p} {C_v} \$
1-atom gases \$ \ frac {3} {2} \ cdot R \$\$ \ frac {5} {2} \ cdot R \$\$ \ frac {5} {3} = 1 {,} \ overline {6} \$
2-atom gases \$ \ frac {5} {2} \ cdot R \$\$ \ frac {7} {2} \ cdot R \$\$ \ frac {7} {5} = 1 {,} 4 \$
3-atom gases \$ \ frac {6} {2} \ cdot R \$\$ \ frac {8} {2} \ cdot R \$\$ \ frac {4} {3} = 1 {,} \ overline {3} \$

example - Hydrogen (\$ H_2 \$) at room temperature: \$ f_ \ mathrm {trans} = 3; \$ \$ f_ \ mathrm {red} = 2; \$ \$ 2f_ \ mathrm {vib} = 2 \$

The specific gas constant \$ R_s = \ tfrac {R} {M} \$ is used:

without vibrations: \$ c_V = \ frac {5} {2} R_s = 10 {,} 3 \, \ mathrm {\ frac {kJ} {kg \, K}} \$ and \$ c_p = \ frac {7} {2 } R_s = 14 {,} 4 \, \ mathrm {\ frac {kJ} {kg \, K}} \$
with vibrations: \$ c_V = \ frac {7} {2} R_s = 14 {,} 4 \, \ mathrm {\ frac {kJ} {kg \, K}} \$ and \$ c_p = \ frac {9} {2 } R_s = 18 {,} 6 \, \ mathrm {\ frac {kJ} {kg \, K}} \$

By comparing these numbers with the measured values ​​\$ c_V = 10 {,} 1 \, \ mathrm {kJ / (kg \, K)} \$ and \$ c_p = 14 {,} 24 \, \ mathrm {kJ / (kg \ , K)} \$ you can see that vibrations are not excited here.

### Isentropic exponent

The isentropic exponent \$ \ kappa \$ of the ideal gas is:

\$ \ kappa = \ frac {C_p} {C_V} = \ frac {C_ {mol, p}} {C_ {mol, V}} = \ frac {\ tfrac {f} {2} R + R} {\ tfrac {f} {2} R} = \ frac {f + 2} {f} \$

If the isentropic exponent \$ \ kappa \$ of the ideal gas is known, the heat capacities can be calculated from the combination of \$ C_ {mol, p} - C_ {mol, V} = R \$ and the following formulas:

\$ \ kappa = \ frac {C_p} {C_V} \ quad \ Rightarrow \ quad C_V = \ frac {C_p} {\ kappa} \ quad \ Rightarrow \ quad R = C_p - C_V = C_p - \ frac {C_p} {\ kappa} = C_p \ left (1 - \ frac {1} {\ kappa} \ right) \ quad \ Rightarrow \ quad C_p = \ frac {\ kappa \, R} {\ kappa - 1} \$

### Negative capacity (stars) This article or the following section is not adequately provided with supporting documents (e.g. individual evidence). The data in question are therefore possibly soon removed. Please help Wikipedia by researching the information and adding good evidence. More details may be given on the talk page or in the version history. Finally, please remove this warning mark.

Most physical systems show one positive Heat capacity. Even if the opposite seems absurd at first, systems can also have a negative heat capacity. These include gravitating objects such as collapsing stars, which heat up when they contract (internal energy is reduced), or very small systems Cluster) from a few hundred atoms close to a phase transition.This strange property is related to thermodynamic stability. Only systems with a positive heat capacity can be regarded as stable in the thermodynamic sense and these can therefore also be regarded as extensive quantities.

### literature

• G. R. Stewart: Measurement of low-temperature specific heat. In: Rev. Sci. Instrum. No. 54, 1983, pp. 1–11 (doi: 10.1063 / 1.1137207)
• Michael Exchange: Chemistry SII, substance - formula - environment. C.C. Buchners Verlag, Bamberg 1993, ISBN 978-3766164537
• 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 9780849304866, pages D-210, D-211.
• Calling: Thermodynamics and an Introduction to Thermostatistics. Wiley & Sons. ISBN 978-0471862567