Heat capacity

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Corresponding Wikipedia article: Heat capacity


The Debye phonons are the photons, which have in addition to two orthogonal waves a third "degree of freedom" in a form of the longitudinal magnetic wave, which exists within a body. The phonons are not the conventional mechanical waves, as Debye thought, but they obey the photons (bosons) statistics.

The particle wave oscillations obviously exist at the magnetic wavelengths more than the doubled distance \((\lambda_{MIN}⁄2)\) between the particles. Therefore, the number of standing waves per volume unit has a limit of \((8⁄\lambda_{MIN}^3)\), that is the approximate number of particles per the volume unit: \[n=\frac{8}{\lambda_{MIN}^3}=\frac{\omega_{MAX}^3}{\pi^3c^3}\tag{1}\] \[\omega_{MAX}=c\pi\sqrt[3]{n}\tag{2}\] The value \(\omega_{MAX}\), which is used in the Debye model, with difference of 24% from (2), is also obtained by integrating the spectral density of the standing waves ("Waves", 7): \[n=\int_0^{\omega_{MAX}}\frac{\mathrm{d}n}{\mathrm{d}\omega}\mathrm{d}\omega=\int_0^{\omega_{MAX}}\frac{\omega^2\mathrm{d}\omega}{2\pi^2c^3}=\frac{\omega_{MAX}^3}{6\pi^2c^3}\tag{3}\] \[\omega_{MAX}=c\sqrt[3]{6\pi^2n}\tag{4}\] The average particle energy \(\hbar\omega_{MAX}\) corresponds to the experimentally determined Debye temperature \(\Theta\), which is constant for a particular substance, because the body is solid at this temperature, and the concentration of particles in a volume depends almost only on the temperature. The equation is: \[\hbar\omega_{MAX}=k\;\Theta\tag{5}\] The volumetric density of the system internal energy, as the integral of the energy spectral density ("Quantum mechanics", 6) of the phonons with three "degrees of freedom", is: \[u=3\int_0^{\omega_{MAX}}\frac{\omega^2}{2\pi^2c^3}\frac{\hbar\omega}{\exp(\hbar\omega/kT)-1}\mathrm{d}\omega\tag{6}\] Applying the substitutions \(x=\hbar\omega/kT\) and \(ω_{MAX}=c\sqrt[3]{6\pi^2n}\), the (6) becomes: \[u=3\frac{k^4T^4}{2\pi^2\hbar^3c^3}\int_0^{\Theta/T}\frac{x^3}{e^x-1}\mathrm{d}x=\frac{9nkT^4}{\Theta^3}\int_0^{\Theta/T}\frac{x^3}{e^x-1}\mathrm{d}x\tag{7}\] The internal energy of a substance mole is determined by replacing \(nk\) with \(N_Аk=R\): \[U_M=\frac{9RT^4}{\Theta^3}\int_0^{\Theta/T}\frac{x^3}{e^x-1}\mathrm{d}x\tag{8}\] The molar heat capacity has the form: \[C_M=\frac{\mathrm{d}U_M}{\mathrm{d}T}=\frac{9RT^3}{\Theta^3}\int_0^{\Theta/T}\frac{e^xx^4}{(e^x-1)^2}\mathrm{d}x\tag{9}\] The (8) at \(T\gg\Theta\) approaches to the Dulong–Petit law, which is valid for the high temperatures: \[U_M\approx\frac{9RT^4}{\Theta^3}\int_0^{\Theta/T}\frac{x^3}{x}\mathrm{d}x=3RT\tag{10}\] \[C_M\approx 3R\tag{11}\] Thus, the theory of heat capacity is produced without the equipartition theorem for the degrees of freedom, but only on the basis of the quantum-wave theory.

The metals and other monohydric substances (or those which can be considered such) have the molar heat capacity law, which is close to the Debye model. The deviations are caused by the mechanical phenomena, such as:

  • Non-spherical molecules have a rotational momentum, which is not shown in the waves, but it increases the heat capacity. In Lomonosov’s opinion, the nature of heat is a gyration of the particles.
  • Crystals are very rigid, and this prevents the elastic waves propagation and reduces both the entropy and the heat capacity. For example, the diamond has a very low molar heat capacity.

Adiabatic process

The adiabatic process is a thermodynamic process without a heat exchange between the system and the environment. Such process most effectively converts the gas thermal energy into the kinetic energy of expansion, and vice versa, the compression energy into the thermal energy.

As commonly understood since the times of acceptance of the Lomonosov’s kinetic theory of heat, the received or released heat of a system, according to the law of conservation of energy, is divided into a change in the internal energy and a change in the pressure energy, which performs a mechanical work: \[\Delta Q=\Delta U+P\Delta V\tag{12}\] The energy distribution depends on the process conditions. In the XIX century a physician J. von Mayer, who was a part-time physicist, together with Helmholtz and Joule had uniquely solved this question by substituting the molar quantities \(U=CT\), \(PV=RT\): \[\Delta Q=(C+R)\Delta T\tag{13}\] And he defined also the usual isochoric \(C_V\) (without a work) and the special isobaric \(C_P\) (with a work) molar heat capacity of gases: \[C_V+R=C_P=S\tag{14}\] The ratio of these quantities, which called the adiabatic index, in the experimental adiabatic processes is typically within: \[\gamma=\frac{C_P}{C_V}\tag{15}\] \[1<\gamma<5/3\tag{16}\] The adiabatic index depends on the temperature, the gas chemical compound, and it is maximal for an ideal gas. Further, the molar heat capacities of an ideal gas were determined, which are much smaller (up to 2 times) than the true heat capacity, which was determined by Dulong and Petit (in the XIX century): \[C=С_V=\frac{3C_P}{5}=\frac{3(C_V+R)}{5}=\frac{3R}{2}\tag{17}\] \[С_P=C_V+R=\frac{5R}{2}\tag{18}\] Further, the application of the equipartition theorem for the degrees of freedom gives a fixed energy per any degree of freedom of a particle: \[\xi=\frac{kT}{2}\tag{19}\] As shown in the derivation of the Debye and Dulong-Petit formulas, the theory of particles-oscillators with degrees of freedom is not required to explain the heat laws.

The mechanical work, in general, is accompanied by an equal change in enthalpy (total energy) of a particle system ("Particle system", 1), which does not disappear after the performed work and constrains the reciprocating engines efficiency. In case of the chemical reaction, this work is the Gibbs free energy ("Chemical interaction", 1) absolute value. For example, when the process is isobaric, then the gas energy density (pressure) remains constant, but the particle system is changing in volume.

The correct equation of the law of conservation of energy looks like: \[\Delta Q=\Delta H+A=\Delta U+P\Delta V+A=\Delta U+2P\Delta V\tag{20}\] The reciprocating engine efficiency cannot exceed 50%, because a useful thermal energy is expended fifty-fifty on the gas expansion and on the useful mechanical work. Examples:

The work for a substance mole: \[A=P\Delta V=R\Delta T\tag{21}\] The total system heat capacity, which does a mechanical work: \[\frac{\mathrm{d}Q}{\mathrm{d}T}=C_M+2R\tag{22}\] The adiabatic index or the work effectiveness, as a ratio of the total heat capacity to the gas heat capacity is: \[\gamma=\frac{\mathrm{d}Q}{\mathrm{d}U}=1+\frac{2R}{C_M}\tag{23}\] Since \(C_M\geq 3R\) for most of gases, \(1<\gamma<5⁄3\).

Speed of longitudinal waves

The adiabatic process equation is: \[C_M\Delta T+2P\Delta V=0\tag{24}\] The equation of state for an ideal gas mole and its total differential is: \[PV=RT\tag{25}\] \[P\Delta V+V\Delta P=R\Delta T\tag{26}\] Substitution of \(\Delta T\) from (24) into (26) produces an equation, which is expressed by an adiabatic index (23): \[\gamma\frac{\Delta V}{V}+\frac{\Delta P}{P}=0\tag{27}\] When \(\Delta\rho\ll\rho\) the following transition from a molar volume to a density is valid: \[\frac{\Delta V}{V}=\frac{V_2-V_1}{V}=\frac{\rho}{\rho_2}-\frac{\rho}{\rho_1}\approx-\frac{\rho_2-\rho_1}{\rho}=-\frac{\Delta\rho}{\rho}\tag{28}\] \[\gamma\frac{\Delta\rho}{\rho}=\frac{\Delta P}{P}\tag{29}\] Here’s a known formula for a speed of longitudinal waves ("Waves", 5): \[c=\sqrt{\frac{\Delta P}{\Delta\rho}}\tag{30}\] The expression of (29) by this speed and multiplication by a molar volume \(V\) produces the formula of sound speed, where \(M\) is a molar mass: \[\gamma P=\rho c^2\tag{31}\] \[\gamma PV=Mc^2\tag{32}\] \[c=\sqrt{\frac{\gamma RT}{M}}\tag{33}\] Substitution for the air \(\gamma\) = 1,4; \(M\) = 0,03 kg/mole; gives the sound speed of 337 m/s at 20°C.

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