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In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure (CP) to heat capacity at constant volume (CV). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas[note 1] or κ (kappa), the isentropic exponent for a real gas. The symbol γ is used by aerospace and chemical engineers.

where C is the heat capacity, C \displaystyle \bar C the molar heat capacity (heat capacity per mole), and c the specific heat capacity (heat capacity per unit mass) of a gas. The suffixes P and V refer to constant-pressure and constant-volume conditions respectively.

To understand this relation, consider the following thought experiment. A closed pneumatic cylinder contains air. The piston is locked. The pressure inside is equal to atmospheric pressure. This cylinder is heated to a certain target temperature. Since the piston cannot move, the volume is constant. The temperature and pressure will rise. When the target temperature is reached, the heating is stopped. The amount of energy added equals CV ΔT, with ΔT representing the change in temperature.

The piston is now freed and moves outwards, stopping as the pressure inside the chamber reaches atmospheric pressure. We assume the expansion occurs without exchange of heat (adiabatic expansion). Doing this work, air inside the cylinder will cool to below the target temperature.

To return to the target temperature (still with a free piston), the air must be heated, but is no longer under constant volume, since the piston is free to move as the gas is reheated. This extra heat amounts to about 40% more than the previous amount added. In this example, the amount of heat added with a locked piston is proportional to CV, whereas the total amount of heat added is proportional to CP. Therefore, the heat capacity ratio in this example is 1.4.

Another way of understanding the difference between CP and CV is that CP applies if work is done to the system, which causes a change in volume (such as by moving a piston so as to compress the contents of a cylinder), or if work is done by the system, which changes its temperature (such as heating the gas in a cylinder to cause a piston to move). CV applies only if P d V = 0 \displaystyle P\,\mathrm d V=0 , that is, no work is done. Consider the difference between adding heat to the gas with a locked piston and adding heat with a piston free to move, so that pressure remains constant.

In the second case, the gas will both heat and expand, causing the piston to do mechanical work on the atmosphere. The heat that is added to the gas goes only partly into heating the gas, while the rest is transformed into the mechanical work performed by the piston.

In the first, constant-volume case (locked piston), there is no external motion, and thus no mechanical work is done on the atmosphere; CV is used. In the second case, additional work is done as the volume changes, so the amount of heat required to raise the gas temperature (the specific heat capacity) is higher for this constant-pressure case.

For an ideal gas, the molar heat capacity is at most a function of temperature, since the internal energy is solely a function of temperature for a closed system, i.e., U = U ( n , T ) \displaystyle U=U(n,T) , where n is the amount of substance in moles. In thermodynamic terms, this is a consequence of the fact that the internal pressure of an ideal gas vanishes.

For a diatomic gas, often 5 degrees of freedom are assumed to contribute at room temperature since each molecule has 3 translational and 2 rotational degrees of freedom, and the single vibrational degree of freedom is often not included since vibrations are often not thermally active except at high temperatures, as predicted by quantum statistical mechanics. Thus we have

As noted above, as temperature increases, higher-energy vibrational states become accessible to molecular gases, thus increasing the number of degrees of freedom and lowering γ. Conversely, as the temperature is lowered, rotational degrees of freedom may become unequally partitioned as well. As a result, both CP and CV increase with increasing temperature.

Despite this, if the density is fairly low and intermolecular forces are negligible, the two heat capacities may still continue to differ from each other by a fixed constant (as above, CP = CV + nR), which reflects the relatively constant PV difference in work done during expansion for constant pressure vs. constant volume conditions. Thus, the ratio of the two values, γ, decreases with increasing temperature.

However, when the gas density is sufficiently high and intermolecular forces are important, thermodynamic expressions may sometimes be used to accurately describe the relationship between the two heat capacities, as explained below. Unfortunately the situation can become considerably more complex if the temperature is sufficiently high for molecules to dissociate or carry out other chemical reactions, in which case thermodynamic expressions arising from simple equations of state may not be adequate.

Values for CP are readily available and recorded, but values for CV need to be determined via relations such as these. See relations between specific heats for the derivation of the thermodynamic relations between the heat capacities.

For an imperfect or non-ideal gas, Chandrasekhar[3] defined three different adiabatic indices so that the adiabatic relations can be written in the same form as above; these are used in the theory of stellar structure:

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A diatomic gas is hydrogen gas. It has 5 degrees of freedom in terms of vibration. The adiabatic index of hydrogen gas, like that of any other diatomic gas, is 1.4. The adiabatic index is a measure of a gas's ability to extract work adiabatically using its own internal energy. Despite its similarities to other diatomic gases, H2 has 14 times the specific heat of N2. The reason for this is that 2 grams of H2 gas contain the same number of molecules as 28 grams of N2, 6.203x10^23 molecules. H2 gas, as a molecule, can store the same amount of energy as any other diatomic gas molecule. On a mass basis, however, H2 gas has fourteen times the capacity to store heat.

In the graph, only H2 has the z >1 and it is across all pressure that shows a positive deviation from the other ideal gases. All other gases, if we take a cut-off point of 200 atm show negative deviation from ideal gas behavior.

This is essentially the compression of H2 and that raises the temperature. Therefore, you need to cool H2 gas below the inversion temperature and bring the molecules closer before compression. While cooling you essentially slow down the molecules

Partially ionized plasmas are found in many different astrophysical environments. The study of partially ionized plasmas is of great interest for solar physics because some layers of the solar atmosphere (photosphere and chromosphere) as well as solar structures, such as spicules and prominences, are made of these kinds of plasmas. To our knowledge, despite it being known that the adiabatic coefficient, γ, or the first adiabatic exponent, Γ1, depend on the ionization degree, this fact has been disregarded in all the studies related to magnetohydrodynamic waves in solar partially ionized plasmas. However, in other astrophysical areas, the dependence of γ or Γ1 on the plasma ionization degree has been taken into account. Therefore, our aim here is to study how, in a plasma with prominence physical properties, the joint action of the temperature, density, and ionization degree modifies the numerical values of the first adiabatic exponent Γ1 which affects the adiabatic sound speed and the period of slow waves. In our computations, we have used two different approaches; first of all, we assume local thermodynamic equilibrium (LTE) and, later, we consider a non-local thermodynamic equilibrium (non-LTE) model. When comparing the results in the LTE and non-LTE cases, the numerical values of Γ1 are clearly different for both and they are probably strongly dependent on the assumed model which determines how the ionization degree evolves with temperature. Finally, the effect of the ionization degree dependence of Γ1 on the period of slow waves has been determined showing that it can be of great importance for seismological studies of partially ionized solar structures.

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The thermodynamics of partially ionized gas (Ballester et al. 2018a) differs from that of fully ionized or fully neutral gas. The differences are due to the number of free particles as well as to the fact that the energy associated with ionization and recombination becomes an extra energy source or sink for the gas. Then, in the most general case, in the expression of the specific internal energy, apart from kinetic energy, the energies stored (or released) from ionization (recombination), excitation (de-excitation), dissociation, etc. should also be taken into account. The adiabatic coefficient, γ, depends on the state of the plasma through the internal energy; therefore, it depends on the ionization fraction and its ionization is due to the fact that part of the energy input is invested in ionization, instead of an increasing gas temperature. Therefore, the consideration of a constant value for γ would overestimate the gas temperature and it is very relevant to determine how the numerical value of the adiabatic coefficient changes, for instance, with the ionization degree (Clayton 1984; Prialnik 2000; Hansen et al. 2004; Leake & Arber 2006; Priest 2014; Ballester et al. 2018b). Then, under some assumptions, a general expression for the adiabatic coefficient, γ, as a function of the ionization degree can be obtained (Prialnik 2000).

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