page_title: Molar heat capacity
section_title: Frozen and active degrees of freedom
hierarchical_section_title: Molar heat capacity / Physical basis of molar heat capacity / Molar heat capacity of polyatomic gases / Frozen and active degrees of freedom
caption_attribution_description: An idealized plot of the dimensionless specific heat of a diatomic gas. Below Trot, the molecules behave as point particles with no internal degrees of freedom. Between Trot and Tvib, the particle behaves as a rigid body with two rotational degrees of freedom. Above Tvib, the two vibrational degrees of freedom become active.
context_page_description: The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of a sample of the substance divided by the amount of substance of the sample; or also the specific heat capacity of the substance times its molar mass. The SI unit of specific heat is joule per kelvin per mole, J⋅K⁻¹⋅mol⁻¹.
  Like the specific heat, measured the molar heat capacity of a substance, especially a gas, may be significantly higher when the sample is allowed to expand as it is heated than when is heated in a closed vessel that prevents expansion. The ratio between the two, however, is the same heat capacity ratio obtained from the corresponding specific heat capacities.
  This property is most relevant in chemistry, when amounts of substances are often specified in moles rather than by mass or volume. The molar heat capacity generally increases with the molar mass, often varies with temperature and pressure, and is different for each state of matter.
context_section_description: According to classical mechanics, a diatomic molecule like nitrogen should have more degrees of internal freedom, corresponding to vibration of the two atoms that stretch and compress the bond between them.
  For thermodynamic purposes, each direction in which an atom can independently vibrate relative to the rest of the molecule introduces two degrees of freedom: one associated with the potential energy from distorting the bonds, and one for the kinetic energy of the atom's motion.  In a diatomic molecule like N
  ₂, there is only one direction for the vibration, and the motions of the two atoms must be opposite but equal; so there are only two degrees of vibrational freedom.  That would bring f up to 7, and cV,m to 3.5 R.
  The reason why these vibrations are not absorbing their expected fraction of heat energy input is provided by quantum mechanics.  According to that theory, the energy stored in each degree of freedom must increase or decrease only in certan amounts (quanta).  Therefore, if the temperature T of the system is not high enough, the average energy that would be available for some of the theoretical degrees of freedom (kT/f) may be less than the corresponding minimum quantum.  If the temperature is low enough, that may be the case for practically all molecules.  One then says that those degrees of freedom are "frozen". The molar heat capacity of the gas will then be determined only by the "active" degrees of freedom — that, for most molecules, can receive enough energy to overcome that quantum threshold.
  For each degree of freedom, there is an approximate critical temperature at which it "thaws" ("unfreezes") and becomes active, thus being able to hold heat energy. For the three translational degrees of freedom of molecules in a gas, this critical temperature is extremely small, so they can be assumed to be always active.  For the rotational degrees of freedom, the thawing temperature is usually a few tens of kelvins (although with a very light molecule such as hydrogen the rotational energy levels will be spaced so widely that rotational heat capacity may not completely "unfreeze" until considerably higher temperatures are reached). Vibration modes of diatomic molecules generally start to activate only well above room temperature.
  In the case of nitrogen, the rotational degrees of freedom are fully active already at −173 °C (100 K, just 23 K above the boiling point).  On the other hand, the vibration modes only start to become active around 350 K (77 °C)  Accordingly, the molar heat capacity cP,m is nearly constant at 29.1 J⋅K⁻¹⋅mol⁻¹ from 100 K to about 300 °C.  At about that temperature, it starts to increase rapidly, then it slows down again.  It is 35.5 J⋅K⁻¹⋅mol⁻¹ at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.  The last value corresponds almost exactly to the predicted value for f = 7.
  The following is a table of some constant-pressure molar heat capacities cP,m of various diatomic gases at standard temperature (25 °C = 298 K), at 500 °C, and at 5000 °C, and the apparent number of degrees of freedom f* estimated by the formula f* = 2cP,m/R − 2:
  (*) At 59 C (boiling point)
  The quantum harmonic oscillator approximation implies that the spacing of energy levels of vibrational modes are inversely proportional to the square root of the reduced mass of the atoms composing the diatomic molecule. This fact explains why the vibrational modes of heavier molecules like Br
  ₂ are active at lower temperatures.  The molar heat capacity of Br
  ₂ at room temperature is consistent with f = 7 degrees of freedom, the maximum for a diatomic molecule. At high enough temperatures, all diatomic gases approach this value.
caption: Constant-volume specific heat capacity of a diatomic gas (idealised). As temperature increases, heat capacity goes from 3/2R (translation contribution only), to 5/2R (translation plus rotation), finally to a maximum of 7/2R (translation + rotation + vibration)
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