Nitrogen, an inert and non-explosive substance, is widely used in scientific research and for industrial purposes. The knowledge of the thermodynamic properties of this gas is essential for the design of equipment for the chemical industry. A wide range of temperature and pressure has been used in the investigation of the thermal properties of nitrogen. Instead of showing tables with almost infinite amounts of data for specific conditions, this articles aims at explaining some of the concepts of thermodynamics and how they could be used for studying the properties of the nitrogen.

## Thermodynamics

Thermodynamics can be defined as the study of the transformations and of the interactions of energy with matter. A molecule is not a rigid body, but rather a space occupied by particles and filled with an electron cloud which can suffer variations in its quantum states. (Think this is difficult? Keep reading; it gets simpler at the end.) Molecules are commonly modeled as a group of spheres of various sizes connected to each other via springs, which represent the forces of attraction and repulsion between the "spheres."

**The Concept of a System**

In thermodynamics, a system is a region with a well-defined boundary. Statistical thermodynamics calculates the functions of a system that consists of a large number of molecules. In this sense, one focuses on the macroscopic properties of the system by calculating averages and probabilities over the dynamical microscopic states. The Boltzmann distribution law expresses the frequency of occurrence of the individual states.

**The Boltzmann Distribution Law**

Considering the energy ε associated with a state of a system, the frequency that this state occurs (probability of occurrence) is proportional to

e ^{-ε/K.T}

The term K.T is known as thermal energy, where K is the Boltzmann constant and T is the absolute temperature (in Kelvin) for the system.

**The Partition Function**

The partition function (q) is a measure of the number (in average) of the possible states of a system that is at a given temperature, at equilibrium. It can be expressed as a sum over the energy levels or states:

q = ∑_{j} e ^{-εj / K.T}

where j represents each state. For a mol of an ideal **diatomic** gas the partition function can be expressed as:

q = (1 / N_{0}!) . (q_{nu} . q_{tr} . q_{in})^{N0}

where N_{0} is the Avogadro's number and q_{nu}, q_{tr} and q_{in} refer to nuclear, translational, and internal contributions (degrees of freedom) to the function, respectively.

At low temperatures, the vibrational-rotational energy — ε^{n}_{v,j} — for **diatomic** molecules can be usually expressed by:

ε^{n}_{v,j }= hc. {(ν + 1/2).ω_{e}^{n} + j.(j+1).Β_{e}^{n }– {(ν + 1/2)^{2}.ω_{e}.χ_{e}^{n} – j^{2}.(j+1)^{2}.D^{n} – (ν + 1/2).j.(j+1).α_{e}^{n}}

where *h* is Planck's constant, *c* is the speed of light, ω_{e}^{n} is the fundamental vibrational frequency of the *nth* electronic state, *D* is the centrifugal stretching constant and *α* is the vibration-rotation coupling constant.

However, this equation cannot be applied at high temperatures. In this case, the thermodynamic properties can be obtained through the second virial coefficient, *B*, for two interacting atoms.

**The Virial Coefficient**

Virial coefficients are used in the virial expansion of the pressure of a system and provide better approximation than the ideal gas law.

Let's consider:

*2N(g) ↔ *N_{2}*(g)*

This reaction has the following equilibrium constant:

K_{p} = (K_{c} / R.T) = (1 / K.T).{ [q(N_{2}) / V] / [q(N) / V]^{2} }

where K_{p} is the equilibrium constant related to the pressure, K_{c} is the equilibrium constant related to the concentration, *V* is the volume of the system, *q(N)* and *q*(N_{2})* *are the partition functions for *N* and N_{2}, respectively.

The virial coefficient *B(T)* is given by:

B(T) = – K_{c}

## References

PHAIR, R. et al. Thermodynamic Properties of Nitrogen Molecules at High Temperatures. *International Journal of Thermophysics*, vol.11, n. 1, 1990.

SYCHEV, V. *Thermodynamic Properties of Nitrogen*. Hemisphere Publishing Corporation, 1987.

JACOBSEN, R. Thermodynamic Properties of Nitrogen from the Freezing Line to 2000K at Pressures to 1000 Pa. *J. Phys. Chem. Ref. Data*, vol. 15, n. 2, 1986.

Nitrogen Gas for the Steel Industry and Metal Production Industry. Praxair