Degrees of Freedom and Molar Specific Heats

Molecule	Example	Degrees of Freedom			Predicted Molar Specific heats
		Translational	Rotational	Total (f)	Cv	Cp = Cv + R
Monatomic	He	3	0	3	3/2 R	5/2 R
Diatomic	O₂	3	2	5	5/2 R	7/2 R
Polyatomic	CH4	3	3	6	3 R	4 R

The above table shows the prediction that Cv = 3/2R agrees with experiment for monatomic gases but fails for diatomic and polyatomic gases. Let us try to explain the discrepancy by considering the possibility that molecules with more than one atom can store internal energy in forms other than translational motion.

The above figure shows common models of helium (a monatomic molecule, containing a single atom), oxygen (a diatomic molecule, containing two atoms) and methane (a polyatomic molecule). From such models, we would ensure that all three types of molecules had translational motion (say, moving left-right and up-down) and rotational motion (spinning about an axis like a top). In addition, we assume that the diatomic and polyatomic molecules have oscillatory motion, with the atoms oscillating slightly away from one another, as if attached to opposite ends of a spring.

To keep account of the various ways in which energy can be stored in a gas, James Clerk Maxwell introduced the theorem of the Equipartition of energy.

Every kind of molecule has a certain number of degrees of freedom, which are independent ways in which the molecule can store energy. Every such degree of freedom has, associated with it, on an average, energy of kT per molecule ( RT per mole)

Sub Topics

Degrees of freedom
Concept of mean free path
Avogadro's number
One mole is the number of atoms in a 12g sample of carbon-12
Zeroth law of thermodynamics

Degrees of freedom

The total number of coordinates or independent quantities required to completely specify the position and configuration (arrangement of constituent particles in space) of a dynamic system is the degrees of freedom of that system.

The total number of possible independent ways in which the position and configuration of a mechanical system may change, is the degrees of freedom of that system.

The total number of independent ways in which the particles of a system can take up energy, is the degrees of freedom of that system.

Consider a particle whose motion is confined along a straight line, say along X-axis. Its position at any instant is completely known by its displacement along X-axis. Therefore, it has one degree of freedom.

If a particle is free to move in a plane, its position at any instant is completely known by its displacement along X-axis and Y-axis. Therefore, it has two degrees of freedom.

If a particle is constrained to move in space, its motion can be resolved along the three rectangular axes. Therefore, the position of the particle at any instant could be determined by its displacement along the three axes. Thus, the particle has three degrees of freedom.

If a system consists of two free particles, then the system has six degrees of freedom. This is because each free particle has three degrees of freedom. If the distance between the two particles remains fixed, then there is a definite relation between them. In this case, the number of coordinates necessary for describing the position and configuration of the system completely, is reduced by one. Therefore, the system has five degrees of freedom.

In general, the number of degrees of freedom of a mechanical system is equal to the total number of coordinates required to specify the positions of all the constituent particles minus the number of independent relations between the constituent particles. If A be the number of particles in a system and R the number of independent relations between them, then the number N of degrees of freedom is given by

N = 3A - R

Concept of mean free path

According to the assumptions of kinetic theory of gases, the free path traveled between two successive collisions will be a straight line with constant velocity. This means the path of a single gas molecule consists of a series of short zigzag path of different lengths. The mean of the length of these paths is the mean free path.

If x₁, x₂, x₃...........x_n are the successive path lengths traveled by a gas molecule in a total time t, then

and n the number of collisions in time t sec.

mean free path

It can be shown

Where D is the diameter of the molecule, P is pressure, T the temperature, K the Boltzmann Constant. The concept of mean free path is of importance in understanding phenomena like diffusion, viscosity and thermal conduction.

Avogadro's number

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It makes sense to measure the sizes of our samples in moles. If we do so, we can be certain that we are comparing samples that contain the same number of atoms or molecules. The mole is one of the seven SI base units and is defined as follows:

One mole is the number of atoms in a 12g sample of carbon-12

The number N_A, given by N_A = 6.023x10²³mol^-¹ is called Avogadro's number, named after Italian scientist Amadeo Avogadro (1776-1856), who suggested that all gases contain the same number of atoms or molecules when they occupy the same volume under the same conditions of temperature and pressure.

The number of moles, n contained in a sample of any substance is equal to the ratio of the number of molecules N in the sample to the number of molecules N_A in 1 mole:

-----(i)

In the equation (i), we used the fact that the mass M of 1 mole is the product of the mass m of one molecule and the number of molecules N_A in 1 mol:

-----(ii)

Zeroth law of thermodynamics

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Two bodies are in thermal equilibrium if no transfer of heat takes place when they are in contact. We can now state the Zeroth law of thermodynamics as follows:

If two bodies A and B are in thermal equilibrium and A and C are also in thermal equilibrium, then B and C are also in thermal equilibrium.

It is a matter of observation and experience. But, in the Zeroth law, it is not obvious. For example, if two people, A and B know each other and A and C know each other, it is not necessary that B and C know each other.

The Zeroth law allows us to introduce the concept of temperature to measure the hotness or coldness of a body. All bodies in thermal equilibrium are assigned equal temperature. A hotter body is assigned a higher temperature than a colder body. Thus, the temperatures of two bodies decide the direction of heat flow when the two bodies are in contact. Heat flows from the body at higher temperature to the body at lower temperature.