Gaseous State B.Sc. 2nd Semester Notes

Critical Phenomenon and Critical Constants

Derivation of Critical Constants

Law of Corresponding State

Postulates of Kinetic Theory of Gases

Derivation of Kinetic Theory of Gases

Law of Equipartition Energy

Viscosity and Coefficient of Viscosity

Effect of Temperature on Viscosity

Effect of Pressure on Viscosity

Surface Tension

Effect of Temperature on Surface Tension

Derivation of Real Gas Equation or Vander Waal Equation

Van der Waals observed that the failure of PV = nRT equation for real gasses is due to the neglect of the following two factors-
1. Volume of gas molecules is negligible in comparison with the total volume of the gas.
2. The gas molecules exert no attraction upon one another. Van der Waals also made necessary corrections in the ideal gas equation, keeping above facts in mind as-

1. Volume Correction

The inner volume (V) of the container in gas is placed, is taken as the volume of the gas in which gas molecules are free to move. Since gas molecules have actually some volume, a part of volume V called the excluded volume is not available for free movement of molecules. If the excluded volume for one mole of a gas be 'b'(called van der Waals constant), then actual volume of the gas will be (V-b) instead of V. Then the real gas equation for one mole becomes-
P(V - b) = RT
and for n moles, it becomes-
P(V – nb) = nRT.

2. Pressure Correction

We know that the pressure of a gas is the force per unit area exerted when the random moving molecules collide with the inner walls of the container. The inter molecular attractive forces become more significant at higher pressure when the gas molecules come closer and closer to one another. Then the colliding molecules must differ in the the velocities. Therefore, molecules dont strike the inner walls with the same force consequently pressure decreases so observed pressure will be less than the ideal pressure.
The attractive force was found to be proportional to the square of the number of moles (n) per unit volume i.e. n²V².
Or the attractive force = an²V² where 'a' is the proportionality constant called vander waal constant.
Hence, for 'n' and one mole of real gas, the vander waal equation having necessary correction in ideal gas equation becomes-
This equation is called Vander Waal Equation.

Deviation of Real Gas From Ideal Gas Behavior

The deviation of real gas from ideal gas behaviour occurs due to the postulates of kinetic theory of gases. Two posulates of the kinetic theory do not hold good. These are-
1. There is no force of attraction between the molecules of a gas. and
2. Volume of the molecules of a gas is negligibly small in comparison to the space occupied by the gas.
If postulate '1' is correct, the gas will never liquify. However, we know that gases do liquify when cooled and compressed. Also, liquids formed are very difficult to compress. This means that forces of repulsion are powerful enough and prevent squashing of molecules in tiny volume.
If postulate '2' is correct, the pressure vs volume graph of experimental data (real gas) and that theoritically calculated from Boyles law (ideal gas) should coincide.
Real gases show deviations from ideal gas law because molecules interact with each other. At high pressures molecules of gases are very close to each other. Molecular interactions start operating. At high pressure, molecules do not strike the walls of the container with full impact because these are dragged back by other molecules due to molecular attractive forces. This affects the pressure exerted by the molecules on the walls of the container. Thus, the pressure exerted by the gas is lower than the pressure exerted by the ideal gas.
P_ideal = P_real + an²/V²
Where, a is a constant.
Repulsive forces also become significant. Repulsive interactions are short-range interactions and are significant when molecules are almost in contact. This is the situation at high pressure. The repulsive forces cause the molecules to behave as small but impenetrable spheres. The volume occupied by the molecules also becomes significant because instead of moving in volume V, these are now restricted to volume (V – nb) where nb is approximately the total volume occupied by the molecules themselves. Here, b is a constant. Having taken into account the corrections for pressure and volume is Ideal Gas Equation becomes-
(P + an²/V²) (V − nb) = nRT -----(equation-1)
(equation-1) is known as van der Waals equation. In this equation n is number of moles of the gas. Constants a and b are called van der Waals constants and their value depends on the characteristic of a gas. Value of ‘a’ is measure of magnitude of intermolecular attractive forces within the gas and is independent of temperature and pressure.
The deviation from ideal behaviour can be measured in terms of compressibility factor Z, which is the ratio of product PV and nRT.
Z = PV/nRT
when-
Z = 1 (Ideal Gas at all Temperature)
Z = >1 (Positive deviation from Ideal Gas Above their Boyle Temperature)
Z = < 1 (negative deviation from Ideal Gas below their Boyle Temperature)

Maxwell law of distribution of velocities for gas molecules

We know that gases molecules can not maintain the same velocity for any length of time as it collides with another molecules. However, there must be a fraction of the total number of gas molecules that have a particular velocity at any time. This fraction was obtained by Maxwell and Boltzman using probability considerations.
They have shown that the distribution of molecular velocities depends upon the temperature and molecular weight of gas molecules is given by the expression-
dnc/n = 4π(M/2πRT)^3/2 e^{−(Mc2/2RT).c2dc}
where, dnc is the number of molecules out of a tota number of 'n' molecules which have velocities between c and c+dc, T is temperature and M is the molecular weight of the gas. The ration dn_c/n gives the fraction of the total number of molecules having velocities between c and c+dc.
Dividing both sides in the above equation by dc we get-
1/n(dnc/dc) = 4π(M/2πRT)^3/2e^{−(Mc2/2RT).c2}
The LHS expression give the probability of finding the molecules with velocity c. If the fraction of the total number of molecules for any gas of known molecular weight at any perticular temperature is plotted against the velocities, a distribution curve is looks like-


The curve shows that the fraction of molecules having velocities between c and c+dc is given by the arc under the strip of the curve within the velocity range. The area between two ordinates separated by dc is equal to dnc/n, hence, the area under the whole curve is equal to the total number of molecules and the height of curve says that the fraction of molecules having too low or too high velocities is very small and the majority of the gas molecules have some intermediate velocity with a small range of variation more or less around the peak, known as the most probable velocity.

Effect of Temperature on the distribution of molecular velocities

Root Mean Square Velocity(C_rms)

It is defined as the square root of the mean value of the squares of the velocities of all the molecules at a particular temperature.
example: If a container contains a gas molecules having n molecules and at a particular moment the velocities of the molecules are C₁, C₂, C₃ ... C_n
then-

From kinetic gas equation-
C²=3PV/M=3RT/M
or, C_rms=√3RT/M
where M is the molecular weight of the gas.

Average Velocity (C_avg)

It is defined as the mean of the velocities of all molecules at a particular temperature.
C_avg=√8RT/πM

Most Probable Velocity (C_mpv)

The velocity possessed by the largest number of gas molecules at a particular temperature is called most probable velocity.
C_mpv=√2RT/M
where M is the molecular weight of the gas.

Relationship among the RMSV, AV and MPV

From the equation
C_mpv : C_avg : C_rms=1 : 1.128 : 1.225
Average Velocity=0.921 X C_rms
C_mpv=0.816 X C_rms

Equipartition Energy

Law of equipartition energy states that for a dynamical system in thermal equilibrium the total energy of the system is shared equally by all the degrees of freedom.
The RMS velocity of gas molecules can be resolved into its components along x, y and z axis as -
c² = c_x² + c_y² + c_z²
multiplying by 1/2m, we get-
1/2mc² = 1/2mc_x² + 1/2mc_y² + 1/2mc_z²
or, E = E_x + E_y + E_z -----(1) (as K.E.(E) = 1/2mc²)
We know that gas molecules do not have any preferential direction. So, velocities along all three axes are equally probable-
c_x = c_y = c_z
or, E_x = E_y = E_z
or, E = 3E_x -----(2)
Each energy term in component contributes equally to the total energy. This is called the law of equipartition of energy.
We know that-
PV = 1/3 mnc²
PV = 2/3 X 1/2 mnc²=2/3 K.E.
or, K.E.=3 X 1/2 PV
or, E = 3 X 1/2 RT (as PV = RT for one mole)
From equation (2), we have-
3 E_x = 3 X 1/2 RT
so, E_x = E_y = E_z = 1/2 RT
Hence, the average kinetic energy possed by molecules in each component per degree of freedom is 1/2 RT per mole.

Mean Free Path

The gas molecules are always in state of rapid motion colliding with each other. The distance travelled by a gas molecules before colliding with another molecules is called free path and the average length of a large number of such paths is called mean free path. It is denoted by l and is directly proportional to temperature and inversely proportional to pressure.

Collision Number

Collision number is defined as the "number of collisions" per unit time.A collision is an interaction between two or more movable bodies.The collision number is indicated by the symbol of Z. When one molecule involved, the average number of collisions per unit time nearly one second per moles of reactant between reacting molecules is called collision number.

Collision Frequecny

Number of collisions suffered by gas per second per cubic meter of a gas is called collision frequency. It is equal to the ratio of the rms velocity and mean free path of a gas. i.e.
Collision frequency=Number of collision per second
or, Collision frequency=C_rms / mean free path

Compressibility Factor (z)

The compressibility factor is a correction factor which describes the deviation of real gas from ideal gas behaviour. It is useful thermodynamic property for modifying the ideal gas law to account for real gas behaviour.
Z = PV / nRT
If the gas behaves ideally, the value of Z is one. The extent to which Z differs from one is a measure of the extent to which the gas is behaving nonideally.
The term compressibility factor can also be defined as the ratio of the actual volume of a gas to the ideal volume of gas for a given temperature and pressure.
Z = V_actual / V_ideal.
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