Revision: Kinetic Theory of Gases JEE Main Kinetic Theory of Gases

A gas that follows all gas laws (Boyle's law, Charles' law, Avogadro's principle) and gas equations at every possible temperature and pressure is called an ideal or perfect gas.

The average distance travelled by the molecule between collisions is called mean free path (λ).

λ = `"kT"/(sqrt(2)π"d"^2"p")`

The average distance travelled by a gas molecule between two successive collisions is called the mean free path.

OR

The distance travelled by a gas molecule between two successive collisions, during which it moves in a straight line with constant velocity, is called free path.

The law which states that for any system in thermal equilibrium, the total energy is equally distributed among all its degrees of freedom, with energy \[\frac {1}{2}\]kT associated with each degree of freedom per molecule, is called the Law of Equipartition of Energy.

The amount of heat required to raise the temperature of one mole of an ideal gas by one degree Celsius (or Kelvin) at constant pressure is called specific heat at constant pressure (C_p).

The amount of heat required to raise the temperature of one mole of an ideal gas by one degree Celsius (or Kelvin) at constant volume is called specific heat at constant volume (C_v).

The energy possessed purely by the motion of molecules in an ideal gas, where the molecules are non-interacting and hence there is no potential energy term, making the internal energy of the gas entirely kinetic in nature, is called the kinetic energy (internal energy) of an ideal gas.

The square root of the mean of squares of the speeds of all the molecules of a gas at a given temperature is called root mean square speed.

The certain minimum value of temperature below which an object cannot be cooled, since the average kinetic energy of molecules has a minimum possible value of zero at this point, is called absolute zero.

The number of collisions per second per molecule is called collision frequency.

The total kinetic energy of a gas associated with the translational motion of all its molecules in a volume V is called translatory kinetic energy.

\[E_T=\frac{1}{2}Mv_{rms}^2=\frac{3}{2}PV\]

The total number of coordinates or independent quantities required to describe the position and configuration of a system completely is called degrees of freedom (dof).

OR

The total number of independent modes (translational, rotational, vibrational) in which a system can possess energy — i.e., the number of independent ways in which a molecule or atom can exhibit motion — is called the degree of freedom.

The minimum number of independent coordinates needed to specify the position and configuration of a thermo-dynamical system in space is called the degree of freedom of the system.

The maximum three degrees of freedom corresponding to translational motion is called translational degree of freedom.

The number of degrees of freedom that depends on the structure of the molecule, corresponding to rotational motion, is called rotational degree of freedom.

The degree of freedom exhibited at high temperatures corresponding to vibrational motion is called vibrational degree of freedom.

A temperature scale with absolute zero (zero kelvin) as the starting point is called the absolute scale or the kelvin scale.

The volume of a given mass of a dry gas varies inversely as the pressure and directly as the absolute temperature.

V ∝ \[\frac {1}{P}\] × T or \[\frac {PV}{T}\] = k (constant)

If volume changes from V₁​ to V_2​, pressure from P_1​ to P₂​, and temperature from T₁ to T₂​, then:

\[\frac {P_1V_1}{T_1}\] = \[\frac {P_2V_2}{T_2}\] = k (constant)

The energy possessed by a body due to its state of motion is called its kinetic energy.

The motion of a body in a straight line path is called translational motion.

The kinetic energy of the body due to motion in a straight line is called translational kinetic energy.

If a body rotates about an axis, the motion is called rotational motion.

The kinetic energy of the body due to rotational motion is called rotational kinetic energy or simply rotational energy.

If a body moves to and fro about its mean position, the motion is called vibrational motion.

The kinetic energy of the body due to its vibrational motion is called vibrational kinetic energy or simply vibrational energy.

PV = nRT = Nk_B​T

Unit: Pa . m³

Let λ₁, λ₂, λ₃,…λ_n​ be the distances travelled by a gas molecule during nn collisions respectively, then the mean free path is:

f = 3A − B

where:

• A = number of atoms in the molecule
• B = number of bonds between atoms

K = \[\frac {1}{2}\] mv²

Kinetic Energy = \[\frac {1}{2}\] mass × (velocity)²

Statement:
For a gas in thermal equilibrium at temperature TT, the average energy associated with each molecule for each quadratic term (degree of freedom) is:

where k_B = 1.38 × 10⁻²³ J/K and T = absolute temperature.

Energy Expressions for Different Types of Motion:

1. Translational K.E.:
   ​\[\frac{1}{2}mv_x^2+\frac{1}{2}mv_y^2+\frac{1}{2}mv_z^2\] (3 degrees of freedom — along x, y, z axes)

2. Rotational K.E.:
   \[\frac{1}{2}I\omega_x^2+\frac{1}{2}I\omega_y^2+\frac{1}{2}I\omega_z^2\] (up to 3 degrees of freedom — rotation about x, y, z axes)

3. Vibrational K.E.:
   \[\frac{1}{2}m\dot{u}^2+\frac{1}{2}kr^2\] (2 terms — kinetic and potential energy of vibration)

Each quadratic term contributes \[\frac {1}{2}\]k_BT to the total average energy of the molecule.

Statement: The pressure exerted by a mixture of non-reactive gases is equal to the sum of partial pressures of each component gas present in the mixture.

• Each gas in a mixture exerts the same pressure as if it alone occupied the container.
• Applies only to non-reactive gas mixtures.

Statement: At the same temperature and pressure, the rate of diffusion of gas is inversely proportional to the square root of the density of the gas.

Since v_rms ∝ \[\frac {1}{\sqrt ρ}\]​, rate of diffusion ∝ v_rms.

Statement: If pressure remains constant, the volume of a given mass of gas increases or decreases by 1/273.15 of its volume at 0°C for each 1°C rise or fall in temperature.

• Also: \[\frac {V}{T}\] = \[\frac {m}{ρT}\] = constant and ρT = constant, ρ₁T₁ = ρ₂T₂​.
• V-T graph: straight line; V vs 1/T: hyperbola.

The average energy per molecule of an ideal gas is directly proportional to the absolute temperature T of the gas:

The total pressure of a gaseous mixture equals the sum of the partial pressures of all individual gases.

Rate of diffusion of a gas is inversely proportional to the square root of its molar mass.

\[\frac{r_1}{r_2}=\sqrt{\frac{M_2}{M_1}}\]

\[\text{Rate of diffusion}=\frac{\text{Volume of gas diffused}}{\text{Time required for diffusion}}\]

Statement:

According to the work-energy theorem, the increase in kinetic energy of a moving body is equal to the work done by a force acting in the direction of the moving body.

Proof:

Let a body of mass m be moving with an initial velocity u. When a constant force F is applied to the body along its direction of motion, it produces an acceleration a, and the body's velocity increases from u to v over a distance S.

Force,

F = ma

Work done by the force,

W = F × S

From the equation of motion,

\[v^2=u^2+2aS\Rightarrow S=\frac{v^2-u^2}{2a}\]

Substituting equations (i) and (iii) into (ii):

W = \[ma\times\frac{v^2-u^2}{2a}=\frac{1}{2}m(v^2-u^2)\]

Now,
Initial kinetic energy, K_i = \[\frac {1}{2}\]mu²
Final kinetic energy, K_f = \[\frac {1}{2}\]mv²

Therefore,

W = K_f − K_i

Conclusion:

Work done on the body = Increase in its kinetic energy.
Hence, the work-energy theorem is proved.

• In gases, the intermolecular forces are very weak, causing the molecules to move apart in all directions.
• Gases have no fixed shape and no fixed size — they can be obtained in a vessel of any shape or size.
• Gases expand indefinitely and uniformly to fill any available space.
• Gases exert pressure on their surroundings.