Revision: Behaviour of Perfect Gases and Kinetic Theory of Gases Physics (Theory) ISC (Science) ISC Class 11 CISCE

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 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 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 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 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 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 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.

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 volume is called specific heat at constant volume (C_v).

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

λ = `"kT"/(sqrt(2)π"d"^2"p")`

f = 3A − B

where:

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

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

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:

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.