RBSE Class 11 Physics Notes Chapter 13 Kinetic Theory

These comprehensive RBSE Class 11 Physics Notes Chapter 13 Kinetic Theory will give a brief overview of all the concepts.

RBSE Class 11 Physics Chapter 13 Notes Kinetic Theory

→ Boyle’s law:
It states that at constant temperature the volume of a given mass of gas is inversely proportional to its pressure
V ∝ \(\frac{1}{P}\) or PV = constant
or P₁V₁ = P₂V₂

→ Charles’ law:
It states that if the pressure remains constant, then the volume of a given mass of a gas increases or decreases by 1/273.15 of its volume at 0°C for each 1°C rise or fall in temperature. Mathematically,
V_t = V₀(1 + \(\frac{t}{273.15}\))
or \(\frac{V}{T}\) = constant
or it states that at constant pressure, the volume of a given mass of a gas is directly proportional to its absolute temperature.

→ Gay Lussac’s law:
At constant volume, the pressure of a given mass of a gas is directly proportional to its absolute temperature.
or \(\frac{P}{t}\) = contains

→ Ideal gas equation: For n moles of a gas,
PV = nRT or \(\frac{P_1 V_1}{T_1}=\frac{P_2 V_2}{T_2}\)
For 1 mol of a gas
PV = RT

 

→ Pressure exerG by a gas:
According to the kinetic theory of gases the pressure exerted by a gas of mass M and volume V or density p is given by
P = \(\frac{1}{3} \frac{M}{V} \bar{v}^2=\frac{1}{3} Iv^2=\frac{1}{3} m n \bar{v}^2\)
Here n is the number of molecules per unit volume, m is the mass of t ^ach molecule and \(\bar{v}^2\) is the root mean square speed.

→ Average kinetic energy of a gas:
Let M be the molecular mass and V the molt ir volume of a gas. Let m be the mass of each mole cule. Then
(i) Mean K.E. per mole of a gas
E = \(\frac{1}{2} m \bar{v}^2=\frac{3}{2} P V=\frac{3}{2} R T=\frac{3}{2} k_B N_A T\)
(ii) Mean K.E. per molecule of a ga s,
\(\bar{E}=\frac{1}{2} m \bar{v}^2=\frac{3}{2} k_B T\)
This is the kinetic interpretation of tei iperature.

→ Avogadro’s law:
It states that equal volume of all gases under similar conditions of temperature and pressure contains equal number of molecules.

→ Graham’s law of diffusion:
It states that the rate of diffusio: n of a gas is inversely proportional to the square r oot of its density.
\(\frac{r_1}{r_2}-\sqrt{\left(\frac{\rho_2}{\rho_1}\right)}\)

→ Dalton’s law of partial pressures:
It states that the total pr essure exerted by a mixture of non reacting gase s occupying a given volume is equal to the sum c if the partial pressures which gas would exert if it alone occupied the same volume at the given temperature.
P = P₁ + P₂ + P₃ + ...

→ Average speed:
It i s defined as the airthmetic mean of the speeds of the molecules of a gas at a given temperature.
\(\vec{v}=\sqrt{\frac{8 k_B T}{\pi m}}=\sqrt{\frac{8 R}{\pi N} \frac{T}{T}}=\sqrt{\frac{8 P V}{\pi M}}\)

→ Root mean square speed:
It is defined as the mean square root of the squares of the speeds of the individual molecules ol F a gas
\(v_{\mathrm{rms}}=\sqrt{\frac{3 k_B T}{m}}=\sqrt{\frac{3 R T}{M}}=\sqrt{\left(\frac{3 P V}{M}\right)}\)

→ Most probable speed:
It is defined as the speed possessed by the maximum number of molecules in a gas sample at a given temperature.
\(v_{m p}=\sqrt{\frac{2 k_B T}{m}}=\sqrt{\frac{2 R T}{M}}=\sqrt{\frac{2 P V}{M}}\)

→ Mean free path:
It is the average distance covered by a molecule between two successive collisions. It is given by
\(\bar{\lambda}=\frac{1}{\sqrt{2} \pi n d^2}\)
where n is the number density and d is the diameter of the molecule.

→ Brownian motion:
The irregular motion of the suspended particles in colloidal solution is known as Brownian motion.

→ Ideal gas equation,
PV = nRT
where, molar number n = \(\frac{N}{N_A}\)

→ Boltzmann constant
k_B = \(\frac{R}{N_A}\) = 1.38 × 10^-28JK^-1

→ Boyle’s law:
PV = constant

→ Pressure of the gas
P = \(\frac{1}{3} \frac{m n \bar{v^2}}{V}=\frac{1}{3} \frac{M}{V} \bar{v}^2=\frac{1}{3} \rho \bar{v}^2\)

→ Root meran square speed of molecules
\(v_{\mathrm{rms}}=\bar{v}=\sqrt{\frac{3 P}{\rho}}=\sqrt{\left(\frac{3 R T}{M}\right)}\)

→ Charles’ law, V ∝ T
or \(\frac{V_1}{T_1}=\frac{V_2}{T_2}\)
where P = constant

→ Gav Lussac’s law P ∝ T or \(\frac{P_1}{T_1}=\frac{P_2}{T_2}\)
where V = constant

→ Graham’s diffusion law, r ∝ \(\frac{1}{\sqrt{\rho}}\)
\(\frac{r_1}{r_2}=\sqrt{\left(\frac{\rho_2}{\rho_1}\right)}\)

→ Dalton’s law of partial pressure
P = P₁ + P₂ + P₃ + .......... + P_N

→ Total kinetic energy of 1 mole gas
E = \(\frac{3}{2}\)RT
\(\frac{E_1}{E_2}=\frac{T_1}{T_2}\)

→ Kinetic energy of 1 molecule
E_k = \(\frac{3}{2}\)k_BT

→ Kinetic energy of any gas molecule
E_k =\(\frac{f n R T}{2}\)
where, f = degree of freedom
n = number of moles

→ Kinetic energy of 1 molecule k_E = \(\frac{f k_B T}{2}\)

→ Specific heat C = \(\frac{Q}{m \Delta T}\)

→ Specific heat at constant volume
C_V = \(\left(\frac{d Q}{d T}\right)_V\)

→ Specific heat at constant pressure
C_p = \(\left(\frac{d Q}{d T}\right)_P\)

→ γ = \(\frac{C_P}{C_V}=\left(1+\frac{2}{f}\right)\)

→ Mean free path λ = \(\frac{1}{\sqrt{2} \pi d^2 n}=\frac{k_B T}{\sqrt{2} \pi d^2 p}\)

→ Ideal gas:
A gas which obeys the ideal gas equation PV = nRT, at all temperatures and pressures is called an ideal gas or perfect gas.

→ Absolute zero:
It is that temperature at which all molecular motion stops.

→ Degrees of freedom:
Total number of independent ways in which the particles of the system can absorb energy.

→ Law of equipartition of energy:
It states that in any dynamical system in thermal equilibrium, the energy is equally distributed amongst its various degrees of freedom and the energy associated with each degree of freedom per molecule is \(\frac{1}{2}\)k_BT.

→ Debye temperature:
The temperature at which the molar specific heat of a solid at constant volume becomes equal to 3R is called Debye temperature.

→ Mean free path:
The mean free path of a gas molecule is defined as the average distance travelled by the molecule between two successive collisions.