Revision: Class 11 >> Kinetic Theory of Gases NEET (UG) 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 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")`

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 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 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).

Specific heat capacity is defined as the amount of heat per unit mass absorbed or given out by the substance to change its temperature by one unit (one degree) 1 °C or 1K.

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

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 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 reactant which is completely used up in a reaction is known as Limiting reagent or Limiting reactant.

An atom is the smallest particle of an element that can take part in a chemical reaction; however, it may or may not exist independently. 

A molecule is the smallest particle of an element or a compound that can exist by itself; it never breaks up except for taking part in a chemical reaction.

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:

P = \[\frac{1}{3}\frac{MN}{V}v_{rms}^2=\frac{1}{3}\rho v_{rms}^2\]

s = \[\frac {ΔQ}{(m · ΔT)}\]

Where,

When the amount of substance is measured in moles (μ) instead of kilograms, we use molar specific heat capacity (C):

C = \[\frac {1}{μ}\] · \[\frac {ΔQ}{ΔT}\]

SI Unit: J · mol⁻¹ · K⁻¹

f = 3A − B

where:

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

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.

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.

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: For a given mass of gas at constant temperature, the volume of a gas is inversely proportional to its pressure.

• Constants held: mass of gas, temperature.
• P-V graph: Hyperbolic curve; P vs 1/V graph: straight line through origin.
• P increases → V decreases proportionally.

"When gases combine or are produced in a chemical reaction, they do so in a simple ratio by volume, provided all gases are at the same temperature and pressure."

• Proposed by Gay-Lussac in 1808.
• e.g. 100 mL H₂ + 50 mL O₂ → 100 mL H₂O vapour (ratio = 2 : 1 : 2).
• The volume ratio of gaseous reactants to products agrees with their molar ratio.
• Volume of a gas is directly proportional to the number of moles (not inversely).

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

where β = pressure expansion coefficient = 1/273 per °C.

"Equal volumes of all gases at the same temperature and pressure contain equal numbers of molecules."

• Proposed by Avogadro in 1811.
• 1 mole of any gas at STP = 22.4 L (at 0°C, 1 atm) or 22.71 L (at 0°C, 1 bar — new IUPAC STP).
• 1 mole of any substance = 6.022 × 10²³ particles.

Avogadro's Law (Volume–Moles Relationship):

At constant temperature (T) and pressure (P), volume is directly proportional to number of moles.

• Specific heat capacity (s) measures how much heat per unit mass is needed to change a substance's temperature by 1°C (or 1 K).
• Formula: ΔQ = m · s · ΔT — applies only when there is no phase change.
• Water has the highest specific heat (4186 J/kg·K) among common substances — this is why it's used in cooling systems and retains heat well.
• Molar specific heat (C) measures heat per mole. For gases, it splits into Cₚ (constant pressure) and Cᵥ (constant volume), with Cₚ > Cᵥ always.
• Specific heat is intrinsic; it depends on the material, not on the amount of substance taken.
• The concept explains everyday phenomena such as coastal climate moderation, engine cooling, cooking times, and therapeutic hot water bags.