Molecular Velocity and its Distribution

Molecular Velocity and its Distribution

• Molecular velocity is defined as the speed of an individual molecule within a gas sample.
• The distribution of molecular velocities was studied by Maxwell and Boltzmann.
• This distribution depends on temperature and the molecular weight of the gas.
• The distribution function in three dimensions is expressed by the equation: f(v) = (m/(2πkT))^3/2 × e^(-mv2/2kT) × 4πv², where:
• f(v) is the probability distribution function.
• m is the mass of a molecule.
• k is the Boltzmann constant.
• T is the absolute temperature.
• v is the molecular speed.
• The constant k equals R/NA (R is the ideal gas constant, NA is Avogadro's number)
• A simplified form of the equation is: f(v) = (M/(2πRT))^3/2 × e^(-Mv2/2RT) × 4πv², where M is the molar mass of the gas.

Features of the Molecular Velocity Distribution Curve

• Probability of molecular speed increases with increasing speed, peaks, then decreases.
• As temperature rises the most probable speed also rises and the curve broadens, but retains its shape.
• At the same temperature, lighter molecules have a higher fraction of molecules with higher speeds.

Effect of Temperature on the Curve

• Higher temperatures result in broader curves, with the peak shifting towards higher speeds. This indicates that as temperature increases, more molecules have higher speeds.

Effect of Molecular Weight on the Curve

• Heavier molecules have a lower fraction of high-speed molecules compared to lighter molecules at the same temperature.

Velocities Associated with the Distribution Curve

• Most Probable Velocity (vp): The velocity possessed by the maximum number of molecules at a given temperature. Mathematically, vp = √(2RT/M).
• Mean Velocity (v): The average of all velocities of molecules in a sample. Mathematically, v = √(8RT/πM).
• Root-Mean-Square Velocity (vrms): The square root of the average of the squares of the molecular velocities. Mathematically, vrms = √(3RT/M).

Worked Example (Methane)

• Methane (CH4)'s most probable (vp), mean (v), and root-mean-square (vrms) velocities were calculated at 40°C using relevant formulas.

Kinetic Energy and Temperature

• The product of pressure and volume (PV) of a gas is directly proportional to its average kinetic energy. The formula PV = (1/3)(n/N_A)mv² links PV to kinetic energy.
• Another formulation of the relationship between pressure-volume and kinetic energy is PV = (2/3)(nRT), where "n" is the number of moles. 
• Average kinetic energy is directly proportional to absolute temperature (KE = (3/2)kT) and kinetic energy is limited to translational motion at absolute zero.

Deductions from KE equation (15)

• Average kinetic energy is directly proportional to temperature.
• At absolute zero temperature, molecular motion ceases.
• The average kinetic energy is limited to translational motion at absolute zero.