Probability Density Function (PDF) Concept & Molecular Speed Distribution: Class 11 Physics Notes

Understanding Probability Density Function (PDF) in Physics

In physics, especially when dealing with systems containing a vast number of particles like gases, individual particle behavior is often too complex to track. Instead, we focus on the statistical distribution of properties. A Probability Density Function (PDF), often denoted as f(x) or P(x), describes the relative likelihood for a random variable (like speed, energy, or position) to take on a given value. It's not a direct probability itself, but rather a function such that the probability of the variable falling within a certain range (e.g., between x and x + dx) is given by f(x)dx. For a continuous variable, the sum of probabilities over all possible values must be 1, meaning the area under the PDF curve is always 1. In Class 11, the most relevant application is the Maxwell-Boltzmann Speed Distribution, which is a specific type of PDF for molecular speeds in an ideal gas. It tells us how many molecules are likely to have speeds within a particular range v to v + dv at a given temperature.

Key Definitions

Probability Density Function (PDF)

A function that describes the relative likelihood for a continuous random variable to take on a given value. The probability of the variable being in an interval is found by integrating the PDF over that interval. For molecular speeds, it's f(v).

Maxwell-Boltzmann Distribution

A statistical distribution that describes the speeds of particles in an idealized gas, where the particles move randomly but follow certain probability patterns based on temperature and mass.

Most Probable Speed (vₚ)

The speed possessed by the largest number of molecules in a gas at a given temperature; corresponds to the peak of the Maxwell-Boltzmann distribution curve.

Average Speed (vₐᵥ)

The arithmetic mean of the speeds of all molecules in a gas at a given temperature. Calculated as the sum of all speeds divided by the number of molecules.

Root Mean Square Speed (vᵣₘₛ)

The square root of the average of the squares of the speeds of all molecules. It's a measure of the typical speed of molecules and is directly related to the kinetic energy of the gas.

Kinetic Theory of Gases

A model that describes the macroscopic properties of gases (like pressure, temperature) in terms of the motion of their constituent molecules.

Maxwell-Boltzmann Speed Distribution: The PDF for Molecular Speeds

Must Remember: Key Speed Formulas

• Most Probable Speed (vₚ): The speed at which the PDF curve peaks. Formula: vₚ = √(2RT/M) where R is the gas constant, T is absolute temperature, M is molar mass.
• Average Speed (vₐᵥ): Mean speed of all molecules. Formula: vₐᵥ = √(8RT/πM) = √(2.54RT/M) (approximately).
• Root Mean Square Speed (vᵣₘₛ): Related to kinetic energy. Formula: vᵣₘₛ = √(3RT/M) = √(3P/ρ) where P is pressure, ρ is density.
• Relationship between Speeds: vₚ : vₐᵥ : vᵣₘₛ = √2 : √(8/π) : √3 ≈ 1 : 1.128 : 1.224. Thus, vₚ < vₐᵥ < vᵣₘₛ.
• Units: Always use SI units: Temperature (T) in Kelvin, Molar Mass (M) in kg/mol (convert g/mol to kg/mol by dividing by 1000), R = 8.314 J mol⁻¹ K⁻¹.
• Kinetic Energy: Average translational KE per molecule = (3/2)kT where k is Boltzmann constant. Total KE of 1 mole = (3/2)RT.
• The PDF curve shows that very few molecules have extremely low or extremely high speeds; most molecules have speeds clustered around the most probable speed.

Worked Examples

• {"title":"Example 1: Calculating RMS Speed","bodyMarkdown":"Calculate the RMS speed of oxygen molecules (O₂) at 27°C.\n Given: T = 27°C = 300 K. Molar mass of O₂ (M) = 32 g/mol = 0.032 kg/mol. R = 8.314 J mol⁻¹ K⁻¹.\n Formula: vᵣₘₛ = √(3RT/M)\n Calculation: vᵣₘₛ = √(3 8.314 J mol⁻¹ K⁻¹ * 300 K / 0.032 kg/mol)\n vᵣₘₛ ≈ √(2494.2 / 0.032) ≈ √77943.75 ≈ 279.18 m/s"}
• {"title":"Example 2: Comparing Speeds","bodyMarkdown":"If the most probable speed of a gas is 400 m/s, what would be its approximate RMS speed?\n Relationship: vᵣₘₛ / vₚ = √3 / √2 ≈ 1.224 / 1.0 = 1.224 (or simply vᵣₘₛ = vₚ √(3/2))\n Calculation: vᵣₘₛ = 400 m/s √(3/2) = 400 * 1.2247 ≈ 489.88 m/s"}

Section 6

Board Exam Trap: Students often forget to convert molar mass from g/mol to kg/mol and temperature from °C to Kelvin. Always double-check units before plugging values into formulas. Also, clearly distinguish between vₚ, vₐᵥ, and vᵣₘₛ – they are distinct values. A common conceptual question asks how the distribution curve changes with temperature or molecular mass; draw and explain the shift and flattening.

Practice Questions with Solutions

• Q1: How does the Maxwell-Boltzmann distribution curve change if the temperature of the gas is increased? A1: The peak shifts to the right (higher speeds), and the curve becomes broader and flatter, indicating a wider range of speeds and fewer molecules at the most probable speed.
• Q2: Which speed is the highest: most probable, average, or RMS speed? A2: The Root Mean Square (RMS) speed (vᵣₘₛ) is the highest, followed by average speed (vₐᵥ), and then most probable speed (vₚ).
• Q3: What does the area under the Probability Density Function (PDF) curve represent? A3: The total area under any PDF curve for a continuous variable is always 1 (or 100%), representing the sum of probabilities for all possible values of the variable.
• Q4: If two gases, Hydrogen (H₂) and Oxygen (O₂), are at the same temperature, which will have a higher RMS speed and why? A4: Hydrogen (H₂) will have a higher RMS speed because vᵣₘₛ ∝ 1/√M. Since Hydrogen has a much smaller molar mass (M) than Oxygen, its molecules will move faster on average at the same temperature.

Frequently Asked Questions