The Maxwell–Boltzmann Distribution ✏ Chemistry In a Nutshell

Rucete ✏ Chemistry In a Nutshell

1. Key Concepts of the Maxwell–Boltzmann Distribution

• Not all gas molecules have the same kinetic energy at a given temperature.
• The velocity distribution follows a bell-shaped curve (some move slower, some faster).
• Higher temperatures shift the distribution to higher speeds and flatten the curve (broader range of speeds).

2. Root Mean Square (RMS) Velocity ($u_{rms}$)

• Represents the average speed of gas molecules.
• Formula:$$u_{rms}=\sqrt{\frac{3RT}{M}}$$
• $R$ = 8.314 J/(K·mol) (gas constant)
• $T$ = Temperature (Kelvin)
• $M$ = Molar mass (kg/mol)

3. Kinetic Energy & Temperature Relationship

• Kinetic energy per molecule:$$K{E}_{\text{avg}}=\frac{1}{2}m{v}^2$$
  
• Kinetic energy per mole of gas:$$K{E}_{\text{avg}}=\frac{3}{2}RT$$
  
• Key takeaway: Temperature is directly proportional to the average kinetic energy of gas particles.

In a nutshell

• Higher temperature → More molecules have higher speeds.
• Lighter molecules (lower molar mass) → Move faster on average.
• Heavier molecules (higher molar mass) → Move slower on average.