9.2.1 Energy Distribution of Particles

• What does the Maxwell-Boltzmann Distribution describe?

  Particle speeds in a gas
• The Maxwell-Boltzmann Distribution formula includes variables such as speed, mass, Boltzmann constant, and temperature
• Higher temperatures cause the Maxwell-Boltzmann distribution to shift to higher speeds.
• Match the temperature with its effect on the Maxwell-Boltzmann distribution:

  Low temperature ↔️ Lower probable speed and narrow spread

  High temperature ↔️ Higher probable speed and wide spread
• What does the x-axis of the Maxwell-Boltzmann curve represent?

  Particle speed
• The y-axis of the Maxwell-Boltzmann curve represents the probability density
• The area under the Maxwell-Boltzmann curve indicates the fraction of particles with speeds within a certain range.
• Match the axis of the Maxwell-Boltzmann curve with its representation:

  x-axis ↔️ Particle speed

  y-axis ↔️ Probability density
• What is the most probable speed on the Maxwell-Boltzmann curve?

  $v_{p}$
• The area under the Maxwell-Boltzmann curve represents the total number of particles
• The integral of the Maxwell-Boltzmann distribution over all speeds equals the total number of particles.
• Arrange the features of the Maxwell-Boltzmann curve from most to least likely to be observed:

  1️⃣ Bell-shaped curve

  2️⃣ Most probable speed

  3️⃣ Spread of speeds

  4️⃣ Area under the curve

  5️⃣ Shift with temperature
• What happens to the Maxwell-Boltzmann curve when the temperature increases?

  It shifts to higher speeds
• The spread of the Maxwell-Boltzmann curve indicates the range of particle speeds
• What does the Maxwell-Boltzmann curve depict?

  Distribution of particle speeds
• The Maxwell-Boltzmann curve peaks at the most probable speed
• The spread of the Maxwell-Boltzmann curve indicates the range of particle speeds
• What does the area under the Maxwell-Boltzmann curve represent?

  Total number of particles
• Higher temperatures shift the Maxwell-Boltzmann curve to higher speeds
• What mathematical concept is used to describe the area under the Maxwell-Boltzmann curve?

  Integral
• The average kinetic energy of particles is proportional to the absolute temperature
• What is activation energy ($E_{a}$)?


  Minimum energy to react
• The area under the Maxwell-Boltzmann curve to the right of the activation energy indicates the fraction of particles with sufficient energy to react
• What does the Maxwell-Boltzmann Distribution formula describe?

  Particle speed probabilities
• The Maxwell-Boltzmann Distribution formula includes the Boltzmann constant
• What does the x-axis of the Maxwell-Boltzmann curve represent?

  Particle speed
• The y-axis of the Maxwell-Boltzmann curve represents the probability density
• What does the Maxwell-Boltzmann curve show?

  Particle speed distribution
• The most probable speed $(v_{p})$ corresponds to the highest point on the curve
• At higher temperatures, the Maxwell-Boltzmann curve shifts to the right
• What does the area under the Maxwell-Boltzmann curve represent mathematically?

  Total number of particles
• Order the characteristics of the Maxwell-Boltzmann curve based on increasing temperature.

  1️⃣ Low temperature, lower most probable speed, narrow spread

  2️⃣ High temperature, higher most probable speed, wide spread
• What is the mathematical relationship between average kinetic energy and temperature?

  $E_{k} =$$\frac{3}{2}kT$
• Activation energy is the minimum energy required for reactants to overcome the energy barrier
• What equation relates activation energy to reaction rate?

  Arrhenius equation
• On the Maxwell-Boltzmann curve, the area to the right of activation energy indicates the fraction of particles with enough energy to react.
• Higher temperatures increase the number of particles with sufficient kinetic energy, leading to a higher reaction rate
• What is the gas constant $R$ in the Arrhenius equation?


  8.314 J/mol·K
• The reaction rate is directly proportional to the number of particles with energy greater than activation energy.
• The Arrhenius equation is expressed as $k =$$Ae^{ - \frac{E_{a}}{RT}}$, where $k$ is the reaction rate