Maxwell Boltzmann Distribution Pogil Answer | Key Extension Questions
Answer: At very low speeds, very few molecules have exactly zero velocity because kinetic energy is quantized in terms of molecular motion; also, the probability density function ( f(v) \propto v^2 e^-mv^2/(2kT) ) gives ( f(v) \to 0 ) as ( v \to 0 ).
Prompt: At the same temperature, compare the M-B distribution for ( Cl_2 ) (M=70 g/mol) and ( Cl ) (M=35 g/mol). Which has a higher fraction of molecules exceeding a given velocity ( v )?
Answer: The lighter molecules (Cl atoms) have a higher fraction exceeding any given velocity.
Reasoning: Average kinetic energy ( \frac12mv^2 = \frac32kT ) is the same for both at a constant T. Because ( KE ) is the same, lighter particles must move faster on average to achieve that same energy. Their distribution curve is shifted to the right and is broader. Therefore, for a fixed velocity threshold, more light particles exceed it. This explains the kinetic isotope effect in reactions.
The extension questions in the Maxwell-Boltzmann Distribution POGIL typically focus on the mathematical relationships between temperature, molar mass, and molecular speed.
Here are the conceptual explanations for the common extension questions found in this activity: 1. The Effect of Temperature on the Peak Answer: At very low speeds, very few molecules
As temperature increases, what happens to the height of the peak and its position on the x-axis? As temperature increases, the peak (the most probable speed ) shifts to the (higher velocity). Simultaneously, the height of the peak (flattens). Reasoning:
Since the total area under the curve represents 100% of the molecules, if the distribution spreads out to include higher speeds, the peak must lower to maintain the same total area. 2. Comparing Different Gases (Molar Mass) If you have Nitrogen ( cap N sub 2 ) and Helium (
) at the same temperature, which will have a broader distribution? will have the broader, flatter distribution. Reasoning:
At a constant temperature, all gases have the same average kinetic energy ( ). Because Helium has a much smaller mass ( ), it must have a much higher velocity (
) to maintain that energy. Lighter gases spread out more across the velocity axis. 3. Activation Energy and Reaction Rates Mark a line for "Activation Energy" ( cap E sub a Suggested extension: Have students derive (v_mp) from the
) on the graph. How does increasing temperature affect the number of molecules capable of reacting?
Increasing the temperature significantly increases the area under the curve to the right of the cap E sub a Reasoning:
Even a small shift in the average temperature leads to a disproportionately large increase in the fraction of molecules with enough energy to overcome the activation barrier, which is why reaction rates increase so sharply with heat. 4. Mathematical Proportions How does the root-mean-square speed ( v sub r m s end-sub ) change if the Kelvin temperature is quadrupled? Reasoning: According to the formula , the velocity is proportional to the square root of the temperature ( 5. Area Under the Curve
What does the total area under any Maxwell-Boltzmann curve represent? The total number of particles (or 100% of the sample). Reasoning:
The statement is approximately true but not strictly true for a real gas. Answer: At very low speeds
This question tests whether students confuse the distribution of energy with the required threshold.
The Distinction:
Graphical Interpretation:
The Consequence: While both methods increase the number of successful collisions, a catalyst does so without increasing the average speed of molecules. This means a catalyst avoids side reactions that occur at high temperatures (e.g., decomposition of reactants).
POGIL Acceptable Answer: "A catalyst does not alter the Maxwell-Boltzmann distribution (the curve does not change). It lowers the activation energy threshold, so a larger fraction of the existing molecules have sufficient energy to react. Temperature changes the shape of the distribution curve itself."
Question:
For a reaction with activation energy ( E_a ), how does increasing temperature affect the fraction of molecules with kinetic energy ( \ge E_a )?
Reasoning & Answer:
