Proportionality in Rates

Worked Example: Concentration Calculation

A reaction with $0.2\text{ mol/dm}^3$ hydrochloric acid has a rate of $0.04\text{ g/s}$. Analyse the expected rate if the concentration is increased to $0.6\text{ mol/dm}^3$.

Step 1: Identify the proportionality factor.

• The concentration increases from $0.2\text{ mol/dm}^3$ to $0.6\text{ mol/dm}^3$.
• This is a $3\text{x}$ increase ($0.6 / 0.2 = 3$).

Step 2: Apply the factor to the rate.

• Because $Rate \propto Concentration$, the rate must also increase by a factor of 3.

Step 3: Calculate the new rate.

• New rate = $0.04 \times 3 = 0.12\text{ g/s}$.

Surface Area and Proportionality

• You can easily snap a thin twig with two fingers, but breaking a thick branch of the same wood requires a saw.
• For solid reactants, the rate of reaction is directly proportional to the exposed surface area. Dividing a large solid chunk into smaller pieces increases the surface area to volume ratio.
• Doubling the available surface area of a solid reactant exactly doubles the number of exposed particles. This mathematically results in a doubling of the collision frequency and a doubling of the reaction rate.

Worked Example: Surface Area Calculation

A solid cube with $40\text{ mm}$ sides is cut into eight smaller cubes with $20\text{ mm}$ sides. Analyse how this mathematical change affects the rate of reaction.

Step 1: Calculate the surface area of the original cube.

• Area of one face = $40 \times 40 = 1,600\text{ mm}^2$.
• Total surface area = $1,600 \times 6 = 9,600\text{ mm}^2$.

Step 2: Calculate the total surface area of the eight smaller cubes.

• Area of one small face = $20 \times 20 = 400\text{ mm}^2$.
• Surface area of one small cube = $400 \times 6 = 2,400\text{ mm}^2$.
• Total surface area = $8 \times 2,400 = 19,200\text{ mm}^2$.

Step 3: State the proportionality.

• The total surface area has doubled (from $9,600\text{ mm}^2$ to $19,200\text{ mm}^2$).
• Therefore, the collision frequency and rate of reaction will also double.

Why Temperature is Different

• Unlike concentration, pressure, and surface area, heating up a reaction does not follow a simple proportional doubling rule.
• Temperature does not show direct proportionality; instead, it has an exponential, non-linear relationship. A rough "10-degree rule" states that for every $10^\circ\text{C}$ (or $10\text{K}$) increase in temperature, the rate of reaction approximately doubles.
• Temperature increases the rate through a dual mechanism. While particles do move faster (slightly increasing collision frequency), the major effect is that a much higher proportion of particles exceed the activation energy, meaning collisions are far more likely to be successful.

Extracting Rate from Curved Graphs

• How do you find the exact rate of reaction when the speed is constantly changing over time?
• On a curved concentration-time graph, the rate is not constant. To analyse the rate at a specific instant, you must draw a tangent (a straight line that touches the curve at exactly one point).
• You then calculate the gradient of this tangent using the formula $m = \frac{\Delta y}{\Delta x}$, which provides the rate at that specific time.
• In contrast, if you are looking at a Rate vs. Concentration graph and see a perfectly horizontal line, it indicates that changing the concentration has zero effect on the reaction rate.