Interpreting Rate of Reaction Graphs for the ESAT

The Fundamentals of Rate Graphs

In chemical kinetics, we monitor how a reaction progresses by measuring the quantity of a substance over time. This quantity can be the mass of a reactant, the volume of a gaseous product, or the concentration of a species in solution. When these data points are plotted on a graph, time is always placed on the $x$ axis, while the quantity being measured is placed on the $y$ axis.

There are two primary types of rate graphs:

1. Product against Time: These graphs start at the origin $(0,0)$ because no product exists at the start. The curve rises as the reaction progresses and eventually plateaus when a limiting reactant is exhausted.
2. Reactant against Time: These graphs start at a maximum value on the $y$ axis. The curve falls as the reactant is consumed and levels off when the reaction stops.

Calculating Rate from the Gradient

The most important principle in interpreting these graphs is that the gradient equals the rate. Because the rate of reaction usually changes as the reaction progresses, we distinguish between three types of measurements:

1. Average Rate

To find the average rate over a specific time interval, you calculate the total change in $y$ divided by the total change in $x$ for that period. For example, if $20\text{ cm}^3$ of gas is produced in the first $40\text{ s}$, the average rate is:

$\text{Average Rate} = \frac{20\text{ cm}^3}{40\text{ s}} = 0.5\text{ cm}^3\text{ s}^{-1}$

2. Instantaneous Rate

To find the rate at one specific moment in time, you must draw a tangent to the curve at that point. A tangent is a straight line that touches the curve at the chosen point and has the same gradient as the curve at that exact moment. You then calculate the gradient of this straight line using two points on the line:

$\text{Gradient} = \frac{y_2 - y_1}{x_2 - x_1}$

3. Initial Rate

The initial rate is the instantaneous rate at $t = 0$. This is often the fastest part of the reaction because the concentrations of reactants are at their highest. On a graph, this is found by drawing a tangent at the origin and calculating its gradient.

The Three Stages of a Rate Curve

A typical rate curve for a reaction that goes to completion exhibits three distinct phases:

1. The Steep Start: The gradient is at its maximum. This indicates a high rate of reaction due to high collision frequency between reactant particles.
2. The Curve Flattens: As reactants are converted into products, their concentration decreases. This leads to fewer successful collisions per second, so the gradient becomes less steep, indicating a slowing rate.
3. The Plateau: The curve becomes a horizontal line (gradient is zero). This means the rate is zero because the reaction has finished, usually because one of the reactants has been completely used up.

Worked Example: Calculating Instantaneous Rate

Question: A student plots a graph of the volume of hydrogen gas produced against time for the reaction between zinc and hydrochloric acid. At $t = 60\text{ s}$, the student draws a tangent. The tangent passes through the points $(20\text{ s}, 10\text{ cm}^3)$ and $(100\text{ s}, 50\text{ cm}^3)$. Calculate the rate of reaction at $60\text{ seconds}$.

Step 1: Identify the coordinates for the gradient calculation from the tangent. $x_1 = 20, y_1 = 10$ $x_2 = 100, y_2 = 50$

Step 2: Apply the gradient formula. $\text{Rate} = \frac{50 - 10}{100 - 20} = \frac{40}{80} = 0.5\text{ cm}^3\text{ s}^{-1}$

Result: The instantaneous rate of reaction at $60\text{ s}$ is $0.5\text{ cm}^3\text{ s}^{-1}$.

Comparing Graphs under Different Conditions

ESAT questions frequently ask you to predict how a curve changes when variables like temperature or concentration are altered.

• Increased Concentration or Pressure: The curve will be steeper at the start because the initial rate is higher. If the amount of limiting reactant remains the same, the curve will plateau at the same final $y$ value but at an earlier time.
• Increased Temperature or Use of a Catalyst: The initial gradient will be steeper. Like concentration, the plateau occurs earlier. The final volume of product remains the same unless the number of moles of reactants was changed.
• Increased Surface Area (e.g., powder instead of lumps): This increases the initial rate, making the starting gradient steeper. The curve reaches the plateau faster but at the same final value.

Worked Example: Changing Reactant Amounts

Question: A reaction between $1.0\text{ g}$ of calcium carbonate lumps and excess $1.0\text{ mol dm}^{-3}$ HCl is plotted. A second experiment is performed using $0.5\text{ g}$ of calcium carbonate powder and the same volume/concentration of HCl. Contrast the two curves.

Analysis:

1. Steepness: The second curve will be steeper because the powder has a larger surface area, increasing the rate of reaction.
2. Final Plateau: The second curve will level off at half the height of the first curve. Since calcium carbonate is the limiting reactant and there is only half the mass, only half the volume of carbon dioxide gas will be produced.