# Concentration-Time Graphs ## Determining Reaction Order from Concentration-Time Graphs - Concentration-time graphs plot how the concentration of a reactant changes as the reaction progresses - The **shape** of the graph reveals the order of reaction with respect to that reactant - Three main patterns exist: zero, first, and second order ### Zero-Order Reactions - In a zero-order reaction, concentration decreases **linearly** with time - The graph is a **straight line** with a negative gradient - The reactant is consumed at a constant rate

For zero-order reactions, the gradient of the concentration-time graph equals both the rate constant $k$ and the rate of reaction

**Key characteristics:** - Straight line decreasing to zero - Constant gradient throughout - Rate of reaction = $k$ (independent of concentration) **Example interpretation:** Imagine a chemical reaction where a metal surface is completely saturated with reactant molecules. Adding more reactant won't speed things up - the reaction proceeds at a fixed rate determined by the available surface area. This gives a zero-order pattern.

### First-Order Reactions - In a first-order reaction, concentration decreases **exponentially** with time - The graph is a **downward curve** that gradually levels off - The rate of concentration decrease slows as concentration falls **Key characteristics:** - Curved line (exponential decay) - Steeper at the beginning, gradually flattening - Rate proportional to concentration: Rate = $k[\text{A}]$ **Example interpretation:** Think of radioactive decay - the number of atoms decaying per second depends on how many radioactive atoms remain. Similarly, in a first-order chemical reaction, the speed at which reactant is consumed depends directly on how much is left.

### Second-Order Reactions - In a second-order reaction, concentration decreases **more steeply** than first-order - The graph is a **steeper downward curve** - Concentration falls rapidly at first, then slows significantly **Key characteristics:** - Steeper curve than first-order - More dramatic change in gradient over time - Rate proportional to concentration squared: Rate = $k[\text{A}]^2$ **Example interpretation:** Consider a reaction requiring two molecules to collide simultaneously. When concentration is high, collisions happen frequently. As concentration drops, finding collision partners becomes dramatically harder - the rate plummets faster than in first-order reactions.

### Comparing All Three Orders The chart below shows all three reaction orders together, making it easier to distinguish their characteristic shapes:

## Summary Table of Concentration-Time Graphs | Order | Graph Shape | Gradient Behaviour | Rate Equation | |-------|-------------|-------------------|---------------| | Zero | Straight line (downward) | Constant | Rate = $k$ | | First | Downward curve (exponential) | Decreases steadily | Rate = $k[\text{A}]$ | | Second | Steep downward curve | Decreases rapidly | Rate = $k[\text{A}]^2$ | ## Half-Life and Reaction Order

Half-life ($t_{1/2}$) is the time taken for the concentration of a reactant to decrease to half its initial value

- Half-life patterns provide another way to identify reaction order - By measuring successive half-lives from a concentration-time graph, you can determine the order ### Half-Life in Zero-Order Reactions - Successive half-lives **decrease** with time - Each half-life is shorter than the previous one - Takes less time to halve the concentration as the reaction progresses **Pattern:** - 1st half-life: longer - 2nd half-life: shorter - 3rd half-life: even shorter **Explanation:** In a zero-order reaction, the same amount of reactant is consumed per unit time. When concentration is high, halving it requires removing a large amount. When concentration is already low, halving it requires removing less - which takes less time at the constant consumption rate. ### Half-Life in First-Order Reactions - Half-life remains **constant** throughout the reaction - Each half-life takes exactly the same amount of time - This is the characteristic signature of first-order kinetics

Constant half-life is the defining feature of first-order reactions

**Pattern:** - 1st half-life: $t$ seconds - 2nd half-life: $t$ seconds - 3rd half-life: $t$ seconds **Relationship with rate constant:** $$k = \frac{\ln 2}{t_{1/2}}$$ Where: - $k$ = rate constant - $t_{1/2}$ = half-life - $\ln 2 \approx 0.693$ **Example calculation:** If a first-order reaction has a half-life of 50 seconds: $$k = \frac{0.693}{50} = 0.0139 \text{ s}^{-1}$$ **Visualizing constant half-life:** The chart below demonstrates how a first-order reaction exhibits equal time intervals for successive halvings:

Notice how each 50-second interval results in the concentration halving - this constant half-life confirms first-order kinetics. ### Half-Life in Second-Order Reactions - Successive half-lives **increase** with time - Each half-life is longer than the previous one - Takes progressively more time to halve the concentration **Pattern:** - 1st half-life: shorter - 2nd half-life: longer - 3rd half-life: even longer **Explanation:** In a second-order reaction, the rate depends on concentration squared. When concentration is high, the reaction is fast and halving happens quickly. As concentration drops, the rate slows dramatically, so subsequent halvings take much longer. ## Comparing Half-Life Patterns | Order | Half-Life Pattern | Time Trend | |-------|------------------|------------| | Zero | Decreasing | $t_{1/2}(1) > t_{1/2}(2) > t_{1/2}(3)$ | | First | Constant | $t_{1/2}(1) = t_{1/2}(2) = t_{1/2}(3)$ | | Second | Increasing | $t_{1/2}(1) < t_{1/2}(2) < t_{1/2}(3)$ |

Always check multiple half-lives when determining order - measuring just one half-life is insufficient to identify the pattern

## Practical Application **To determine reaction order from experimental data:** 1. Plot concentration vs time from your measurements 2. Observe the overall shape: - Straight line → zero-order - Gentle curve → likely first-order - Steep curve → likely second-order 3. Measure at least three successive half-lives to confirm: - Mark where concentration halves (e.g., from 1.0 to 0.5 mol dm$^{-3}$) - Mark the next halving (0.5 to 0.25 mol dm$^{-3}$) - Mark another halving (0.25 to 0.125 mol dm$^{-3}$) 4. Compare the time intervals between halvings 5. Apply the half-life pattern to confirm the order **Example scenario:** A student monitors the decomposition of hydrogen peroxide and records concentrations every 30 seconds. After plotting the data, they observe a smooth downward curve. Measuring half-lives, they find: - Time to drop from 2.0 to 1.0 mol dm$^{-3}$: 180 seconds - Time to drop from 1.0 to 0.5 mol dm$^{-3}$: 180 seconds - Time to drop from 0.5 to 0.25 mol dm$^{-3}$: 180 seconds The constant half-life confirms this is a **first-order** reaction.