Proportionality in Rates

The reaction is therefore 2.5 times as fast. Factors allow changes in rate to be compared fairly even when the original values differ.

Apply the same calculation to the condition being changed:

For example, increasing an acid concentration from $0.50$ to $1.00\ \text{mol/dm}^3$ is an increase by a factor of 2. If the measured rate also doubles, the results are consistent with rate being directly proportional to concentration under those experimental conditions.

What direct proportion means

If rate $r$ is directly proportional to a factor $x$, then

and this can be written as

where $k$ is the constant of proportionality. Dividing each rate by its corresponding value of $x$ should give the same value of $k$:

A doubling of $x$ then doubles the rate, while a tripling of $x$ triples the rate. On a graph of rate against $x$, a directly proportional relationship appears as a straight line through the origin.

Data for Recognising direct proportion in reaction-rate data

A straight line that does not pass through the origin is not evidence of direct proportion. Nor is a curved graph. In both cases, increasing the factor may still increase the rate, but not in a constant proportion.

Applying the comparison to concentration

A more concentrated solution contains more reacting particles in the same volume. The particles do not move faster simply because concentration has increased. Instead, collisions occur more frequently, producing more successful collisions each second and increasing the reaction rate.

Imagine that doubling acid concentration from $0.50$ to $1.00\ \text{mol/dm}^3$ changes the rate from $0.12$ to $0.24\ \text{cm}^3/\text{s}$. Both factors are 2, so this pair of results is consistent with direct proportion.

If doubling the concentration instead changed the rate from $0.24$ to $0.40\ \text{cm}^3/\text{s}$, the rate factor would be $0.40\div0.24\approx1.67$. Concentration still increased the rate, but these results would not show direct proportion because the two factors are unequal.

This distinction matters: an increase is not automatically a proportional increase. Experimental data must be used to test the relationship.

Pressure as a concentration change for gases

Increasing the pressure of reacting gases by compressing them forces the same number of particles into a smaller volume. There are then more particles per unit volume, so collisions occur more frequently and the reaction becomes faster.

For example, halving a gas volume from $200\ \text{cm}^3$ to $100\ \text{cm}^3$, while keeping the number of particles and temperature constant, doubles the number of particles per unit volume. This predicts an increased rate. Whether the rate itself doubles must be checked from rate measurements rather than assumed.

Rate and the time taken

Average rate is calculated from

When the same fixed quantity of reactant is used or product is formed each time, rate is inversely proportional to the time taken:

Thus, if the same change is completed in half the time, the average rate is doubled. It would be wrong to say that rate is directly proportional to time: faster reactions take less time, not more.

Making a valid comparison

Only the factor being investigated should be deliberately changed. If concentration is changed, the acid volume, temperature, mass of solid and surface area of the solid should be controlled. Rate can then be obtained from quantities such as gas volume produced or mass lost per unit time.

The same mathematical approach can be used when comparing any factor that affects rate:

1. calculate the factor by which the condition changed;
2. calculate the factor by which the measured rate changed;
3. compare the two factors;
4. use several results or a graph before concluding that the relationship is directly proportional.

Real experimental values may not agree perfectly because of measurement uncertainty. Results can be described as approximately proportional when the ratios are close and the plotted points lie close to a straight line through the origin.