Chemistry 7.3-7.5 - Collision theory and rate graphs

What reaction rate measures

A reaction rate describes how quickly a reactant is used up or a product is formed. It needs two measurements:

1. a quantity that changes because of the reaction, such as gas volume, mass or concentration;
2. the time over which that change happens.

This gives a bridge between three views of chemistry. At the macroscopic level, a gas syringe fills or the mass on a balance falls. At the particle level, reactant particles are being converted into product particles. At the symbolic level, the rate is a change divided by a time.

The unit depends on what was measured. Gas volume in $\mathrm{cm^3}$ over seconds gives $\mathrm{cm^3\,s^{-1}}$; mass in $\mathrm g$ over seconds gives $\mathrm{g\,s^{-1}}$; concentration in $\mathrm{mol\,dm^{-3}}$ over seconds gives $\mathrm{mol\,dm^{-3}\,s^{-1}}$. A numerical answer without the matching unit does not fully identify the rate.

Worked example: mean gas-production rate

A gas volume rises from $18\,\mathrm{cm^3}$ at $20\,\mathrm s$ to $66\,\mathrm{cm^3}$ at $50\,\mathrm s$.

1. Change in volume: $66 - 18 = 48\,\mathrm{cm^3}$.
2. Time interval: $50 - 20 = 30\,\mathrm s$.
3. Mean rate: $48 / 30 = 1.6\,\mathrm{cm^3\,s^{-1}}$.
4. Sense-check: a little more than $1.5\,\mathrm{cm^3}$ each second for $30\,\mathrm s$ should make about $48\,\mathrm{cm^3}$, so the result is plausible.

A mean rate covers an interval. The rate can change during that interval, so it does not claim that exactly $1.6\,\mathrm{cm^3}$ formed in every individual second.

Collision theory and changing conditions

Collision theory is a model that links what we measure to what particles do. Reactant particles must collide, and the collision must have enough energy for the reaction to occur. A faster reaction therefore means more reaction-causing collisions in each second.

The phrase collision frequency means the number of collisions per unit time. Saying only "there are more collisions" is incomplete: a reaction that lasts longer may have many collisions overall while still being slow. Rate depends on how frequently suitable collisions happen.

For surface area, compare the same mass of solid. Crushing does not create more substance or make its particles smaller; it exposes a larger total surface. Only particles at the solid's exposed surface can collide directly with particles in a liquid or gas.

Changing a condition can alter rate without altering the final amount of product. If every run starts with the same mass of the gas-forming solid and all of that solid can react, a faster reaction can reach the same final gas volume sooner. Rate and total product are different quantities.

Surface area and ratios

A cube of side $2\,\mathrm{cm}$ has surface area $6\times2^2=24\,\mathrm{cm^2}$ and volume $2^3=8\,\mathrm{cm^3}$, giving surface area to volume ratio $24/8=3\,\mathrm{cm^{-1}}$. Cutting it into eight cubes of side $1\,\mathrm{cm}$ keeps the total volume at $8\,\mathrm{cm^3}$, but gives total area $8\times6=48\,\mathrm{cm^2}$. The ratio doubles to $6\,\mathrm{cm^{-1}}$. Real chips are irregular, but the same reasoning explains why smaller pieces react faster at the same total mass.

Testing proportionality

An increase is not automatically a directly proportional relationship. Suppose doubling acid concentration from $0.50$ to $1.00\,\mathrm{mol\,dm^{-3}}$ doubles the measured initial rate from $0.80$ to $1.60\,\mathrm{cm^3\,s^{-1}}$. These results are consistent with direct proportionality over the tested conditions: rate divided by concentration stays constant. Test more concentrations; a straight best-fit line through the origin supports $\text{rate}\propto\text{concentration}$. A line with an intercept or a curve does not. Do not assume that doubling temperature doubles rate, or that every reaction follows the same concentration relationship.