Lesson 4 of 9 / Practical and data skills
Read a rate from a changing quantity
Why does total product divided by total time miss the initial rate?
I can calculate mean and instantaneous rates and distinguish rate from accumulated product.
About 6 min
Make a guess. You are not marked.
A reaction makes 40 cm3 of gas in 10 minutes, but most of it in the first 2 minutes. Is 4 cm3 per minute the initial rate?
Show the answer
The initial rate is the starting gradient.
An average over the whole run hides the fast start. Draw a tangent at time zero.
The key idea
A rate is change per unit time. The gradient at the start measures initial rate; a whole-run average measures something different.
Work with the evidence
Compare the start with the whole run
Gas volume / cm3
Initial rate: 12/20 = 0.60 cm3 s-1. Use two points on the starting tangent, not two later points on the curve.
Original schematic curve, with the stated tangent and endpoint. A shallower later curve means a lower rate of further product formation.
Explanation
If a reaction produces 12 cm3 of gas in 30 s, its mean rate over that interval is 12/30 = 0.40 cm3 s-1. Use the change in gas volume, not its final value alone, if the interval begins after time zero. Rate units combine the measured quantity with inverse time.
A product-versus-time curve may become less steep because substrate is depleted or other conditions change. Its gradient represents rate. To estimate the rate at a particular time, draw a tangent to the curve at that time, choose two well-separated points on the tangent and calculate change in volume divided by change in time.
The initial rate is the gradient near the start, before substrate depletion substantially changes the conditions being compared. Dividing final product by the duration of a long run gives an average that includes later slower periods. It is not generally the initial rate.
In a fixed-endpoint assay, every run may require the disappearance of the same amount of substrate. Then 1/time can provide a relative rate, in s-1, when the endpoint and starting amount are comparable. It does not directly measure moles per second unless the amount transformed is also known.
Percentage change is another useful transformation: (final - initial)/initial x 100. It makes changes from differently sized starting specimens more comparable, but still depends on a valid controlled design. Keep a negative sign when a specimen loses mass.
Step by step
- 1
Name the requested rate
Distinguish an interval mean, an initial gradient and a fixed-endpoint relative rate.
- 2
Choose matching values
Use both ends of the interval or two points on the tangent.
- 3
Calculate with units
Divide the changes and retain a sensible number of significant figures.
Worked example
Work through the evidence
A tangent at the start of a gas-production curve passes through (0 s, 0 cm3) and (20 s, 12 cm3). At 60 s the experiment has produced 18 cm3. Calculate the initial and whole-run mean rates.
One way to explain it
Initial rate = 12/20 = 0.60 cm3 s-1. The mean over 60 s = 18/60 = 0.30 cm3 s-1. The lower whole-run mean is consistent with the curve becoming less steep as the run proceeds.
Why this answer works
- The first calculation uses the tangent.
- The second uses total change over the whole run.
- Different intervals or gradients answer different questions.
Watch out for this
A student says: "A plateau on a product-time curve means product is disappearing." What is wrong with this?
Show the answer
A horizontal section means accumulated product is approximately constant, so net product formation is approximately zero. A falling curve would indicate a decrease.
Check your understanding
An endpoint is reached in 50 s in trial A and 100 s in trial B, with the same initial amount and endpoint. How do their relative rates compare?
Show the answer
Trial A is twice as fast.
(1/50)/(1/100) = 2.
Quick recall
Why does total product divided by total time miss the initial rate?