Science calculations - calculating rates
Rate changes show how quickly something is happening. The rate of change can be calculated using this equation:
The rate of decay of milk can be calculated. As milk spoils or decays, it becomes more acidic.
Milk was incubated at three different temperatures and the pH was recorded every 24 hours.
| 0 hours | 24 hours | 48 hours | 72 hours | |
| 5掳颁 | 6.5 | 6.4 | 6.4 | 6.0 |
| 20掳颁 | 6.5 | 6.1 | 5.5 | 4.8 |
| 35掳颁 | 6.5 | 5.1 | 4.8 | 4.8 |
| 5掳颁 | |
|---|---|
| 0 hours | 6.5 |
| 24 hours | 6.4 |
| 48 hours | 6.4 |
| 72 hours | 6.0 |
| 20掳颁 | |
|---|---|
| 0 hours | 6.5 |
| 24 hours | 6.1 |
| 48 hours | 5.5 |
| 72 hours | 4.8 |
| 35掳颁 | |
|---|---|
| 0 hours | 6.5 |
| 24 hours | 5.1 |
| 48 hours | 4.8 |
| 72 hours | 4.8 |
To calculate the rate of change we first need to know the change, or difference, from the previous point for each value. These calculations and answers are shown in the table below.
| 0 hours | 24 hours | 48 hours | 72 hours | |
| 5掳颁 | 6.5 - 6.4 = 0.1 | 6.4 - 6.4 = 0 | 6.4 - 6.0 = 0.4 | |
| 20掳颁 | 6.5 - 6.1 = 0.4 | 6.1 - 5.5 = 0.6 | 5.5 - 4.8 = 0.7 | |
| 35掳颁 | 6.5 - 5.1 = 1.4 | 5.1 - 4.8 = 0.3 | 4.8 - 4.8 = 0 |
| 5掳颁 | |
|---|---|
| 0 hours | |
| 24 hours | 6.5 - 6.4 = 0.1 |
| 48 hours | 6.4 - 6.4 = 0 |
| 72 hours | 6.4 - 6.0 = 0.4 |
| 20掳颁 | |
|---|---|
| 0 hours | |
| 24 hours | 6.5 - 6.1 = 0.4 |
| 48 hours | 6.1 - 5.5 = 0.6 |
| 72 hours | 5.5 - 4.8 = 0.7 |
| 35掳颁 | |
|---|---|
| 0 hours | |
| 24 hours | 6.5 - 5.1 = 1.4 |
| 48 hours | 5.1 - 4.8 = 0.3 |
| 72 hours | 4.8 - 4.8 = 0 |
Because we started at pH 6.5 at zero hours there is no rate change for this time. Similarly, because the milk did not change from pH 4.8 in the last 24 hours at 35掳颁 there is no rate change here.
We now need to divide each change in value by the change in time (24 hours).
| 24 hours | 48 hours | 72 hours | |
| 5掳颁 | 0.1 梅 24 = 0.0042 | 0 梅 24 = 0 | 0.4 梅 24 = 0.017 |
| 20掳颁 | 0.4 梅 24 = 0.017 | 0.6 梅 24 = 0.025 | 0.7 梅 24 = 0.029 |
| 35掳颁 | 1.4 梅 24 = 0.058 | 0.3 梅 24 = 0.013 | 0 梅 24 = 0 |
| 5掳颁 | |
|---|---|
| 24 hours | 0.1 梅 24 = 0.0042 |
| 48 hours | 0 梅 24 = 0 |
| 72 hours | 0.4 梅 24 = 0.017 |
| 20掳颁 | |
|---|---|
| 24 hours | 0.4 梅 24 = 0.017 |
| 48 hours | 0.6 梅 24 = 0.025 |
| 72 hours | 0.7 梅 24 = 0.029 |
| 35掳颁 | |
|---|---|
| 24 hours | 1.4 梅 24 = 0.058 |
| 48 hours | 0.3 梅 24 = 0.013 |
| 72 hours | 0 梅 24 = 0 |
The greatest rate of change is shown in bold above.
Rate of change can also be calculated from graphs. Here we use this equation:
The rate of change can be calculated from the above graph by finding the gradient of the trendline, using the equation above.
For example after 0 hours in the above graph, the pH is 6.6; after 50 hours it is 5.4.
\(\text{rate of change (gradient)}=\frac{\text{vertical change (y axis)}}{\text{horizontal change (x axis)}}\)
= \(\frac{\text{5.4 - 6.6 pH units}}{\text{50 - 0 hours}}\)
= \(\frac{\text{-1.2 pH units}}{\text{50 hours}}\)
= \(\text{-0.024 pH units/hour}\) (to 2 significant figures)
The '-' sign means that the pH went down and the milk became more acidic.
The calculated rate of -0.024 pH units/hour is the mean of the rates shown by the red line in the first graph.
Using the gradient of a graph to find a rate of change is easier than calculating rate when there is a lot of data, or when the data shows a great deal of variability.