Graphs are one of the best ways we have to determine the quantitative relationship between an observed (plotted on the y-axis) and a controlled variable (plotted on the 𝑥-axis).
Let’s take the reaction where A + B ⇾ C.
We could measure how the concentration of reactant A varies over time and when we plot a graph of our results, it might look a little like this:
If we were to take this investigation a step further, we might repeat our experiment with different starting concentrations of reactant A, and determine the initial rate of the reaction in each experiment (this would be the gradient of the curve at time = 0). We could then plot a graph of initial rate of reaction against the concentration of A:
Of course, its also possible that there is no relationship between two variables, for example if the rate of reaction did not depend on the concentration of reactant B, and then the graph would look like:
The equation of a straight line is always
Returning to our reaction, we know that the rate of the reaction depends on the concentration of A but not B, in which case the rate equation for the reaction would be
rate = k[A]
where k is the rate constant.
We need to be able to recognise this as a form of y = m𝑥 + c
But what if we get a curve when we plot the observed variable against the controlled variable?
Finding the equation for a curved line is not so straight forward (and the gradient depends on the point of the curve at which a tangent is drawn – a curve has an infinite number of gradients!). It’s far easier to re-plot our data in a form that gives us a straight line from which we can determine a gradient …
How to linearise a data set to give us a straight line graph
- Re-write the equation as that of a straight line, y = m𝑥 + c
E.g. determining the relationship between Gibbs energy, ΔG, and temperature, T, as given by the Gibbs-Helmholtz equation, for the reaction between ammonia and oxygen
A plot of ΔG against temperature is not linear but a smooth curve suggesting that there is a relationship between the two variables, it’s just not as simple as ΔG ∝ T.
We need to re-write the equation in the form of y = m𝑥 + c
A plot of (ΔG / T) against (1/T) gives us a straight-line and the gradient gives us the enthalpy, ΔH, for the reaction in question.
- Where our equation is in the form y = ab𝑥 (which is surprisingly common), we can use natural logarithms to convert the equation into that of a straight line graph.
The first step is to take the natural logarithm of each side, and then apply the first law of logarithms:
ln y = ln (ab𝑥)
ln y = ln a + ln b𝑥
The second step is to simplify using the third law of logarithms:
ln y = ln a + 𝑥 . ln b
Now our equation is in the form of y = c + 𝑥 . m
The third step is to plot a graph of ln y against 𝑥. The gradient will be ln b (b = egradient) and the intercept is ln a (a = eintercept).
Let’s look at an example to help make some sense of this! You’ll be familiar with the idea that pH is a measure of the hydrogen ion concentration of a solution:
pH = -log10 [H+]
and the inverse of this equation is:
[H+] = 10-pH
y = a b𝑥
where [H+] is the observed variable, pH is the controlled variable and in this case, a = 1.
Following the method above we need to take the natural logarithm of each side and then simplify:
ln [H+] = ln (10-pH)
ln [H+] = pH . – ln 10
We can now plot a graph of ln [H+] on the y-axis against pH on the 𝑥-axis and the gradient will be – ln 10.
- Where our equation is in the form y = a 𝑥b we can adapt the method above using natural logarithms once again to convert the equation into that of a straight line graph.
The first step is to take the natural logarithm of each side and apply the first law of logarithms:
ln y = ln (a𝑥b )
ln y = ln a + ln 𝑥b
We can simplify this equation by applying the third law of logarithms:
ln y = ln a + b . ln 𝑥
y = c + m . 𝑥
Finally we can plot a graph of ln y against ln 𝑥. The gradient will be b and the intercept is ln a (a = eintercept).