Using Graphs to Make Predictions Chemistry Tutorial
Key Concepts
- Graphs are useful in Chemistry because they allow us to make predictions.
- Predictions can be made about what happens:
(a) between two known points on the graph (interpolation)
(b) before the first known point on the graph (extrapolation)
(c) after the last known point on the graph (extrapolation)
- Interpolate means to insert points between known points on the graph.
- Extrapolate means to insert points either before the first known point, or, after the last known point on the graph.
- Interpolated lines on a graph are drawn as solid lines between plotted points.
- Extrapolated lines on a graph are draw as dotted lines (or sometimes dashed lines) beyond the known plotted points.
- There are limits to how far a line on a graph should be extrapolated.
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Why Draw a Graph?
There are two really good reasons for drawing a graph to represent numerical data:
- A graph conveys a lot of information very quickly.
- A graph can be used to make predictions.
Sometimes these predictions can be the difference between life and death!
For example, as a diver dives further down in the ocean, pressure increases. For every 10 m the diver descends, the pressure increases by about 1 atmosphere.
As the pressure increases, more nitrogen gas dissolves in the diver's blood and tissues.
When the diver ascends, the nitrogen gas becomes less soluble.(1)
If the diver's ascent is slow, the nitrogen gas slowly leaves the blood and tissues. But, if the rate of ascent is too fast, bubbles of gas become trapped, causing tremendous pain (and in the worst cases, even death).
The solution to this problem would seem to be obvious, just re-dissolve the nitrogen gas bubbles by increasing the pressure again.
But, how much nitrogen is already dissolved? And how much pressure needs to be applied?
| The table on the right shows the solubility of nitrogen gas in water(2) at various pressures.
Looking at the data in the table we can see that as the pressure increases from 1 to 5 atmospheres, the solubility of nitrogen gas also increases from 2.33 to 9.33 mg of N2(g) per 100 g of water.
If our diver descended to 20 m where the pressure is 2 atmospheres, then the table shows that 3.73 mg of nitrogen gas would be dissolved per 100 g of water in the diver's body.
Similarly, if we want to know how much pressure is required to dissolve 5.60 mg of N2(g) in 100 g of water, the table tells us the pressure required is 3.0 atmospheres (equivalent to diving 30 m down in the ocean).
|
| N2(g) solubility |
|---|
| pressure / atm |
solubility / mg per 100 g |
|---|
| 1.0 | 1.87 |
| 2.0 | 3.73 |
| 3.0 | 5.60 |
| 4.0 | 7.47 |
| 5.0 | 9.33 |
|
But, what if the diver descends 2.3 m where the pressure is 23 atmospheres, how much nitrogen gas will be dissolved then?
Or, what if a diver returns to the surface suffering from decompression sickness, with blood that contains 14.6 mg nitrogen per 100 g of water, what pressure needs to be applied to treat the diver and remove the pain?
Graphing the data in the table will give us a clearer picture of the relationship between pressure and solubility, and allow us to make predictions using the techniques of interpolation and extrapolation.
Interpolation
Consider the first question we asked above, if the diver descends 2.3 m where the pressure is 23 atmospheres, how much nitrogen gas will be dissolved then?
| The data from the table above has been plotted on the graph on the right, and the line of best fit has been drawn through the points.
The relationship between pressure and N2(g) solubility is very clearly a linear relationship over this range of pressures.
This graph can now be used to predict the solubility of N2(g) at 2.3 atmospheres pressure because the same linear relationship between pressure and solubility exists for any point between 1 and 5 atmospheres.
If you potision your mouse on 2.3 atm on the horizontal axis (x axis) and then move it up vertically until you meet the line, you should see a blue circle marking the position on the line and box to the right telling you that the solubility at this point is 4.29 mg per 100 g (that is, the reading on the vertical, or y, axis).
At a depth of 2.3 m, 23 atmospheres pressure, the concentration of nitrogen gas in the diver's body is 4.29 mg per 100 g.
|
Solubility
/ g per 100 g |
N2(g) solubility
Pressure / atm |
|
Predicting the value of a point on the graph that occurs between two known points on the same graph is known as interpolation.
In our example the known points (2.0 atm, 3.73 mg/100 g) and (3 atm, 5.60 mg/100 g) from the data table lie on a straight line on the graph.
The unknown point occurs where the pressure (the x value) is 2.3 atmospheres, which lies between the 2 known points.
By interpolation we find that the solubility of nitrogen gas at this point (the y value) is 4.29 mg per 100 g.
Interpolation assumes that the overall relationship described for the known points is also true between known points.
This is almost always a good assumption, so there is very little risk in making predictions from a graph using interpolation.
Extrapolation
Consider the second question we asked, what if a diver returns to the surface suffering from decompression sickness, with blood that contains 14.6 mg nitrogen per 100 g of water, what pressure needs to be applied to treat the diver and remove the pain?
The last known point on the graph occurs when the nitrogen gas solubility is 9.33 mg per 100 g.
We need to find pressure for a solubility of 14.6 mg per 100 mg, which occurs after the last known point on the graph so the technique of interpolation, as described above, cannot be used here.
Instead, we will assume that the linear relationship holds true for values greater than 5.0 atmospheres pressure.
This assumption enables us to extend, or extrapolate, the line of best fit past the last known point.
It is important that we distinguish the "known" parts of the graph from the "extrapolated" parts of the graph because we really can't be sure if our assumption is good or not.
When you draw a graph, you will use a solid line for the parts of the graph that are "known", and dashed or dotted lines for those parts of the graph which are extrapolated past the known points.
The graph on the right is the same as the graph used above, except that it has been extended past the last known point, or extrapolated, to 15.0 mg per 100 g.
The known part of the graph is shown in orange.
The unknown part, or extrapolation, of the graph is shown in blue.
The extrapolated part of the line can now be used to predict how much pressure is required to dissolve 14.6 mg of nitrogen gas per 100 g of water in the body.
Place your mouse at the point where the x axis and the y axis meet, then move up the vertical (y axis) until you reach 14.6 mg per 100 g.
Move your mouse horizontally until you meet the line, and a red circle appears around the point on the line.
A small box should appear to the right telling you that the solubility at this point is 14.6 mg per 100 g, and the pressure is 7.83 atmospheres (the reading on the horizontal, or x, axis).
We can treat the diver in a decompression chamber by applying a pressure of 7.83 atmospheres to re-dissolve the nitrogen bubbles in the blood, and then slowly reducing the pressure until we reach atmospheric pressure.
|
Solubility
/ g per 100 g |
N2(g) solubility
Pressure / atm |
|
Predicting the value of a point on the graph that occurs either before the first known point on the graph or after the last known point on the graph is known as extrapolation.
In our example the last known point on the graph was (5.0 atm, 9.33 mg per 100g). We needed to know the pressure corresponding to 14.6 mg per 100 g which represents a point past the last known point. The line was therefore extrapolated past the last known point to include the point we were interested in. The pressure (x axis) value could then be read off the extrapolated line using the solubility (y axis) value.
Extrapolation assumes that the overall relationship described for the known points is also true for points before the first known value and points after the last known value.
Often this assumption is not good in chemistry. There is a lot of risk in making predictions from a graph using extrapolation. There are limits to how far many graphs can be extrapolated in chemistry.
For this reason, you should never extrapolate a line too far from the known points.
Extrapolation Limits
Extrapolation assumes that the relationship between known or measured x values and y values will also be true for unknown or unmeasured x values and y values.
This is not always a good assumption.
We have to use our "chemical sense" to decide if we can extrapolate the line on a graph or not.
Often we want to extrapolate a known line back until it meets either the x axis (0,y), or the y axis (x,0), or the origin (0,0).
When we do this, we should ask ouselves if this makes sense from a "chemistry point of view".
| Consider the example used above of nitrogen gas dissolving in a diver's body.
When we extrapolate the line graph back to the origin, as shown on the right, we see that if the pressure is 0 atmospheres then no nitrogen gas will be dissolved (0 g per 100 g).
From a "chemistry point of view", this makes sense. If the pressure on the system is 0 atmospheres there is no force acting to prevent any gas from escaping, so the concentration of any gas in solution would be expected to be 0.
|
Solubility
/ g per 100 g |
N2(g) solubility
Pressure / atm |
|
A diving bell can descend to ocean depths of about 1,000 m where the pressure is about 100 atmospheres.
The graph on the right now shows the line extrapolated to 100 atmospheres in blue.
At 100 atmospheres, we predict from the graph that 187 mg of nitrogen gas will dissolve in 100 g of water.
A bathyscaphe can dive to depths of 10,000 m where the pressure is about 1,000 atmospheres.
The graph on the right now shows the line extrapolated to 1000 atmospheres in red.
At 1000 atmospheres, we predict from the graph that 1870 mg of nitrogen gas will dissolve in 100 g of water.
But, is it reasonable to keep on extrapolating the line to even greater and greater pressures?
|
Solubility
/ g per 100 g |
N2(g) solubility
Pressure / atm |
|
Our "chemical sense" warns us that there is a limit to how much gas you can dissolve in 100 g of water, even under immense pressure.
At some point, we expect the water to become saturated so that no more gas will dissolve.
And at some point, we expect the attraction between gas molecules to have increased sufficiently to interfere with their ability to be dissolved.
Furthermore, there will come a point when the attraction between gas molecules is so great that we expect them to come together and form a liquid.
Therefore, we conclude that at "some point" the linear relationship between solubility and pressure will no longer be true.
We do not know what this point is without conducting further experiments and calculations and drawing a new graph(3).
Our "chemical sense" tells us that while we might be happy to extend the known line graph from the last known point of (5.0 atm, 9.33 mg per 100 g) to (6.0 atm, 11.22 mg per 100 g) because this point is still quite close to the last known point, we would be concerned about extrapolating the line any further past this point because it is too far away from the last known point and we don't know the point at which the relationship ceases to be linear.
Footnotes:
(1) If you would like an explanation of this, go to Solubility and le Chatelier's Principle.
(2) The human body is mostly water, so we will use this as an approximation for how much nitrogen gas can be dissolved in the blood and tissues of the diver's body.
(3) For those who are interested, at 25°C nitrogen gas starts deviating from this linear relationship at about 100 atmospheres pressure.
