Graphing in Chemistry (SL IB Chemistry)

Sketch graphs

• Sketch graphs are a way to represent qualitative trends where the variables shown are often proportional or inversely proportional
• Sketch graphs do not have scales or data points, but they must have labels as these examples from the Gas Laws show:

Sketch graphs of Boyle's Law

Sketched graphs show relationships between variables

Graphical Relationships

• In the first sketch graph above you can see that the relationship is a straight line going through the origin
  • This means as you double one variable the other variable also doubles so we say the independent variable is directly proportional to the dependent variable
• The second sketched graph shows a shallow curve which is the characteristic shape when two variables have an inversely proportional relationship
• The third sketched graph shows a straight horizontal line, meaning as the independent variable (x-axis) increases the dependent variable does not change or is constant

General guidance on drawing graphs

• The types of graphs that students are expected to be able to draw include:
  • Bar charts
  • Histograms
  • Scatter graphs
  • Line / curve graphs
• Graphs need to have:
  • Clear title
  • Labelled axes
  • Units on the axes
  • Appropriate linear scales without any jumps
    • This means the plotted graph must occupy at least half or more of the sheet or grid
    • A rough rule of thumb is that if you can double the scale and still fit all the points on, then your scale is not appropriate
  • Clearly shown data points
    • The most common convention is to use small crosses to show the data points

Graph of concentration versus time

Graphs must show appropriate scales, labelling and units. The independent variable usually goes on the x-axis and the dependent variable on the y-axis

• Remember: The independent variable is the one you control or manipulate and the dependent variable is the one that changes as a result of your manipulation
• Always draw data points in pencil as it makes it easier to make corrections and adjustments

Best Fit Lines

• Students often confuse the term lines of best fit with straight lines
• Lines of best fit can be straight lines or curves (just like the example above) and:
  • They show the trend of the data
    • It does not have to go through all the points, but shows the general trend
  • They must go through the majority of the points
  • Where the data is scattered the points should be evenly distributed on either side of the best fit line

Extrapolation and interpolation

• Extrapolation is the term used to describe the process of extending a line of best fit 

Extrapolation on a temperature correction graph

This temperature correction graph from a calorimetry investigation shows how the two best fit lines are extrapolated to find the maximum temperature rise

• Interpolation is the term used to describe the process of assuming a trend line applies between two points

Extrapolation and interpolation on a graph

Interpolation uses the line of best fit within the plotted points and extrapolation extends the best fit line beyond the plotted points

Other features of graphs

Gradient

• The gradient of a graph can be found by:
  • For straight-line graphs
    • Draw a triangle
    • Then, use the equation for a straight line
  • For curves
    • Draw a tangent to the graph, using a ruler to line up against the curve at the point where the gradient is to be measured
    • Then, use the equation for a straight line to calculate the gradient

How to draw a tangent to a curve

Lining up a ruler against the curve is essential to drawing a tangent accurately

• The triangle should be as large as possible to minimise precision errors
• The equation for a straight line is y = mx + c, where:
  • y = dependent variable
  • x = independent variable
  • m = slope
  • c = y-intercept
• Therefore, the gradient or slope, m = ∆y / ∆x

Changes in gradient

• Graphs with curves of best fit have changing gradients
• This means that multiple gradients can be calculated to show:
  • The progressing rate of a reaction
  • The effects of factors, such as concentration, on the rate of reaction 

A rate kinetics graph illustrating the calculation of rates from a curve

The gradient can be found at different points on a curve. Each rate has been multiplied by 60 to convert it from minutes^-1 to seconds^-1

Intercepts

• Intercepts are the points where a line / curve of best fit crosses an axis on a graph
• The most common use of intercepts is in graphs of free energy versus temperature

Graph of free energy versus temperature for the synthesis of ammonia

The x-intercept shows the temperature when the reaction ceases to be feasible, in this case at 460 K (187 ^oC) and the y-intercept shows that the reaction is exothermic as the extrapolated value is approximately -46 kJ mol ^-1 

Maxima and minima

• The maxima and minima are the highest and lowest points respectively
  • Maxima - the gradient goes from positive to 0 to negative
  • Minima - the gradient goes from negative to 0 to positive
• A literal application of maxima and minima is in energy profile diagrams

The energy profile diagram for H₂ + Cl₂ → 2HCl

Maxima represent the energy of the transition state and minima represent the energy of the reactants and products

Areas

• A common use of areas of graphs in chemistry is Maxwell-Boltzmann curves

A Maxwell-Boltzmann distribution curve

The total area under the curve represents the total number of molecules / particles, while the E_a line further divides this area into particles with and without the minimum activation energy to react

Uncertainty bars

• The uncertainty in a measurement can be shown on a graph as an uncertainty bar
  • This bar is drawn above and below the point and shows the uncertainty in that measurement
• Uncertainty bars are plotted on graphs to show the absolute uncertainty of values plotted
  • Usually, these bars will be in the vertical direction, for y-values but they can be plotted horizontally, for x-values
• The size of the uncertainty bar can be used as an indication of the amount of error / uncertainty in the measurement

Graph to show uncertainty bars

Uncertainty bars show the error / uncertainty in a specific measurement