Applying General Mathematics in Chemistry (DP IB Chemistry: SL): Revision Note

Applying general mathematics in Chemistry

Using arithmetic and algebraic calculations in chemistry

• Chemistry often requires the use of calculations, which can include:

• Decimals

  • Most chemical calculations use decimals, e.g. the concentration of a chemical

• Fractions

  • These are most commonly used in uncertainty calculations

  • Most scientific calculators will initially give answers as fractions

    • Make sure you know where the S⇔D button is so that you convert the fraction into a decimal

• Percentages

  • There are many percentage calculations, including percentage yield, percentage atom economy, percentage change, percentage difference, percentage error and percentage uncertainty

• Ratios

  • These are commonly used in moles calculations where the stoichiometry of the balanced chemical equation is not 1 : 1, e.g.

H₂SO₄ + 2NaOH → Na₂SO₄ + 2H₂O

• Reciprocals

  • These are most obvious in gas laws, using 1 / V, and concentration versus rate graphs, using 1 / T

    • For more information about gas laws, see our revision notes in the Ideal Gases topic

    • For more information about concentration and rates, see our revision notes in the How Fast? The Rate of Chemical Change topic

• Logarithmic functions

  • These are most obvious in pH and Arrhenius calculations

    • For more information about pH, see our revision note on The pH Scale

    • For more information about Arrhenius, see our revision note on The Arrhenius Equation

• Exponential functions (Additional Higher level)

  • This is most obvious in Arrhenius calculations

• Section 1 of the Data Booklet contains a list of the mathematical equations that you are expected to be able to manipulate and work with

  • Careful: This is not an exhaustive list - there are other equations that you will be expected to know that are not given in Section 1 of the Data Booklet, e.g. percentage yield

What is the mean average?

• The mean average is often just called the “average”

• It is the total of all the values divided by the number of values, i.e. add all the numbers together and divide by how many there are

• For example, two common isotopes of chlorine are chlorine-35 and chlorine-37, which exist in a 3 : 1 ratio

  • The information shows that there are three chlorine-35 atoms for every one chlorine-37 atom

  • Overall, there is a total of four atoms to be considered in the calculation

  • So, the mean average = $\frac{35+35+35+37}{4}$ = 35.5

• Problems with the mean average occur when there are anomalous results 

  • These should not be included in a mean average calculation

What mean average calculations are in chemistry?

• Two main calculations in chemistry that involve mean average calculations are:

  • Calculating relative atomic mass from isotopic abundance data

    • For more information, see our revision note on Isotopes

  • Calculating average titres as part of a titration calculation

    • These calculations will typically have values that should not be considered because they are not concordant

Range

• The range can only be applied to numerical data

• It is a measure of how spread out the data is, which means that it is the difference between the highest and lowest values

  • You can remember this as "Hi - Lo"

• It can be expressed as:

  • A range of values

    • e.g. 9.2 - 8.4 

  • A single value

    • e.g. 9.2 - 8.4 = 0.8

• The range can be affected when the highest and / or lowest data are anomalous results themselves

What is scientific notation?

• Scientific notation is also known as standard form

• It is a system of writing and working with very large or very small numbers

  • For example, Avogadro's number 602 000 000 000 000 000 000 000 is 6.02 x 10²³ in scientific notation

• Numbers in scientific notation are written as:

a × 10ⁿ

• They follow these rules:

  • a is a number above 1 and below 10

  • For large numbers, n is an integer that is greater than 0 

    • i.e It shows how many times a is multiplied by 10

  • For small numbers, n is an integer that is less than 0 

    • i.e It shows how many times a is divided by 10

  • n < 0 for small numbers i.e how many times a is divided by 10

Applying scientific notation to numbers

• When rounding a number in standard form to a certain number of significant figures, only the value of a is rounded (the × 10ⁿ value will not be significant)

  • For example, 4.37 × 10⁶ to 2 significant figures is 4.4 × 10⁶

Orders of magnitude

• When a number is expressed to an order of 10, this is an order of magnitude

  • Example: If a number is described as 3 × 10⁸ then that number is actually 3 × 100 000 000

  • The order of magnitude of 3 × 10⁸ is just 10⁸

• Orders of magnitude follow rules for rounding

  • The order of magnitude of 6 × 10⁸ is 10⁹ as the magnitude is rounded up

• A quantity is one order of magnitude larger than another quantity if it is about ten times larger

  • Similarly, two orders of magnitude would be 100 times larger, or 10²

• In chemistry, orders of magnitude can be very large or very small

Approximation and estimation

• Approximation and estimation are both methods used to obtain values that are close to the true or accurate values

  • While they share some similarities, they have distinct characteristics and are used in different contexts

• An approximation is a value that is close to the actual value but not exact.

• Used when:

  • A precise calculation is unnecessary

  • A quick answer is sufficient

  • The exact value is hard or time-consuming to obtain

• Approximations are typically easy and fast to work out

• For example, the pH of a strong acid is generally between pH 1.0 and 3.0

  • A useful approximation is to say pH = 1.0.

  • This is not exact, but it is widely accepted and practical for many uses

• Estimationis making a reasoned guess based on available data.

• Used when the true value is unknown or can't be directly measured.

• For example, estimating percentage yield

  • Start with a known amount of reactant.

  • Calculate the theoretical mass of product.

  • Perform the reaction and measure actual product mass

$\mathrm{Percentage}\mathrm{yield}=\frac{\mathrm{actual}\mathrm{yield}}{\mathrm{theoretical}\mathrm{yield}}\times 100$

Appreciate when some effects can be ignored and why this is useful

• During calculations using acid and base dissociation constants the assumption made is that the value of [H⁺] = [HA] 
  

  • Or the concentration of hydrogen ions is the same as the concentration of the acid

• This is because

  • The weak acid has a low degree of dissociation so you can assume the concentration of the acid is the same value at equilibrium and use it in the expression for K_a
    

    • CH₃COOH $\rightleftharpoons$ CH₃COO^– + H⁺

    • A point to remember is that K_a values are different depending on the acid, so the degree of dissociation also varies. Therefore the error using this assumption for some acids will be larger than others 

  • All the H⁺ ions are assumed to come from the acid as water will contribute a very small number of H⁺ ions

• So the K_a expression:

K_a = $\frac{\left[H^{+}\right]\left[A^{-}\right]}{\left[HA\right]}$

can be simplified to 

K_a = $\frac{{\left[H^{+}\right]}^{\mathbf{2}}}{\left[\mathrm{HA}\right]}$

• Remember to give the full expression in an exam when asked

Percentage change and percentage difference

• Percentage change and percentage difference are commonly used to express the relative change between two values

  • They are useful for comparing experimental results, determining reaction yields and analysing other chemical data

Percentage change

• Percentage change is used to express the relative change between an initial value and a final value

• It is calculated using the following formula:

Percentage Change = $\frac{\mathbf{(}\mathbf{Final}\mathbf{Value}\mathbf{-}\mathbf{Initial}\mathbf{Value}\mathbf{)}}{\mathbf{Initial}\mathbf{value}}\mathbf{\times }\mathbf{100}$

Percentage difference

• Percentage difference is used to compare two values to determine how much they differ from each other as a percentage

• It is calculated using the following formula:

Percentage Difference = $\frac{(\mathrm{value}1-\mathrm{value}2)}{\left(\mathrm{average}\mathrm{of}\mathrm{value}1\mathrm{and}\mathrm{value}2\right)}\mathbf{\times }\mathbf{100}$

Percentage Error

• Percentage error is used to express the difference between a final calculated answer and an accepted or literature value

• It is calculated using the following formula

Percentage error = $\frac{\mathbf{accepted}\mathbf{value}\mathbf{-}\mathbf{experimental}\mathbf{value}}{\mathbf{accepted}\mathbf{value}}\mathbf{\times }\mathbf{100}$

• You should be able to comment on any differences between the experimental and literature values

Percentage uncertainty

• Percentage uncertainties are a way to compare the significance of an absolute uncertainty on a measurement

  • This is not to be confused with percentage error, which is a comparison of a result to a literature value

• It is calculated using the following formula

Percentage uncertainty = $\frac{\mathbf{absolute}\mathbf{uncertainty}}{\mathbf{measured}\mathbf{value}}\mathbf{\times }\mathbf{100}$ 

Diagram with examples of percentage uncertainty calculations for common laboratory apparatus

• For more information about uncertainties, see our revision note on Processing Uncertainties in Chemistry

Mathematical skills linked to graphs and tables

• There are several specification points linked to graphs and tables

  • For more information about working specifically with graphs, see our revision note on Graphing in Chemistry

• These include, but are not limited to, being able to:

  • Distinguish between qualitative and quantitative data, incorporating continuous and discrete variables

  • Understand direct and inverse proportionality, as well as positive and negative correlations between variables

  • Determine rates of change from tabulated data

Qualitative and quantitative data

• Qualitative data usually describes something in words, not numbers 

  • For example:

    • Copper sulfate solution is blue

    • A more dilute copper sulfate solution is pale blue, while a more concentrated copper sulfate solution is a darker blue

• Quantitative data uses numbers to count / measure something

  • For example:

    • The neutralisation of 25.0 cm³ sodium hydroxide by 25.0 cm³ hydrochloric acid increases the temperature of the system by 2.5 ^oC

• Discrete data is quantitative

  • It consists of separate, distinct and countable values

  • For example:

    • The stoichiometric coefficients representing the relative number of molecules or atoms involved in the reaction are discrete values and must be integers

    • Electrons can only occupy certain discrete energy levels, e.g. 1s, 2s, etc

• Continuous data is also quantitative

  • It is based on measurements and can include decimal numbers or fractions

  • This allows for an infinite number of values 

  • For example:

    • The temperature of an exothermic reaction as time progresses

    • The volume of gas produced during the thermal decomposition of calcium carbonate

Direct and inverse proportionality

• There are a number of terms that are commonly applied to trends, particularly in graphs
  

• Directly proportional

  • This applies to a trend that has a clearly linear relationship

  • Mathematically, this can be described as y = kx, where k can be positive or negative

  • In most situations, it is clear that k is positive

  • This means that the relationship can be described as "when one variable increases, the other increases" or "if x doubles, then y doubles"

  • A directly proportional relationship is always a straight line through the origin with a fixed gradient 

• Inversely proportional

  • This can be described as y = $\frac{\mathrm{k}}{\mathrm{x}}$, where k can be positive or negative

  • This means that the relationship can be described as "when one variable increases, the other decreases" or "if x doubles, then y halves" 

  • When plotted, inverse proportionality is not a straight line and does not pass through the origin

• Positive correlation

  • This term is best applied to the gradient of a graph

  • The gradient of the graph is positive / slopes or curves upwards

  • It describes a relationship where as x increases, y also increases

• Negative correlation

  • This term is, also, best applied to the gradient of a graph

  • The gradient of the graph is negative / slopes or curves downwards

  • It describes a relationship where as x increases, y decreases

Determine rates of change from tabulated data

• To determine rates of change from tabulated data, you can use the average rate of change or gradient, if the data has been plotted as a graph

• The average rate of change between two points on a graph or in a table is:

Rate of change = $\frac{\mathbf{the}\mathbf{change}\mathbf{in}\mathbf{the}\mathbf{dependent}\mathbf{variable}\mathbf{(}\mathbf{y}\mathbf{-}\mathbf{axis}\mathbf{)}}{\mathbf{the}\mathbf{change}\mathbf{in}\mathbf{the}\mathbf{independent}\mathbf{variable}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{axis}\mathbf{)}}$