Applying General Mathematics in Chemistry (HL IB 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 = = 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
• For example:

Applying scientific notation to numbers

The scientific notation for numbers greater than 1 has the x 10 raised to a positive power while the scientific notation for numbers less than 1 has the x 10 raised to a negative power 

• 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

Approximation

• Approximation involves finding a value that is close to the actual value of a quantity
  • It may not necessarily be very precise or accurate
• It is often used when an exact calculation is challenging or time-consuming and a reasonably close value is sufficient 
• Approximations are typically quick and easy to calculate
• For example, the pH of a strong acid is commonly accepted as being between pH 1.0 and pH 3.0
  • A reasonable approximation would be to say that the pH of a strong acid is pH 1.0 
  • The approximation is not exact, but it is reasonable as well as being aligned with a value that most people accept as a strong acid

Estimation

• Estimation involves making an educated guess or assessment based on available information or data
• It is used when the true value of a quantity is unknown or cannot be directly measured
• For example, estimating the percentage yield of an industrial reaction involves:
  • Taking a known amount of reactant
  • Calculating the theoretical mass of the product that should be made
  • Completing the reaction and measuring the mass of the product that is made
  • Performing the percentage yield calculation 
  • The percentage yield calculation is correct for that amount of reactant under the specific reaction conditions used but can be applied to the industrial reaction to give an estimated percentage yield 

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  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 = 

can be simplified to 

K_a =

• • 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 =

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 =

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 = 

• 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 =  

Diagram with examples of percentage uncertainty calculations for common laboratory apparatus

The absolute uncertainty for analogue measurements is ± half a division and for digital measurements is ± the last significant division

• 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
  •  Mathematically, this can be described as y = , 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 =