Math Fundamentals for Chemistry

Introduction: Why Start with Math?

Chemistry is a quantitative science. We measure, calculate, and predict phenomena using the shared language of mathematics. This chapter is a review of the essential math skills that form the toolkit for everything else we will do.

Mastering these core concepts now will allow you to focus on the chemistry later, without getting bogged down by the calculations.

Number Notation Formats

Chemists work with numbers that span an enormous range. The number of molecules in a drop of water is about 1,670,000,000,000,000,000,000, while the mass of a single electron is 0.000000000000000000000000000000911 kg. Writing these values in full standard notation is impractical and error-prone.

Scientists use several formats to represent numbers more efficiently. Let’s compare four common notations using the same example: 47,000,000.

Standard Notation47,000,000Unnormalized Scientific470 × 105Normalized Scientific4.7 × 107Engineering Notation47 × 106

The Four Notation Formats

Standard Notation is the familiar way we write numbers with all digits shown (47,000,000). While intuitive, it becomes unwieldy for very large or very small values.

Unnormalized Scientific Notation expresses a number as any coefficient multiplied by a power of 10. In the example above, 470 × 105 is valid unnormalized notation. The coefficient can be any value, and the exponent adjusts accordingly. While rarely used intentionally, unnormalized notation often appears as an intermediate step in calculations before renormalizing to standard form.

Normalized Scientific Notation (also called standard form) follows a specific rule: the coefficient must be between 1 and 10 (i.e., 1 ≤ |M| < 10), with exactly one non-zero digit to the left of the decimal point. This format is written as:

\[ M \times 10^n \]

where M is the coefficient (also called the significand) and n is the exponent. The value 47,000,000 becomes 4.7 × 107, where 4.7 is the coefficient and 7 is the exponent. This is the standard format used throughout chemistry and most scientific fields.

Engineering Notation restricts the exponent to multiples of 3 (103, 106, 109, etc.). This aligns with metric prefixes like kilo (103), mega (106), and giga (109). The value 47,000,000 becomes 47 × 106, which an engineer would read as “47 mega” (47 M). This makes conversions to prefixed units more direct.

Engineering notation is particularly useful in electrical engineering and fields that work with metric prefixes. The restriction to exponents that are multiples of 3 creates a direct correspondence with SI prefix names. Here are some examples:

Notice how the engineering notation exponent directly tells you which metric prefix to use: 103 = kilo, 106 = mega, 10−6 = micro.

NoteCalculator Display: E Notation

Calculators and computer programs cannot display superscripts, so they use E notation to represent “× 10 to the power of”. The letter ‘E’ (or ‘e’) is followed by the exponent.

  • 4.7 × 107 is displayed as 4.7E7 or 4.7e7
  • 6.022 × 1023 is displayed as 6.022E23 or 6.022e23
  • 1.67 × 10−27 is displayed as 1.67E-27 or 1.67e-27

The ‘E’ is not a mathematical variable. It is simply shorthand for the power of ten.

For this chemistry course, we will use standard notation and normalized scientific notation. However, you should recognize the other formats when you encounter them in different contexts.

Converting to Scientific Notation

  • For large numbers (n > 0): Move the decimal point to the left until you have a number between 1 and 10. The number of places you moved is your positive exponent.

    • Example: The speed of light is 299,792,458 m/s. Move the decimal 8 places to the left: 2.99792458 × 108 m/s.

  • For small numbers (n < 0): Move the decimal point to the right until you have a number between 1 and 10. The number of places you moved is your negative exponent.

    • Example: The mass of one proton is 0.00000000000000000000000000167 kg. Move the decimal 27 places to the right: 1.67 × 10−27 kg.

Calculations with Scientific Notation

Your calculator will handle these operations, but you should understand the rules.

  • Multiplication: Multiply the coefficients and add the exponents.
    • Example: \((2 \times 10^3) \times (3 \times 10^2) = (2 \times 3) \times 10^{3+2} = 6 \times 10^5\)
  • Division: Divide the coefficients and subtract the exponents.
    • Example: \((6 \times 10^5) / (3 \times 10^2) = (6 / 3) \times 10^{5-2} = 2 \times 10^3\)
  • Addition/Subtraction: The numbers must first be converted to have the same exponent. Then, add or subtract the coefficients. This is because you can only add or subtract like terms—just as you can’t directly add 2 meters and 3 kilometers without converting them to the same unit first.
    • Example: \((4.0 \times 10^3) + (2.0 \times 10^2)\) First, rewrite the second number: \(2.0 \times 10^2 = 0.20 \times 10^3\). Now, add the coefficients: \((4.0 + 0.20) \times 10^3 = 4.2 \times 10^3\).

The Rules of Calculation: Order of Operations

When you face an equation with multiple steps, the order in which you perform the calculations is critical. A different order can lead to a completely different, incorrect answer. The universally accepted sequence is known by the acronym PEMDAS.

  • Parentheses: Always solve what’s inside parentheses first.
  • Exponents: Next, handle all exponents and roots.
  • Multiplication and Division: Perform all multiplication and division from left to right.
  • Addition and Subtraction: Finally, perform all addition and subtraction from left to right.
Important

Common Mistake: Multiplication does NOT always come before division, and addition does NOT always come before subtraction. Within each pair (M/D and A/S), work strictly from left to right.

Example: \(12 \div 3 \times 2 = (12 \div 3) \times 2 = 4 \times 2 = 8\), NOT \(12 \div (3 \times 2) = 12 \div 6 = 2\)

Example: Solve \(5 + (3 \times 2^2) - 1\)

  1. Parentheses: Inside, we have an exponent.
  2. Exponent: \(2^2 = 4\). The expression is now \(5 + (3 \times 4) - 1\).
  3. Parentheses (continued): Now do the multiplication inside. \(3 \times 4 = 12\). The expression is now \(5 + 12 - 1\).
  4. Addition/ Subtraction (left to right): \(5 + 12 = 17\). Then \(17 - 1 = 16\).
    • The final answer is 16.

Working with Logarithms and Exponentials

Exponentials and logarithms (“logs”) are essential for topics like pH, kinetics, and thermodynamics. A logarithm is simply the inverse of an exponent.

The expression \(log_b(x)\) asks the question: “To what power must I raise base \(b\) to get the number \(x\)?”

The Two Most Common Logs in Chemistry:

  • Common Log (base 10): Written as \(\log(x)\). This is used for the pH scale.
    • Example: \(\log(1000) = 3\), because \(10^3 = 1000\).
  • Natural Log (base e): Written as \(\ln(x)\). The number \(e\) is an important mathematical constant, approximately 2.718. This log appears in many equations related to energy and reaction rates.
    • Example: \(\ln(7.39) \approx 2\), because \(e^2 \approx 7.39\).

Key Logarithm Rules:

  • \(\log(AB) = \log(A) + \log(B)\)
  • \(\log(A / B) = \log(A) - \log(B)\)
  • \(\log(A^n) = n \times \log(A)\)

These rules apply to all logarithms, regardless of the base. Here’s an example:

  • Example: Simplify \(\log(100 \times 50)\)

    Using the product rule: \(\log(100) + \log(50) = 2 + 1.699 = 3.699\)

    Compare to direct calculation: \(\log(5000) = 3.699\) ✓

The Antilogarithm

The “antilog” is how you reverse a logarithm to find the original number. It simply means making the log’s result an exponent.

  • If \(\log(x) = y\), then \(x = 10^y\).
  • If \(\ln(x) = y\), then \(x = e^y\).

The Mantissa in Logarithms

When working with logarithms in chemistry, you’ll encounter the term mantissa—the fractional part of a logarithm (the digits after the decimal point). The mantissa is particularly important when dealing with significant figures.

For example, in log(325) = 2.512:

  • The characteristic is 2 (the whole number part, which indicates the order of magnitude)
  • The mantissa is .512 (the fractional part, which carries the precision of the original measurement)

The same concept applies to antilogarithms. In 102.512 = 325, the mantissa .512 in the exponent determines the significant figures in the result.

The MantissaLogarithmsAntilogarithmslog(325) = 2.512ln(6.8 × 103) = 8.82102.512= 325e8.82= 6.8 × 103

Understanding the mantissa is essential for correctly handling significant figures in pH calculations and other logarithmic operations, which we’ll cover in the Significant Figures chapter.

NoteThe Lost Art of Logarithms

To us, the ‘log’ button on a calculator is instantaneous. But what if I told you that behind that button lies a “lost art” that powered science for over 300 years?

Understanding logarithms isn’t just about memorizing rules; it’s about discovering a beautiful method of thinking that allowed humanity to explore the universe. For more on this history and how logarithms actually work, see the online book The Lost Art of Logarithms.

The Importance of Context: Units

A number in science is meaningless without its unit. If you tell someone a distance is “15,” they don’t know if that means 15 inches, 15 meters, or 15 light-years. Units give numbers their physical meaning.

The SI System

To avoid confusion, scientists around the world agree to use a standard set of units called the International System of Units (SI). There are seven SI base units, but these are the most common in chemistry:

Common Metric Prefixes

The metric system, which the SI system is based on, uses prefixes to scale these base units by powers of ten. You must be familiar with these prefixes:

Quick Reference

The Metric Prefixes can be conveniently accessed in the “Quick Reference” section.

  • Example: 1 kilometer (km) = 1000 meters (m)
  • Example: 1 nanogram (ng) = 10−9 grams (g)

Understanding these foundational concepts is the first step toward building the skills you need to solve real-world chemistry problems, which we will begin in the next sections on significant figures and dimensional analysis.