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Chemistry · Ch 1 — Some Basic Concepts of Chemistry

Scientific Notation

1.4.1

Scientific Notation

Scientific Notation

Chemistry deals with numbers that span an extraordinary range. A single gram of hydrogen contains roughly 602,200,000,000,000,000,000,000 molecules, while the mass of a single hydrogen atom is about 0.00000000000000000000000166 g. Constants like Planck's constant, the speed of light, and the charges on particles fall into the same territory — numbers choked with zeros that make even simple arithmetic a chore.

Writing or counting such numbers is tedious enough, but performing addition, subtraction, multiplication, or division with them directly is genuinely impractical. Scientific notation — also called exponential notation — solves this problem by expressing any number in the compact form N×10nN \times 10^n, where nn is an integer exponent (positive or negative) and NN is a number between 1.000... and 9.999..., called the digit term.

Converting a Number to Scientific Notation

The procedure is straightforward: move the decimal point until only one non-zero digit remains to its left. The number of places you moved the decimal becomes the exponent of 10 — positive if you moved left, negative if you moved right.

Consider 232.508. The decimal is moved two places to the left to get 2.32508, so the scientific notation is 2.32508×1022.32508 \times 10^2. The exponent 2 matches the number of leftward shifts.

Now take 0.00016. The decimal must be moved four places to the right to obtain 1.6, giving 1.6×10−41.6 \times 10^{-4}. The negative exponent signals the original number was less than 1.

Tip

A quick check: if the original number is greater than 10, the exponent will be positive. If it is less than 1, the exponent will be negative. Numbers between 1 and 10 are already in scientific notation with exponent 0.

Multiplication and Division in Scientific Notation

These operations follow the rules for exponential numbers. When multiplying, multiply the digit terms and add the exponents. When dividing, divide the digit terms and subtract the exponents.

For multiplication:

(a×10m)×(b×10n)=(a×b)×10m+n(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}

For division:

a×10mb×10n=ab×10m−n\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}

Watch out

A common error is to multiply or divide the exponents instead of adding or subtracting them. Remember: exponents add for multiplication and subtract for division — they are not themselves multiplied or divided.

Addition and Subtraction in Scientific Notation

These operations require that the numbers first be written with the same exponent. Only then can the digit terms be added or subtracted.

Procedure:

  1. Adjust one or both numbers so that they share the same exponent.
  2. Add or subtract the digit terms.
  3. Keep the common exponent.

Example — Addition:

Add 6.65×1046.65 \times 10^4 and 8.95×1038.95 \times 10^3.

First, make the exponents equal. Convert 8.95×1038.95 \times 10^3 to 0.895×1040.895 \times 10^4 (moving the decimal one place left and increasing the exponent by 1). Then:

(6.65×104)+(0.895×104)=(6.65+0.895)×104=7.545×104(6.65 \times 10^4) + (0.895 \times 10^4) = (6.65 + 0.895) \times 10^4 = 7.545 \times 10^4

Example — Subtraction:

Subtract 4.8×10−34.8 \times 10^{-3} from 2.5×10−22.5 \times 10^{-2}.

Convert 4.8×10−34.8 \times 10^{-3} to 0.48×10−20.48 \times 10^{-2} (moving the decimal one place left and increasing the exponent by 1). Then:

(2.5×10−2)−(0.48×10−2)=(2.5−0.48)×10−2=2.02×10−2(2.5 \times 10^{-2}) - (0.48 \times 10^{-2}) = (2.5 - 0.48) \times 10^{-2} = 2.02 \times 10^{-2}

Important

Never add or subtract digit terms unless the exponents are identical. Doing so is mathematically invalid — it would be like adding 3 metres and 5 centimetres without first converting to a common unit.

Reference Standards for Measurement

While scientific notation handles the writing of numbers, reliable measurement requires agreed-upon reference standards. Scientists have established such standards for fundamental quantities so that all measuring devices can be calibrated consistently.

The Mass Standard

Since 1889, the kilogram has been defined as the mass of a specific platinum-iridium (Pt-Ir) cylinder stored in an airtight jar at the International Bureau of Weights and Measures in Sèvres, France. Pt-Ir was chosen because it resists chemical attack extremely well, so its mass remains stable over very long periods. …

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