Measurement and Problem Solving in Chemistry: Scientific Notation and Uncertainty

Measurement and Problem Solving

Introduction

Accurate measurement and problem solving are foundational skills in chemistry. This section introduces scientific notation, uncertainty in measurements, and the significance of reporting data correctly.

Module 2.1-2.2: Scientific Notation

Objectives

• Express very large and very small numbers using scientific notation.

Key Vocabulary

• Scientific notation: A method for expressing numbers as a product of a decimal part and an exponential part.

• Decimal part: A number between 1 and 10.

• Exponential part: 10 raised to an exponent (n).

• Exponent: The power to which 10 is raised.

Uncertainty in Measurement

All measurements in science have some degree of uncertainty, which is indicated by the last reported digit.

• Example: If average global temperatures have risen by 0.6 degree Celsius in the last century, reporting "0.6 degree Celsius" means 0.6 ± 0.1 degree Celsius.

• The true value could be as much as 0.7 or as little as 0.5 degree Celsius.

• The degree of certainty in measurements can influence important decisions.

Structure of Scientific Notation

A number in scientific notation has two parts:

• Decimal part: A number between 1 and 10.

• Exponential part: 10 raised to an exponent, $n$.

Example: $1.2 \times 10^{-10}$

Writing Very Large and Very Small Numbers

• A positive exponent ($n$) means 1 multiplied by 10 $n$ times.

• A negative exponent ($-n$) means 1 divided by 10 $n$ times.

Examples:

• $10^0 = 1$

• $10^1 = 10$

• $10^2 = 100$

• $10^3 = 1000$

• $10^{-1} = \frac{1}{10} = 0.1$

• $10^{-2} = \frac{1}{10 \times 10} = 0.01$

• $10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001$

Converting a Number to Scientific Notation

• Find the decimal part by moving the decimal point to obtain a number between 1 and 10.

• Multiply that number by 10 raised to the power that reflects the movement of the decimal point.

Example steps:

• Move the decimal point to the left: exponent is positive.

• Move the decimal point to the right: exponent is negative.

Example conversions:

• $5083 = 5.083 \times 10^3$

• $0.00034 = 3.4 \times 10^{-4}$

Practice Problems

• Express 323,000,000 in scientific notation: $3.23 \times 10^8$

• Express 0.0000000007 in scientific notation: $7 \times 10^{-10}$

Converting Scientific Notation to Standard Notation

• Move the decimal point the number of places indicated by the exponent.

• If the exponent is positive, move the decimal to the right; if negative, move it to the left.

Example:

• $4.5 \times 10^{-4}$ mm = 0.00045 mm

• $1.85 \times 10^3$ = 1850

Summary Table: Scientific Notation Conversion

Standard Notation	Scientific Notation
5083	$5.083 \times 10^3$
0.00034	$3.4 \times 10^{-4}$
323,000,000	$3.23 \times 10^8$
0.0000000007	$7 \times 10^{-10}$

Additional info: Later modules cover significant figures, SI units, dimensional analysis, and density, which are also essential for introductory chemistry.