Problem 60 - General Chemistry

Question

The astronomical unit equals the mean distance between the earth and the sun. This distance is \(150,000,000 \mathrm{~km},\) which is precise to three significant figures. Express this in scientific notation to the correct number of significant figures.

Step-by-Step Solution

Answer

1.50 x 10^8 km

Step 1: Identify Significant Figures

The given number is 150,000,000 km. This number has 3 significant figures: 1, 5, and 0 (up to the first 0 in the zeros that follow).

Step 2: Convert to Scientific Notation

Express 150,000,000 km in the form of a decimal number multiplied by a power of 10. Move the decimal point 8 places to the left, which gives you 1.50.

Step 3: Formulate the Scientific Notation

Since we moved the decimal point 8 places to the left, the power of 10 will be 8. Therefore, the scientific notation is 1.50 x 10^8.

Key Concepts

Significant Figures

Understanding significant figures is essential when dealing with measurements. They are important in reflecting the precision of a given number. In the exercise provided, the number 150,000,000 km is said to have three significant figures, which are 1, 5, and the first 0.

• Why Significant Figures Matter: They indicate the certainty of the measurement. More figures suggest more precise measurements.
• How to Count Them: Begin counting from the first non-zero digit. In the number 150,000,000, you start from 1 and count until the first zero.
• Zeroes as Significant Figures: Leading zeroes are never significant, and trailing zeroes may be significant depending on the presence of a decimal point.

Understanding this concept helps in maintaining accuracy when converting large numbers to scientific notation.

Astronomical Unit

An astronomical unit (AU) is a standard unit of measurement used in astronomy to describe vast distances. Specifically, it refers to the average distance between the Earth and the Sun.

• Significance in Astronomy: It provides a convenient way to express and understand cosmic distances within the solar system.
• Actual Distance: One AU is approximately 150,000,000 km.

This measure is utilized to simplify the communication of distances in the universe, instead of using large and cumbersome numbers.

Decimal Notation

Decimal notation is a way to represent numbers using the base ten. This system is familiar as it’s based on powers of ten, making it universally understood. In the example given, we convert 150,000,000 km to scientific notation.

• Standard Format: A regular decimal number system uses digits 0-9 and places them in increasing powers of ten from right to left.
• Conversion to Scientific Notation: Place the decimal after the first significant figure and count the places moved to determine the power of ten.
• Advantages: Decimal notation allows for a clear and straightforward presentation of numbers, which makes calculation and comprehension easier, especially with large numbers.

It's this simplicity that makes decimal notation useful in scientific calculations and expressing large numbers efficiently.