Problem 40 - Chemistry: The Central Science

Question

(a) The diameter of Earth at the equator is \(7926.381 \mathrm{mi}\). Round this number to three significant figures and express it in standard exponential notation. (b) The circumference of Earth through the poles is \(40,008 \mathrm{~km}\). Round this number to four significant figures and express it in standard exponential notation.

Step-by-Step Solution

Answer

(a) The diameter of Earth at the equator rounded to three significant figures and expressed in standard exponential notation is: \(7.93 \times 10^3 \mathrm{mi}\). (b) The circumference of Earth through the poles rounded to four significant figures and expressed in standard exponential notation is: \(4.001 \times 10^4 \mathrm{~km}\).

Step 1: Identify the significant figures

In the given diameter, \(7926.381 \mathrm{mi}\), there are six significant figures: 7, 9, 2, 6, 3, and 8. We need to round to three significant figures.

Step 2: Round to three significant figures

To round to three significant figures, we will keep the first three digits (7, 9, 2) and consider the fourth digit (6). Since the fourth digit is greater than or equal to 5, we need to round up. So, the rounded number is: \(7930 \mathrm{mi}\).

Step 3: Express in standard exponential notation

To convert \(7930 \mathrm{mi}\) to standard exponential notation, write as \(7.93 \times 10^n\), where n is the exponent. Here, n is the number of places we have to move the decimal point to the left, which is 3. So, the final expression in standard exponential notation is: \(7.93 \times 10^3 \mathrm{mi}\). Part (b):

Step 1: Identify the significant figures

In the given circumference, \(40,008 \mathrm{~km}\), there are five significant figures: 4, 0, 0, 0, and 8. We need to round to four significant figures.

Step 2: Round to four significant figures

To round to four significant figures, we will keep the first four digits (4, 0, 0, 0) and consider the fifth digit (8). Since the fifth digit is greater than or equal to 5, we need to round up. So, the rounded number is: \(40,010 \mathrm{~km}\).

Step 3: Express in standard exponential notation

To convert \(40,010 \mathrm{~km}\) to standard exponential notation, write as \(4.001 \times 10^n\), where n is the exponent. Here, n is the number of places we have to move the decimal point to the left, which is 4. So, the final expression in standard exponential notation is: \(4.001 \times 10^4 \mathrm{~km}\). Therefore, (a) The diameter of Earth at the equator rounded to three significant figures and expressed in standard exponential notation is: \(7.93 \times 10^3 \mathrm{mi}\). (b) The circumference of Earth through the poles rounded to four significant figures and expressed in standard exponential notation is: \(4.001 \times 10^4 \mathrm{~km}\).

Key Concepts

Scientific Notation in Chemistry

Scientific notation is a method used to express very large or very small numbers in a more concise and standardized form. This is particularly useful in chemistry where measurements and calculations can span many orders of magnitude.

To write a number in scientific notation, you place a number between 1 and 10 before an exponential term of 10. The exponent indicates how many spaces the decimal point has been moved to the right for numbers greater than 1 or to the left for numbers less than 1.

For example, expressing the thickness of a chemical film that is 0.000007 meters thick would be cumbersome and prone to error. Using scientific notation, this would be written as \(7 \times 10^{-6}\) meters, which is more compact and easier to work with in calculations.

When converting regular numbers to scientific notation, it's important to determine the precision needed, often guided by the number of significant figures in the original measurements.

Measuring Earth's Dimensions

Understanding Earth's dimensions requires precise measurements that often involve very large numbers. These measurements can include the diameter and circumference of the planet. For instance, the diameter at the equator and the circumference through the poles have specific values that can be difficult to visualize and manipulate due to their magnitude.

To tackle this, scientists and mathematicians use scientific notation to make these values more manageable. By expressing Earth's dimensions in this way, it becomes simpler to understand, compare, and use them in further scientific calculations or when comparing with other celestial bodies.

As technology advances, so too does our capacity for measurement accuracy, thus providing a more refined understanding of our planet. For educational purposes, however, we often round these measurements to a certain number of significant figures to streamline communicating and learning the concepts, while maintaining a reasonable level of precision.

Rounding Numbers in Science

Rounding numbers is a vital skill in science to ensure clarity and appropriate precision. Deciding how many significant figures to use is often determined by the precision of the measurement instruments or the necessity of the calculation being performed.

Rounding involves looking at the number to the immediate right of your last significant digit. If this digit is five or greater, you increase the last significant digit by one. If it is less than five, you leave the last significant digit as it is.

For example, if you have a measurement of \(7.536\), and you want to round to three significant figures, you'd round it to \(7.54\) since the third digit after the decimal point is a '6', which is greater than five. This practice of rounding helps maintain the integrity of data in scientific communication by providing a consistent level of detail.