Question

Indicate which of the following are exact numbers: (a) the mass of a piece of paper, \((\mathbf{b})\) the volume of a cup of coffee, \((\mathbf{c})\) the number of inches in a mile, \((\mathbf{d})\) the number of ounces in a pound, (e) the number of microseconds in a week, (f) the number of pages in this book.

Short answer

The exact numbers are: (c) The number of inches in a mile, which is fixed at 63,360 inches. (d) The number of ounces in a pound, which is fixed at 16 ounces. (e) The number of microseconds in a week, which can be calculated using the defined units \(7 \times 24 \times 60 \times 60 \times (10^6)\) microseconds. (f) The number of pages in this book, as it is a result of counting objects with a fixed value.

Step-by-step solution

(a) Mass of a piece of paper

A piece of paper has a mass that can vary depending on its dimensions, production process, or material. The mass of a piece of paper is an inexact number, as it would require measurement and may vary.

(b) Volume of a cup of coffee

The volume of a cup of coffee would depend on the size of the cup, the amount of liquid inside or even the shape of the cup. This is an inexact number because it requires measurement and may not be consistent from one cup of coffee to another.

(c) Number of inches in a mile

The number of inches in a mile is an exact number, because it is a defined value and does not require measurement. A mile is defined as 5280 feet or 63,360 inches.

(d) Number of ounces in a pound

The number of ounces in a pound is an exact number as it is a conversion between two defined units of measurement. There are exactly 16 ounces in a pound.

(e) The number of microseconds in a week

The number of microseconds in a week is an exact number because it results from a defined unit conversion. We know there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 7 days in a week, so we can find the number of microseconds in a week by multiplying these known values: \[ \text{microseconds in a week} = 7 \times 24 \times 60 \times 60 \times (10^6)\]

(f) Number of pages in this book

The number of pages in a book is an exact number because it comes as a result of counting objects. This does not require a measuring process and is a fixed value.

Key concepts

Units of Measurement

Understanding the concept of units of measurement is fundamental in chemistry where precise quantitative information is crucial. Units of measurement serve as a standardized baseline that allows us to quantify the physical properties of substances and objects. These units can include meters for length, kilograms for mass, and liters for volume, among others.

For example, when measuring the mass of chemicals for a reaction, the unit of measurement ensures that the quantities used are accurate and the reaction proceeds as expected. Without this standardization, communication and replication of results in scientific experiments would be challenging. In the textbook exercise, units such as inches, miles, ounces, and pounds are mentioned, which are standard units used to convey specific amounts of distance and weight.

Conversion of Measurement Units

Measurement units often need to be converted from one system to another, or from larger to smaller units and vice versa, especially in a global context where different measuring systems coexist. Conversion of measurement units is a methodical process which utilizes exact numbers defined by relationships between units.

The exercise mentioned conversion between ounces and pounds, which is a defined exact number—there are precisely 16 ounces in a pound. Similarly, knowing that there are 63,360 inches in a mile, allows for direct conversions between these units without any ambiguity. Understanding how to convert units manually is a valuable skill, but nowadays, many online tools and calculators provide instant conversions which can be very helpful in quickly solving scientific problems.

Defining Exact Numbers

Exact numbers in chemistry are considered as having an unchanging value. They are not obtained by measurement but rather through definition or counting. For example, the number of ounces in a pound, which is 16, or the number of inches in a mile, which is 63,360, are both exact numbers because these are defined conversion factors that do not change.

Exact numbers also include counts of discrete objects, like the number of molecules in a sample, or the number of pages in a book. They have an infinite number of significant figures as they do not have uncertainty or variability. Recognizing which numbers are exact is vital in calculations to prevent rounding errors and to maintain precision in quantitative analysis.