Problem 9 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) What is the length of the pencil in the following figure if the ruler reads in centimeters? How many significant figures are there in this measurement? (b) An automobile speedometer with circular scales reading both miles per hour and kilometers per hour is shown. What speed is indicated, in both units? How many significant figures are in the measurements? [Section 1.6]

Step-by-Step Solution

Answer

To measure the length of the pencil, place the ruler alongside the pencil and read the length in centimeters (cm). The number of significant figures depends on the smallest increment on the ruler. For example, if measured to be 12.5 cm, there are three significant figures. To determine the speed indicated on the automobile speedometer, locate the needle pointing to the current speed in both mph and km/h. The number of significant figures depends on the smallest increment shown on the speedometer. For example, if marked with increments of 1 mph and 1 km/h, count the number of digits in both measurements, including any zeros in the middle and end, but excluding any leading zeros.

Step 1: Measuring Length in Centimeters

To measure the length of the pencil, you will need to place the ruler alongside the pencil and read the value that lines up with the end of the pencil. Make sure you start at the zero mark of the ruler. Record the length in centimeters (cm).

Step 2: Identifying the Number of Significant Figures

Significant figures are the digits in a measurement that are known with certainty plus one uncertain digit. For a ruler that shows centimeters, it is essential to identify the smallest increment on the ruler. For example, if the ruler has markings for every millimeter (mm), then the last digit in the measurement will be uncertain by at most a millimeter. Count the number of digits in the measurement, including any zeros in the middle and end, but excluding any leading zeros. This will give you the total number of significant figures. For example, if the length of the pencil is measured to be 12.5 cm, there are three significant figures.

Step 3: Reading the Automobile Speedometer

To determine the speed indicated on the automobile speedometer, locate the needle that points to the current speed in both miles per hour (mph) and kilometers per hour (km/h). Note the values where the needle is pointing in both units.

Step 4: Identifying the Number of Significant Figures in Measurements from the Speedometer

Just like in the case of a ruler, the number of significant figures depends on the smallest increment shown on the speedometer. For example, if the speedometer is marked with increments of 1 mph and 1 km/h, then the last digit in your measurement will be uncertain by at most 1 unit. Count the number of digits in the measurements for both mph and km/h, including any zeros in the middle and end, but excluding any leading zeros. This will give you the total number of significant figures for each measurement.

Key Concepts

Measurement Techniques

Measurement techniques are how we observe and record data about physical quantities. Using precise and accurate tools is crucial to ensure reliable results.

When measuring a length, like that of a pencil, align the object carefully with measurement markings, starting from the zero point for the most accurate reading. If you're using a ruler, ensure the ruler is not damaged or bent, as this can skew results.
- Always use the correct units when recording, like centimeters (cm) for length. - Each measurement can have a degree of uncertainty, depending on the tool's precision. - To improve accuracy, repeat the measurement a few times and use the average value. When reading the measurement tools such as speedometers or rulers, pay attention to the smallest divisions they can accurately indicate. Understanding these divisions allows you to determine the number of significant figures to report. - Be aware that the last digit in a measurement is the least certain, representing the inherent uncertainty in the measurement tool.

Metric System

The metric system is a uniform system of measurement used worldwide, except in a few countries. It is based on multiples of ten, which makes conversions particularly straightforward and reliable.

This system uses base units like meter (m) for length, liter (L) for volume, and gram (g) for weight, with prefixes like kilo- (k) for thousand times, centi- (c) for one-hundredth, and milli- (m) for one-thousandth.
- For example, 1 kilometer (km) is 1,000 meters. - Easy conversion through moving the decimal point simplifies calculations. The universality of the metric system makes it ideal for scientific and international work due to its consistency and ease of transferability. When performing measurements or conversions, always note whether your measurement is in the metric system to avoid confusion.
Bear in mind the importance of significant figures. They help maintain the correct level of precision with metric units, ensuring measurements and subsequent calculations are both accurate and precise. - Using appropriate significant figures supports the accurate comparison and conversion of metric measurements, maintaining reliable results.

Unit Conversion

Unit conversion is the process of changing a measure expressed in one unit to the equivalent value in another unit. This can involve conversions within the same system (like from meters to centimeters) or between different systems (like from kilometers to miles).

To perform a unit conversion, first identify the conversion factor. A conversion factor is a ratio that expresses how many of one unit are equivalent to another. For example:

• 1 inch equals 2.54 centimeters.
• 1 mile equals 1.609 kilometers.

Use these ratios to create fractions equal to 1, allowing you to multiply without changing the value but altering the unit. This is a core principle in converting units. When converting, pay careful attention to significant figures, as they dictate the precision of your calculations. Always round your final answer to the appropriate number of significant figures matching the least precise measurement involved. In summary, mastering unit conversions enhances your ability to work with different measurement systems and ensures that your data is precise and accurate across various scientific, professional, and everyday contexts.