Problem 59 - Chemistry: Atoms First

Question

The label on a soft drink bottle gives the volume in two units: 2.0 \(\mathrm{L}\) and \(67.6 \mathrm{fl}\) oz. Use this information to derive a conversion factor between the English and metric units. How many significant figures can you justify in your conversion factor?

Step-by-Step Solution

Answer

The conversion factor is 0.030 L/fl oz, using two significant figures.

Step 1: Write down the given information

We are given the volume of the soft drink in two different units: 2.0 L and 67.6 fl oz. We will use these values to find the conversion factor.

Step 2: Calculate the conversion factor

To find the conversion factor from fluid ounces to liters, we divide the volume in liters by the volume in fluid ounces. Thus, the conversion factor (C) is obtained by: \( C = \frac{2.0\, L}{67.6\, fl\, oz} \).

Step 3: Perform the division

When we divide 2.0 L by 67.6 fl oz, we get \( C = \frac{2.0}{67.6} = 0.0295857988\, L/fl\, oz \), which we can then round according to the significant figures in the original data.

Step 4: Assess significant figures for conversion factor

The volume in liters is given to two significant figures (2.0 L), and the volume in fluid ounces is given to three significant figures (67.6 fl oz). The conversion factor should be reported with the least number of significant figures, which is two in this case. Therefore, the conversion factor is 0.030 L/fl oz when rounded to two significant figures.

Key Concepts

Significant Figures

In chemistry and other scientific fields, significant figures are a way of indicating the precision of a number. These figures include all the digits that are known with certainty, plus the first uncertain digit. Understanding significant figures is crucial when taking measurements or reporting the results of calculations to ensure that the level of precision is properly communicated.

When determining the significant figures in a number, it is important to identify which digits are significant. For example, in the number 67.6, there are three significant figures: 6, 7, and the final 6, because it is the first uncertain digit. Zero can act both as a placeholder and as a significant figure, depending on its position. In the number 2.0 L, the zeroes indicate significant figures because they come after a decimal point and after a non-zero number.

In calculations, the number with the fewest significant figures determines the precision of the result. When rounding a calculated number, it should have the same number of significant figures as the measurement with the fewest significant figures. In the given exercise, since we have the volume in liters (to two significant figures) and the volume in fluid ounces (to three significant figures), the conversion factor must be rounded to two significant figures, the lesser of the two.

Units Conversion

Converting units is a fundamental skill in chemistry, physics, and other sciences. It allows scientists and students to translate measurements into different systems for comparison or calculation purposes. Units conversion involves using a multiplier, known as a conversion factor, to convert from one unit to another.

To establish a conversion factor like in our exercise, you compare how many of one unit goes into another based on a known equivalency. For instance, if 2.0 L of a substance is equivalent to 67.6 fl oz, you would derive the conversion factor between liters and fluid ounces by dividing the two (as shown in the solved exercise). This conversion factor can then be used to convert any volume between these two units.

A critical step in unit conversion is ensuring that the units you are converting to and from are of the same category, such as volume or length. Incorrectly matching the types of units will lead to incorrect results. For example, you cannot directly convert liters (volume) to meters (length) without additional information about the shape and dimensions of the container.

Metric to English Units

The metric and English systems are two different systems of measurement. The metric system, or the International System of Units (SI), is based on multiples of ten and is used globally for scientific measurement. The English system, mainly used in the United States, is also referred to as the Imperial system.

Common units in the metric system for volume include milliliters (mL) and liters (L), whereas the English system uses ounces (oz), pints (pt), quarts (qt), and gallons (gal). These systems do not directly correlate, which often necessitates the need for a conversion factor to convert measurements from one system to the other.

When deriving a conversion factor between metric and English units, as in our exercise, accuracy is crucial. It requires precise measurements and understanding of how to manipulate significant figures to ensure the conversion is as accurate and useful as possible.

Volume Conversion

Converting volume from one unit to another is often required in many scientific and everyday situations. Volume conversion is simply the process of converting the measure of space occupied by a substance from one unit to another.

When converting volume, you must use a conversion factor that relates the two units of volume, such as liters to fluid ounces. This is achieved by knowing or measuring the equivalent volumes in both units, as we did with the soft drink bottle labeled in both liters and fluid ounces. Then, by dividing or multiplying by the conversion factor, any volume can be converted between these two units.

Remember, when converting volumes, to also account for significant figures — the conversion factor should not imply greater accuracy than the measurements allow. This integrity ensures that the converted volume is consistent with the precision of the original measurement, as we discussed in the exercise.