Ch.1 - Introduction: Matter, Energy, and Measurement

Brown - Chemistry: The Central Science 14th Edition

Brown14th EditionChemistry: The Central ScienceISBN: 9780134414232Not the one you use?Change textbook

Brown 14th Edition

Ch.1 - Introduction: Matter, Energy, and Measurement

Problem 11

Chapter 1, Problem 11

Consider the jar of jelly beans in the photo. To get an estimateof the number of beans in the jar you weigh six beansand obtain masses of 3.15, 3.12, 2.98, 3.14, 3.02, and 3.09 g.Then you weigh the jar with all the beans in it, and obtain amass of 2082 g. The empty jar has a mass of 653 g. Based onthese data, estimate the number of beans in the jar. Justifythe number of significant figures you use in your estimate.

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Calculate the average mass of a single jelly bean by adding the masses of the six beans and dividing by six: \( \text{Average mass} = \frac{3.15 + 3.12 + 2.98 + 3.14 + 3.02 + 3.09}{6} \).

Determine the total mass of the jelly beans by subtracting the mass of the empty jar from the mass of the jar with beans: \( \text{Total mass of beans} = 2082 \text{ g} - 653 \text{ g} \).

Estimate the number of jelly beans by dividing the total mass of the beans by the average mass of a single bean: \( \text{Number of beans} = \frac{\text{Total mass of beans}}{\text{Average mass of a bean}} \).

Consider the significant figures in your calculations: the average mass of a bean should be reported with the same number of decimal places as the least precise measurement, and the total mass of beans should be calculated with the precision of the subtraction operation.

Use the appropriate number of significant figures in your final estimate of the number of beans, based on the precision of the measurements used in the calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Mass and Weight Measurement

In chemistry, mass is a measure of the amount of matter in an object, typically measured in grams. Weight, while often used interchangeably with mass, refers to the force exerted by gravity on that mass. Accurate mass measurements are crucial for calculations in stoichiometry and determining quantities in chemical reactions.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. This concept is essential in scientific measurements, as it reflects the certainty of the data collected. When performing calculations, the result should be reported with the same number of significant figures as the measurement with the least number of significant figures to maintain accuracy.

Estimation and Calculation of Quantity

Estimating the number of jelly beans involves calculating the total mass of the beans and then dividing by the average mass of a single bean. This process requires understanding how to manipulate mass data and apply basic arithmetic to derive a meaningful quantity. The average mass is calculated from the measured masses of the six beans, which is then used to estimate the total number of beans in the jar.

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