Problem 11 - Chemistry The Central Science In Si Units

Question

Consider the jar of jelly beans in the photo. To get an estimate of the number of beans in the jar you weigh six beans and obtain masses of \(3.15,3.12,2.98,3.14,3.02,\) and \(3.09 \mathrm{~g}\). Then you weigh the jar with all the beans in it, and obtain a mass of \(2082 \mathrm{~g}\). The empty jar has a mass of \(653 \mathrm{~g}\). Based on these data, estimate the number of beans in the jar. Justify the number of significant figures you use in your estimate.

Step-by-Step Solution

Answer

There are approximately 463 beans in the jar, rounded to three significant figures.

Step 1: Calculate the Total Mass of Jelly Beans

First, determine the mass of the jelly beans alone by subtracting the mass of the empty jar from the total mass of the jar with the beans. \[\text{Mass of beans} = 2082 \text{ g} - 653 \text{ g} = 1429 \text{ g}\]

Step 2: Calculate the Average Mass of One Bean

To find the average mass of one bean, add together the masses of the six beans and then divide by six (the number of beans weighed).\[\text{Average mass} = \frac{3.15 + 3.12 + 2.98 + 3.14 + 3.02 + 3.09}{6} = \frac{18.50}{6} = 3.0833 \text{ g/bean}\]

Step 3: Estimate the Number of Beans

Use the total mass of the beans and the average mass of a single bean to estimate the number of beans in the jar. Divide the total mass of beans from Step 1 by the average mass from Step 2.\[\text{Number of beans} = \frac{1429 \text{ g}}{3.0833 \text{ g/bean}} \approx 463.37\]

Step 4: Determine Significant Figures

The least precise measurement was the total mass of the beans (1429 g), which has four significant figures. To reflect this precision, round the estimated number of beans to the nearest whole number, which results in 463 beans.

Key Concepts

Average Mass Calculation

When calculating the average mass of an object, like our jelly beans, we take the sum of the individual masses and then divide by the total number of items. This gives us a typical or 'average' value for a single object.
The formula used for average mass calculation is:

• Sum all the individual mass measurements.
• Divide by the number of measurements taken.

In our example, we added the masses of the six beans, which were 3.15, 3.12, 2.98, 3.14, 3.02, and 3.09 grams, totaling 18.50 grams. Then, we divided this sum by six, the number of beans weighed, resulting in an average mass of approximately 3.0833 grams per bean.
Calculating the average helps standardize the measurement, especially if individual mass varies slightly. This is particularly useful in gaining a fair approximation in broader calculations such as estimating total quantities.

Mass Measurement

Measuring mass, especially in experiments involving single objects such as jelly beans, involves careful consideration of both precision and accuracy. Precision refers to the closeness of two or more measurements to each other, while accuracy relates to how close a measurement is to its true value.
In our scenario:

• The mass of one bean is measured multiple times to assure precise results.
• The mass of the jar with and without the beans is measured to find the mass of the beans alone.

Our steps involved subtracting the jar's mass (653 grams) from the mass of the jar filled with beans (2082 grams), resulting in the total mass of beans (1429 grams). Such careful measurement ensures minimal error and provides a base for further analysis, such as estimating the number of beans.

Estimation Techniques

Estimation techniques help approximate values that can be cumbersome to measure directly. By applying a step-by-step approach, we reach a practical answer that serves our immediate needs, even amid potential variations.

• First, precision in measuring components (like the beans' mass) aids in reliable base calculations.
• Second, calculating the average provides a standardized measure to facilitate uniform assumptions.
• Third, dividing the total mass of beans by the average mass of a single bean estimates the overall count effectively.

In practice, after we determined a total bean mass of 1429 grams and an average bean mass of 3.0833 grams, dividing these gives an estimated count of approximately 463.37 beans. This estimated number reflects possible decimal variations but should be practical enough for most uses, thus highlighted with appropriate significant figures.