Problem 110 - Chemistry Matter and Change

Question

Round each figure to three significant figures. \(\begin{array}{ll}{\text { a. } 0.003210 \mathrm{g}} & {\text { d. } 25.38 \mathrm{L}} \\ {\text { b. } 3.8754 \mathrm{kg}} & {\text { e. } 0.08763 \mathrm{cm}} \\ {\text { c. } 219,034 \mathrm{m}} & {\text { f. } 0.003109 \mathrm{mg}}\end{array}\)

Step-by-Step Solution

Answer

a. \(0.003210 g \approx 0.00321 g\) b. \(3.8754 kg \approx 3.88 kg\) c. \(219,034 m \approx 219,000 m\) d. \(25.38 L \approx 25.4 L\) e. \(0.08763 cm \approx 0.0876 cm\) f. \(0.003109mg \approx 0.00311 mg\)

Step 1: a. Rounding 0.003210 g

First, locate the first non-zero digit (the first significant digit). For 0.003210 g, it is the 3. Since we want to round it to three significant figures, we will consider the next two digits as well (2 and 1). We will drop all the rest digits and round based on the next digit. The next digit is 0 so we don't need to increase 1 in the last significant digit. So it becomes: \(0.00321 g \approx 0.00321 g\)

Step 2: b. Rounding 3.8754 kg

Here 3 is the first significant digit, and we need three significant digits, so we look at 8 and 7. We round based on the next digit which is 5, so rounding up 7 to 8, and we get: \(3.8754 kg \approx 3.88 kg\)

Step 3: c. Rounding 219,034 m

For 219,034 m, 2, 1, and 9 are the three significant digits as these are the first three nonzero digits. We round based on the next digit which is 0, so we don't need to round up. Then it becomes: \(219,034 m \approx 219,000 m\)

Step 4: d. Rounding 25.38 L

Here 2, 5, and 3 are the three significant digits, and we round based on the next digit, which is 8, so rounding the last significant digit (3) up by 1, and we get: \(25.38 L \approx 25.4 L\)

Step 5: e. Rounding 0.08763 cm

First, locate the first significant digit which is 8. We require three significant digits, so we will also consider 7 and 6. Based on the next digit, 3, we don't need to round up. So it becomes: \(0.08763 cm \approx 0.0876 cm\)

Step 6: f. Rounding 0.003109mg

For 0.003109 mg, 3, 1, and 0 are the three significant digits as these are the first three nonzero digits. We round based on the next digit which is 9, so rounding the last significant digit (0) up by 1, and we get: \(0.003109mg \approx 0.00311 mg\)

Key Concepts

Scientific Notation

Understanding scientific notation is crucial for handling very large or very small numbers efficiently. This notation expresses numbers as a product of two parts: a coefficient and a power of ten. The coefficient is a number greater than or equal to 1 but less than 10, and it includes all of the significant digits of the original number. The power of 10 component indicates how many places to move the decimal point to recover the original number.

For example, the number 219,034 in scientific notation becomes \(2.19034 \times 10^5\), reflecting the process of shifting the decimal point five places to the right to obtain the original number. When rounding significant figures in scientific notation, you adjust the coefficient to have the desired number of significant digits, while the power of ten remains unchanged. Therefore, for three significant figures, 219,034 becomes \(2.19 \times 10^5\).

This not only helps in maintaining precision, but also simplifies calculations, as working with powers of ten is easily manageable.

Measurement Precision

Measurement precision refers to how closely individual measurements agree with each other. It's important to distinguish between precision and accuracy, where accuracy reflects how close measurements are to the true or accepted value. When dealing with precision in measurement, significant figures play a key role. They indicate the reliable digits in a measurement, including the last digit which is an estimate.

Precision is also reflected in the level of detail we express in a measurement. For example, stating a length as 0.003210 meters suggests a higher precision than 0.00321 meters, although both may originate from similar measurements. When you round to a certain number of significant figures, you're selecting a balance between detail and practicality, ensuring that the figures you use are meaningful and reflective of your measuring capability.

Significant Digits

Significant digits, or significant figures, are the digits within a number that carry meaning towards its precision. They include all nonzero numbers, any zeros between significant digits, and trailing zeros in the decimal portion of a number. Leading zeros, which simply place the decimal point, are not significant.

For example, in the number 0.003210 grams, the significant digits are 3210. The leading zeros are not counted because they aren't measuring anything; they just help us read the scale of the number. When rounding to a specific number of significant figures, as in our exercise where we round to three significant figures, we focus on maintaining only the most meaningful digits of the original number to represent the precision of the measurement. As such, 3.8754 kilograms rounded to three significant figures is 3.88 kilograms – here, the last digit, 8, reflects an estimate based on the next digit (5), which dictates the rounding action.