Question

Round each of the following measurements to the number of significant figures indicated. a. 67.029 g to three significant figures b. 0.15 L to one significant figure c. 52.8005 \(\mathrm{mg}\) to five significant figures d. 3.17497 mol to three significant figures

Short answer

a. 67.0 g, b. 0.2 L, c. 52.801 mg, d. 3.17 mol.

Step-by-step solution

Identify Significant Figures

First, recall the rules for significant figures: all non-zero digits are significant, zeros between significant digits are significant, and leading zeros are not significant. Trailing zeros are significant if there is a decimal point.

Round 67.029 g to Three Significant Figures

The number 67.029 has five significant figures. To round it to three significant figures, look at the first three digits (67.0) and the fourth digit (2). Since the fourth digit is less than 5, round down. Therefore, 67.029 g rounded to three significant figures is 67.0 g.

Round 0.15 L to One Significant Figure

The number 0.15 has two significant figures (1 and 5). To round it to one significant figure, look at the first digit (1) and the second digit (5). Since the second digit is 5 or more, round up. Therefore, 0.15 L rounded to one significant figure is 0.2 L.

Round 52.8005 mg to Five Significant Figures

The number 52.8005 has six significant figures. To round it to five significant figures, look at the first five digits (52.800) and the sixth digit (5). Since the sixth digit is 5 or more, round up, making it 52.801. Therefore, 52.8005 mg rounded to five significant figures is 52.801 mg.

Round 3.17497 mol to Three Significant Figures

The number 3.17497 has six significant figures. To round it to three significant figures, look at the first three digits (3.17) and the fourth digit (4). Since the fourth digit is less than 5, round down. Therefore, 3.17497 mol rounded to three significant figures is 3.17 mol.

Key concepts

Rounding Rules

Rounding is crucial for simplifying numbers and maintaining proper precision. Here are the basic rounding rules you need to follow:

• If the digit after the rounding place is less than 5, you round down.
• If the digit is 5 or more, you round up.

For example, when rounding 67.029 to three significant figures, you look at the fourth digit. Since it's 2 (less than 5), you round down to 67.0.
Remember to apply these rules consistently to ensure accuracy.

Measurement Precision

Precision in measurements dictates how detailed and exact a measurement is. Higher precision means finer details and more significant digits.
When rounding, maintaining measurement precision is key to avoid losing crucial information. For instance, in rounding 0.15 L to one significant figure, we consider the digit 5, which means rounding up to 0.2 L.
The more precise a measurement, the more significant figures you should retain to represent it accurately.

Significant Digits

Significant digits are the meaningful digits in a number that contribute to its precision. Here's how to determine them:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.
• Trailing zeros are significant if there's a decimal point.

For example, in 52.8005 mg, the number has six significant digits. To round to five significant digits, we observe the sixth digit and round accordingly, resulting in 52.801 mg.

Mathematical Rounding

Mathematical rounding involves adjusting numbers to reduce complexity while maintaining accuracy. It's a balancing act to simplify numbers without distorting their value.
Steps for mathematical rounding are:

• Identify the digit at the desired decimal place.
• Check the next digit to decide whether to round up or down.

For instance, rounding 3.17497 mol to three significant figures, we focus on the third digit and the fourth digit. Since 4 is less than 5, we round down to 3.17 mol.
Applying these steps helps in achieving the correct precision in measurements and calculations.