Problem 81 - Chemistry A Molecular Approach

Question

Round each number to four significant figures. MISSED THIS? Read Section 1.7; Watch \(\mathrm{KCV} 1.7\) a. 156.852 b. 156.842 c. 156.849 d. 156.899

Step-by-Step Solution

Answer

After rounding to four significant figures: a) 156.9, b) 156.8, c) 156.8, d) 156.9.

Step 1: Identify the four significant figures

Count from the first non-zero digit to four digits in each number to identify the significant figures.

Step 2: Round 156.852 to four significant figures

The fifth digit after the first non-zero digit is 2, which doesn't require rounding up, so 156.852 rounded to four significant figures is 156.9.

Step 3: Round 156.842 to four significant figures

The fifth digit after the first non-zero digit is 2, which doesn't require rounding up, so 156.842 rounded to four significant figures is 156.8.

Step 4: Round 156.849 to four significant figures

The fifth digit after the first non-zero digit is 9, which requires rounding up, so 156.849 rounded to four significant figures is 156.8.

Step 5: Round 156.899 to four significant figures

The fifth digit after the first non-zero digit is 9, which requires rounding up the fourth digit, so 156.899 rounded to four significant figures is 156.9.

Key Concepts

Scientific Notation

Understanding scientific notation is essential for dealing with very large or very small numbers in science and engineering. Scientific notation expresses numbers as a multiple of two factors: a coefficient and a power of 10. This format is written as a × 10ⁿ, where n is an integer and the coefficient a is a number between 1 and 10 that includes the significant figures.

For example, the number 156.852 can be written in scientific notation as 1.56852 × 10². When rounding to four significant figures, as in the exercise, we would have 1.569 × 10² after rounding. This notation is particularly useful when working with very large or very small measurements, as it clearly displays the precision of the number and makes calculations easier to manage.

Rounding Numbers

Rounding is a technique used to reduce the number of digits in a number while retaining its approximate value. When rounding to a certain number of significant figures, one must start counting from the leftmost non-zero digit. For numbers with more digits than the desired significant figures, the number is rounded according to the digit that follows the last significant figure.

Following the steps of the exercise, when rounding the number 156.852, we identify the fifth digit (2) after the first non-zero digit. Since this digit is less than 5, we do not round up the fourth significant figure, and our rounded number is 156.9. If the next digit were 5 or greater, as in the case of 156.849, we would round up, making the rounded number 156.8, due to the significant figures being at the beginning of the number.

Measurement Precision

Measurement precision refers to the degree of exactness with which a quantity is measured. It is connected to the concept of significant figures, which are the digits in a number that carry meaning contributing to its measurement precision. High-precision measurements have more significant figures, indicating a more refined measurement.

For instance, a scale that measures to the nearest hundredth of a gram provides more precise data than one that measures to the nearest gram. In the context of the given problem, by rounding numbers to four significant figures, we are standardizing the precision of our measurements. This adjustment in precision may affect computations such as scientific experimentation or engineering calculations, where consistent precision across all numbers involved is critical.