Problem 45 - Chemistry The Central Science In Si Units

Question

What is the number of significant figures in each of the following measured quantities? (a) \(902.5 \mathrm{~kg}\), (b) \(3 \times 10^{-6} \mathrm{~m}\) (c) \(0.0096 \mathrm{~L}\), (d) \(2.94 \times 10^{3} \mathrm{~m}^{2}\) (e) \(92.03 \mathrm{~km}\) (f) \(782.234 \mathrm{~g}\).

Step-by-Step Solution

Answer

(a) 4, (b) 1, (c) 2, (d) 3, (e) 4, (f) 6.

Step 1: Identify Significant Figures for Part (a)

The number 902.5 has a non-zero digit 9, a zero that appears between two non-zero numbers (0), a non-zero digit 2, and a non-zero digit 5 at the end. In such a number, all zero and non-zero digits are considered significant. So, 902.5 has 4 significant figures.

Step 2: Evaluate Significant Figures for Part (b)

In 3 \( \times 10^{-6}\), the number before the multiplication is a single digit, 3, which is significant in scientific notation. The exponent does not affect the count of significant figures, so there is 1 significant figure in this number.

Step 3: Determine Significant Figures for Part (c)

The number 0.0096 has leading zeros (0s before the first non-zero digit), which are not significant. Only the non-zero digits (9 and 6) are significant. Therefore, there are 2 significant figures.

Step 4: Calculate Significant Figures for Part (d)

In the number 2.94 \( \times 10^{3} \), all non-zero digits (2, 9, and 4) are significant. The exponent does not affect the total count of significant figures. Therefore, there are 3 significant figures.

Step 5: Count Significant Figures for Part (e)

For 92.03, all digits (9, 2, 0, and 3) are either non-zero or zeros between non-zero digits, which are significant. Hence, the number has 4 significant figures.

Step 6: Assess Significant Figures for Part (f)

The number 782.234 has all non-zero digits (7, 8, 2, 2, 3, and 4), which are significant. Therefore, it contains 6 significant figures.

Key Concepts

Scientific Notation

Scientific notation is a method of writing very large or very small numbers. It helps express numbers in a compact form. This notation uses powers of ten to simplify the presentation. For instance, the number 3,000 can be written as \(3 \times 10^3\). The first part, which contains the digits, is called the "significant part".
The significant part contains the digits that are truly important for the precision of the number. In scientific notation, significant figures matter just as much. When you encounter a number like \(3 \times 10^{-6}\), the digit '3' is the only significant figure here.
Key benefits of scientific notation include:

• It simplifies calculations by focusing on the most important digits.
• It makes comparisons easier between vastly different numbers.
• It's crucial in fields like science and engineering for clarity.

Leading Zeros

Leading zeros can often be confusing when determining significant figures. They are zeros that appear in front of all non-zero numbers in a decimal number.
For example, in the number 0.0096, the zeros before the '9' are leading zeros. Importantly, these zeros do not count as significant figures. They only serve as placeholders.
Consider these fundamental rules about leading zeros:

• Leading zeros appear at the beginning of a number before the first significant figure.
• They do not add any precision to measured quantities, unlike decimal zeros.
• Only zeros between non-zero digits or at the end of a decimal number are significant.

Understanding these basics ensures you can correctly count significant figures in any situation.

Measured Quantities

Measured quantities refer to the values obtained from measurement tools or methods. These measurements are vital in science for experiments and observations. Each measured quantity often has a degree of precision, indicated by its significant figures.
The number of significant figures in a measurement reflects the precision of the tool used. For instance, measuring 902.5 kg implies a higher precision than measuring 900 kg.
Key points to remember about measured quantities:

• The number of significant figures in a measurement shows how accurate the measurement is.
• More significant figures mean more precision.
• Errors and variations should be expected in measured quantities, and understanding significant figures helps manage these.

Significant Digits

Significant digits are the numbers in a measured value that carry meaningful information about its precision. They include all the non-zero numbers, zeros between them, and zeros at the end, if they come after a decimal point.
In determining the number of significant digits, consider the following rules:

• All non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Trailing zeros in a decimal number are significant because they indicate precision.

For example, in 782.234 grams, all six digits are significant. If a number is written as 92.03 km, then all four digits are significant. Understanding these helps in expressing and using numbers accurately in computations and reports.