Problem 78 - Chemistry and Chemical Reactivity

Question

Give the number of significant figures in each of the following numbers: (a) \(0.00546 \mathrm{g}\) (b) \(1600 \mathrm{mL}\) (c) \(2.300 \times 10^{-4} \mathrm{g}\) (d) \(2.34 \times 10^{9}\) atoms

Step-by-Step Solution

Answer

(a) 3 sf, (b) 2 sf, (c) 4 sf, (d) 3 sf.

Step 1: Identify Significant Figures in 0.00546 g

In the number 0.00546, leading zeros are not significant. The significant figures here are '5', '4', and '6'. Thus, the number 0.00546 has 3 significant figures.

Step 2: Identify Significant Figures in 1600 mL

In the number 1600, it is important to determine if zeros are placeholders or significant. Without a decimal point, 1600 generally has 2 significant figures. If the precision of the measurement was greater, it would typically be indicated by writing 1600. or using scientific notation.

Step 3: Identify Significant Figures in 2.300 × 10^{-4} g

In the number 2.300, all the zeros are significant because they are between significant figures or at the end of the number with a decimal point. Therefore, 2.300 has 4 significant figures.

Step 4: Identify Significant Figures in 2.34 × 10^{9} atoms

In the number 2.34, all digits are significant because there are no zeros acting merely as placeholders. Thus, 2.34 has 3 significant figures.

Key Concepts

Scientific Notation

Scientific notation is a method used to express large or small numbers in a more manageable form. This format consists of a number between 1 and 10 multiplied by a power of ten. For example, the number \(2.300 \times 10^{-4}\) is expressed in scientific notation. In this method:

• The number before the multiplication sign is called the significand or coefficient.
• The power of ten indicates how many places the decimal point has moved.
• This is especially useful in fields like chemistry and physics where extremely large or small values are common.

By using scientific notation, we limit confusion and make calculations simpler by reducing the number of zeros we have to work with. It also helps in clearly identifying significant digits.

Significant Digits

Significant digits, also known as significant figures, are crucial in conveying the precision of a numeric value. These digits in a number reflect which are known to be reliable based on the precision of measurement. For example, in the number 0.00546, the significant digits are 5, 4, and 6.

• All non-zero digits are always significant.
• Zeros between non-zero digits are significant.
• Leading zeros, or those before the first non-zero digits, are not significant.
• Trailing zeros in a number with a decimal point are significant.

Understanding significant digits helps ensure data accuracy and consistency, especially when reporting or interpreting results.

Measurement Precision

Measurement precision refers to how finely a measurement is expressed. It indicates the exactness of a measurement value. For instance, the number 2.300 shows that the measurement has been recorded to the thousandths place, indicating high precision. This is crucial because:

• It defines the resolution of a measurement process or tool.
• Precise measurements often incorporate more significant figures.
• Precision is essential when the goal is to track small changes in a value.

Recording measurements with proper precision ensures that the results are reliable and repeatable in scientific and engineering contexts.

Decimal Point Significance

The presence or absence of a decimal point can drastically alter the interpretation of zeros as significant or not. This subtlety is crucial in understanding a number’s accuracy. Consider the numbers 1600 and 1600.:

• In 1600, without a decimal point, the zeros are considered placeholders, giving generally 2 significant figures.
• 1600. has a decimal point, indicating that the zeros are indeed significant, leading to 4 significant figures.

This significance affects how we record and communicate data, as well as perform calculations involving these values, ensuring clarity and precision in various scientific communications.