Problem 9 - Chemistry

Question

How many significant figures does each of the following quantities have? (a) \(76.600 \mathrm{~kJ}\) (b) \(4.50200 \times 10^{3} \mathrm{~g}\) (c) \(3000 \mathrm{~nm}\) (d) \(0.00300 \mathrm{~mL}\) (e) 18 students (f) \(3 \times 10^{-5} \mathrm{~g}\) (g) \(47.60 \mathrm{~mL}\) (h) \(2070 \mathrm{mi}\)

Step-by-Step Solution

Answer

(a) 5, (b) 6, (c) 1 or context-dependent, (d) 3, (e) infinite, (f) 1, (g) 4, (h) 3 (context-dependent).

Step 1: Understanding Significant Figures

Significant figures are the digits in a number that contribute to its precision. They include all the certain digits and the first uncertain digit. Zeros may be significant or not based on their position in the number.

Step 2: Analyze Each Quantity

We'll analyze each quantity individually to determine how many significant figures it has. \(76.600\ \mathrm{kJ}\): All non-zero digits are significant. Zeros between non-zero digits or after a decimal point and a significant figure are significant.**(a) 76.600 kJ:** There are 5 significant figures — the two '6's and zeros after them.**(b) 4.50200 x 10³ g:** Five numbers are given; all non-zero numbers and trailing zeros after a decimal are significant, so there are 6 significant figures.**(c) 3000 nm:** The leading zeros are not significant unless specified by a decimal point. In scientific practice, usually 1 significant figure, but contextually can vary.**(d) 0.00300 mL:** Here, the leading zeros are not significant, but the zeros after the '3' are, so it has 3 significant figures.**(e) 18 students:** This involves a count of discrete objects, so it has an infinite number of significant figures (typically considered exact).**(f) 3 x 10⁻⁵ g:** Any number in front of the scientific notation is significant; here, it is 1 significant figure.**(g) 47.60 mL:** All non-zero digits and zeros between or after non-zero digits with a decimal are significant, so it has 4 significant figures.**(h) 2070 mi:** This depends on whether the zero is considered significant. Without a decimal point, it's assumed 3 significant figures unless specified.

Key Concepts

Precision in Measurements

Precision in measurements refers to how detailed or exact a measurement is. It reflects the level of uncertainty or possible error in a measurement. Understanding precision helps in knowing how reliable a number is when making calculations.

When we talk about the precision of a measurement, it's important to think about these aspects:

• **Resolution of the Measuring Device**: The smallest change that can be measured, or the smallest division on the scale.
• **Repeatability**: The proximity of repeated measurements.
• **Accuracy**: How close a measurement is to the true value.

In any scientific experiment, achieving high precision means that your measurements are consistent and repeatable.

Measurement precision is usually communicated through significant figures, which show the number's precision level.

Scientific Notation

Scientific notation is a way to write very large or very small numbers concisely. It's widely used in science to clearly express these numbers and see their significant figures more clearly.

A number in scientific notation is written as\[a \times 10^b\]where:

• **a** is a number greater than or equal to 1 and less than 10.
• **b** is an integer (positive for numbers larger than 1, negative for those smaller than 1).

Scientific notation makes it easier to handle unwieldy numbers, especially when performing operations like multiplication or division. It's also helpful in identifying significant figures straight away, as all digits in 'a' are significant.

Significant Digit Rules

Significant digit rules are guidelines used to identify which digits in a number are meaningful in expressing its precision. Here are the key rules:

• **Non-zero digits** are always significant. For example, in the number 47.60, all digits are significant.
• **Any zeros between significant digits** are also significant. An example is 4.50200, which has 6 significant figures.
• **Leading zeros** are not significant. For instance, in 0.00300, only the '3' and the trailing zeros are significant, totaling 3 significant figures.
• **Trailing zeros after a decimal point** are significant, as they denote precision.
• **Trailing zeros in a whole number** without a decimal are not considered significant unless specified by a context, like scientific notations or underlined emphasis.

Recognizing significant digits helps you understand the accuracy and precision provided by a number. This understanding is crucial when rounding numbers, performing calculations, or interpreting data in experiments.