Problem 78 - Chemistry: An Atoms-Focused Approach

Question

Which of the following numbers have four significant figures? (a) \(0.0592 ;\) (b) \(0.08206 ;\) (c) \(8.314 ;\) (d) 273.15 (e) \(5.091 \times 10^{3} ;\) (f) 9.490

Step-by-Step Solution

Answer

a) 0.0592 b) 0.08206 c) 8.314 d) 273.15 e) 5.091 × 10³ f) 9.490 Answer: The numbers with four significant figures are (a) 0.0592, (b) 0.08206, (c) 8.314, (e) 5.091 × 10³, and (f) 9.490.

Step 1: (a) \(0.0592\)

For \(0.0592\), the leading zeros before the non-zero digits are not significant. Therefore, we have four significant figures: \(5, 9, 2, 2\).

Step 2: (b) \(0.08206\)

For \(0.08206\), the leading zeros before the non-zero digits are not significant. Therefore, we have four significant figures: \(8, 2, 0, 6\).

Step 3: (c) \(8.314\)

For \(8.314\), all the digits are significant. Therefore, we have four significant figures: \(8, 3, 1, 4\).

Step 4: (d) 273.15

For 273.15, all the digits are significant. Therefore, we have five significant figures: \(2, 7, 3, 1, 5\).

Step 5: (e) \(5.091 \times 10^{3}\)

For \(5.091 \times 10^{3}\), we only consider the digits in the coefficient before the exponential part, and all these digits are significant. Therefore, we have four significant figures: \(5, 0, 9, 1\).

Step 6: (f) 9.490

For 9.490, all the digits are significant, including the trailing zero, because it is a decimal number. Therefore, we have four significant figures: \(9, 4, 9, 0\). In conclusion, the numbers that have four significant figures are (a) \(0.0592\), (b) \(0.08206\), (c) \(8.314\), (e) \(5.091 \times 10^{3}\), and (f) 9.490.

Key Concepts

Decimal Numbers

Decimal numbers are numbers that have a decimal point separating the whole number from the fractional part. These points serve as markers to understand the precision of the number being represented.

• For example, in a number like 12.345, the decimal point indicates the fractional part starts immediately after the number 12.
• Decimals are crucial for representing non-integer values precisely and accurately.

In scientific measurements, decimal numbers help in maintaining accuracy by clearly showing how many places a value extends into the fractional values. All parts of a decimal contribute to the overall numerical value, except for leading zeros, which only serve as placeholders. For instance, in 0.08206, the digits 8, 2, 0, and 6 hold value, but the zero before them only affects the magnitude of the number, not its significant figure count.
Understanding how to read decimals properly is pivotal when learning about significant figures since these numbers help determine how precise a given measurement is.

Precision in Measurement

Precision in measurement refers to the degree to which repeated measurements under unchanged conditions show the same results. It's crucial in scientific experiments and everyday measurements to ensure that repeated trials give similar outcomes.

• Precision is about consistency: the closer multiple measurements are to each other, the more precise they are.
• It doesn't guarantee accuracy—which is about how close the measurement is to the true value—but rather the reliability and reproducibility.

Significant figures play an essential role in expressing precision. They indicate how many digits in a number are known with a certain confidence level. For instance, in the number 273.15, there are five significant figures representing considerable precision. Each digit contributes to the measure's reliability, reinforcing how precisely we understand this value.
Recognizing the number of significant figures is integral for scientists and students, ensuring that all calculations reflect the correct degree of precision needed for results to be trusted and verified.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form, making them easier to read and comprehend. This is especially useful in scientific work where such figures frequently appear.

• Format: A number is typically written as the product of a decimal and a power of ten. For instance, 5.091 in scientific notation is expressed as \(5.091 \times 10^3\).
• This notation simplifies complex calculations and ensures the scale of numbers is immediately apparent.

In calculations involving scientific notation, the significant figures are found in the coefficient (the number before the \( \times 10^n \)), like the four significant figures in 5.091. This cleaves complex numbers down to their essential parts, making comparisons and computations more manageable.
Scientific notation not only simplifies calculations but also openly displays significant figures, ensuring clarity and precision in both mathematical and scientific documentation.