Problem 68 - Chemistry The Science in Context

Question

Which of these uncertain values has the largest number of significant figures? (a) \(1 / 545 ;\) (b) \(1 / 6.4 \times 10^{-3} ;\) (c) \(1 / 6.50\) (d) \(1 / 1.346 \times 10^{2}\)

Step-by-Step Solution

Answer

(a) \(1 / 545\) (b) \(1 / 6.4 \times 10^{-3}\) (c) \(1 / 6.50\) (d) \(1 / 1.346 \times 10^{2}\) Answer: Both (b) and (c) have the largest number of significant figures with 5 significant figures each.

Step 1: Determine the number of significant figures for each value

We will analyze each value separately and determine the number of significant figures they have. (a) \(1 / 545\) The result is approximately \(0.00183\) which has 3 significant figures. (b) \(1 / 6.4 \times 10^{-3}\) The result is approximately \(156.25\) which has 5 significant figures. (c) \(1 / 6.50\) The result is approximately \(0.15385\) which has 5 significant figures. (d) \(1 / 1.346 \times 10^{2}\) The result is approximately \(0.00743\) which has 4 significant figures.

Step 2: Compare the number of significant figures

Here are the number of significant figures for each value: (a) 3 significant figures (b) 5 significant figures (c) 5 significant figures (d) 4 significant figures

Step 3: Identify the value with the largest number of significant figures

The values (b) and (c) have the largest number of significant figures, with 5 significant figures each.

Key Concepts

Uncertainty in Measurements

Whenever we measure something, there's always a bit of blur or fuzziness in our numbers, known as uncertainty. This uncertainty is a natural part of the measuring process. No tool or person can perfectly measure something every time. The key is understanding how much wiggle room there is in our numbers. We express uncertainty using significant figures. These numbers tell us how precise our measurement is. The more significant figures a number has, the more precise it is. For instance:

• When a ruler gives a reading of 5.6 cm, the '6' is a bit fuzzy.
• But if it reads 5.60 cm, there's less room for error. It's more precise.

Uncertainty becomes crucial when making scientific analyses or calculations. It helps in making decisions about whether results are significant or how they compare to other measurements.

Scientific Notation

Scientific notation is a helpful way to handle very large or very small numbers easily. It simplifies numbers by expressing them as a product of a number between 1 and 10 and a power of ten. For example, instead of writing 0.00023, you can write it as:

• \[2.3 \times 10^{-4}\]

This format makes calculations more manageable, especially with numbers that aren't convenient for regular decimal form. Scientific notation is commonly used in science and engineering because it allows scientists to communicate numbers clearly without clutter or confusion. It’s particularly useful in calculations involving multiplication or division, keeping track of the powers makes arithmetic straightforward.

Accuracy in Calculations

Accuracy refers to how close a calculated result is to the true value or how much the result can be trusted. Calculations using measurements must consider the number of significant figures carefully, ensuring the outcome is as accurate as possible. Here are some tips for maintaining accuracy:

• Always use the same number of significant figures as the least precise measurement when multiplying or dividing.
• When adding or subtracting, the result should have as many decimal places as the number with the fewest decimals.

In the original problem, by examining the number of significant figures in each operation, we determine which calculation is the most precise. However, it's equally important to understand that precision is not the same as accuracy. A result can be precise without being accurate if it's consistently off due to systematic error, like using a miscalibrated instrument.