Problem 1 - A Complete Resource Book in Chemistry for JEE Main

Question

The number of significant figures for the numbers \(161 \mathrm{~cm}, 0.161 \mathrm{~cm}, 0.0161 \mathrm{~cm}\) are (a) 2,3 and 4 respectively (b) 3,3 and 3 respectively (c) 3,5 and 4 respectively (d) 3,4 and 4 respectively

Step-by-Step Solution

Answer

The correct option is (b) 3,3 and 3 respectively.

Step 1: Understand Significant Figures

Significant figures in a number are the digits that carry meaning contributing to its precision. These include all non-zero digits, zeroes between non-zero digits, and trailing zeroes in a decimal number.

Step 2: Analyze Each Number

Let's break down each number one by one:1. For \(161 \mathrm{~cm}\), it has three non-zero digits: 1, 6, and 1. Thus, it contains 3 significant figures.2. For \(0.161 \mathrm{~cm}\), the leading zeros are not significant. The digits 1, 6, and 1 are significant. Therefore, it has 3 significant figures.3. For \(0.0161 \mathrm{~cm}\), we again do not count the leading zeros, but digits 1, 6, and 1 are significant, resulting in 3 significant figures.

Step 3: Choose the Correct Answer

After determining the significant figures, we conclude that the number of significant figures for each value is 3, 3, and 3 respectively. Thus, the correct answer is (b) 3,3 and 3 respectively.

Key Concepts

Significant Figures in Measurements

Significant figures are crucial when it comes to measurements, as they give us information about the precision of the numbers involved. The concept simply implies how exact a measurement is, based on the number of meaningful digits it contains. These digits include:

• All non-zero digits
• Zeros that are between non-zero digits
• Trailing zeros in a number with a decimal point

It is important to know which zeros count and which do not. For example, in the number 161 cm, all digits are significant, giving us three significant figures. But in a number like 0.161 cm, the leading zeros are not significant, leaving us with three significant figures as well. Similarly, in 0.0161 cm, both 1, 6, and 1 are counted as significant figures, again resulting in three significant digits.

Precision in Measurements

Precision in measurements refers to the detail and exactness of a measurement. When we talk about precision, we often look at the number of significant figures. More significant figures mean that the measurement is more precise. A precise measurement minimizes rounding errors and conveys how finely tuned the instrument making the measurement is. For instance, a value of 161 cm compared to 0.161 cm shows different levels of precision due to the amount of significant figures considered. Both are interpreted with three significant figures, thus indicating the intended precision of the measurements taken. Precision does not only mean accuracy but reflects how carefully measurements were recorded and how certain we are of their detail.

Analyzing Significant Figures

Analyzing significant figures in a number requires us to understand which parts of a number are meaningful. Let's break down the analysis process:

• Start with non-zero digits: These are always significant.
• Assess zeros: Leading zeros, like in 0.161 cm, are not significant, but zeros between non-zero digits and trailing zeros in decimals are.
• Determine the total count: After identifying the significant components, count these digits to ascertain the number of significant figures.

This analysis helps in reporting results precisely and understanding possible measurement limitations. Thus, when you see a number like 0.0161 cm, analyzing significant figures involves excluding the zero initially and focusing on the recurring pattern of 1, 6, 1, resulting in three significant digits.