Problem 12 - Chemistry for Allied Health

Question

Determine the number of significant figures in each of the following values. a. 406 b. \(3.00\) c. \(3.20\) d. \(0.25\) e. \(0.0689\) f. \(0.002910\) g. 3941 h. \(46.250\) i. \(30.21\) j. \(0.10300\)

Step-by-Step Solution

Answer

a. 3, b. 3, c. 3, d. 2, e. 3, f. 4, g. 4, h. 5, i. 4, j. 5.

Step 1: Determine significant figures for whole numbers

To determine the number of significant figures in the whole number '406', consider all non-zero digits as significant. The number '406' has three non-zero digits: 4, 0, and 6. Thus, all three are significant figures.

Step 2: Identify significant figures in decimal numbers with trailing zeros

For the number '3.00', all non-zero digits are significant, and trailing zeros after a decimal point are also considered significant. Therefore, '3.00' has three significant figures: 3, 0, and 0.

Step 3: Significant figures for decimals with non-zero digits only

In the case of '3.20', the leading digit and all following digits are considered significant, including zeros after a decimal point. Thus, '3.20' contains three significant figures: 3, 2, and 0.

Step 4: Identify significant figures in small decimal numbers

For the number '0.25', ignore leading zeros as they are not significant. Consider only the digits 2 and 5 after the zeros, which makes two significant figures.

Step 5: Consider significant figures in small decimals with non-zero digits

In '0.0689', skip the leading zeros as they do not count. The significant figures are 6, 8, and 9, totaling three significant figures.

Step 6: Count significant figures in small decimals ending with zero

In '0.002910', exclude the leading zeros. The digits 2, 9, 1, and the trailing zero after the number are significant. There are four significant figures here.

Step 7: Whole numbers with all non-zero digits are significant

For the number '3941', since it contains no zeros, all digits are non-zero and therefore significant. This results in four significant figures: 3, 9, 4, and 1.

Step 8: Decimals with final zero are counted as significant

In '46.250', trailing zeros after the decimal point are significant. The digits counted here are 4, 6, 2, 5, and 0. Therefore, it has five significant figures.

Step 9: Observe significant figures in a decimal with varied digits

The number '30.21' includes all digits as significant since it's in decimal form without leading or trailing zeros. Thus, it has four significant figures: 3, 0, 2, and 1.

Step 10: Check significant figures in a decimal with zeros at the end

For '0.10300', ignore leading zeros, but trailing zeros after a decimal are significant. The significant figures are 1, 0, 3, 0, and 0, making a total of five significant figures.

Key Concepts

Whole Numbers

When dealing with significant figures in whole numbers, it's essential to understand that all the digits, including zeros that are not leading zeros, count if they are non-zero themselves. For example, in the whole number '406', all the digits are significant because each digit plays a role in defining the number's value. You simply count each digit:

• Every non-zero digit (4 and 6 in 406) is significant.
• Zeros between non-zero digits are also significant (the 0 in 406).

Thus, '406' has three significant figures. Whole numbers without zeros or with zeros surrounded by non-zero digits straightforwardly yield all significant figures associated with the visible digits.

Decimal Numbers

Decimal numbers often require careful consideration when determining significant figures because both the position of zeros and the presence of a decimal point can affect counting. Decimal numbers that include trailing zeros have different rules from whole numbers:

• All non-zero digits in a decimal are considered significant (e.g., 3 and 2 in 3.20).
• Any zeros to the right of both a decimal point and a non-zero digit are significant (such as both zeros in 3.00).

For example, in 3.00, the zero becomes significant, providing greater precision than simply 3. On the contrary, leading zeros, like those in 0.25, are not considered significant as they merely serve as placeholders.

Trailing Zeros

Trailing zeros can dramatically change the number of significant figures when they follow a decimal point. In mathematics, they often indicate a level of precision in measurement. For numbers like 0.002910 and 46.250:

• Zeros at the end of a number after a non-zero and a decimal point are always significant.
• For instance, in 3.00, both trailing zeros add specificity, leading to three significant figures.

Trailing zeros amplify the precision implied by a measurement, showing that a measurement was likely done to a specific decimal place, even if those places are zero.

Leading Zeros

Leading zeros present before any non-zero digits in a number do not count toward significant figures. Their primary purpose is to set the decimal point, not to convey accuracy or precision of measured values:

• In numbers such as 0.25 or 0.0689, any zeros that precede a non-zero digit are not significant.
• This rule is crucial especially for very small numbers (like those less than 1), making the only significant figures the non-zero digits that come afterward (2 and 5 in 0.25, or 6, 8, and 9 in 0.0689).

This principle helps in focusing on which parts of a number truly contribute to its value and precision.