Problem 77 - Chemistry: An Atoms-Focused Approach

Question

Which of the following numbers have just three significant figures? (a) \(7.02 ;\) (b) \(6.452 ;\) (c) \(302 ;\) (d) \(6.02 \times 10^{23}\) (c) \(12.77 ;\) (f) 3.43

Step-by-Step Solution

Answer

(a) 7.02 (b) 6.452 (c) 302 (d) 6.02 x 10^23 (e) 12.77 (f) 3.43 Answer: (a) 7.02, (c) 302, (d) 6.02 x 10^23, and (f) 3.43 have just three significant figures.

Step 1: Identify Significant Figures

Take each given number and identify the digits and the decimal point (if any). Look for zero digits before the first non-zero digit, and if a zero digit comes after the decimal point, then it's considered as a significant figure.

Step 2: Check the Significant Figures for Each Number

(a) \(7.02\) - There are no leading or trailing zeroes and all digits are non-zero hence it has 3 significant figures. (Answer) (b) \(6.452\) - There are no leading or trailing zeroes and all digits are non-zero hence it has 4 significant figures. (c) \(302\) - leading zero is not significant, thus there are 3 significant figures. (Answer) (d) \(6.02 \times 10^{23}\) - scientific notation usually for measurable quantities, here 6.02 has 3 significant figures. (Answer) (e) \(12.77\) - There are no leading or trailing zeroes and all digits are non-zero hence it has 4 significant figures. (f) \(3.43\) - There are no leading or trailing zeroes and all digits are non-zero hence it has 3 significant figures. (Answer) The numbers (a) \(7.02\), (c) \(302\), (d) \(6.02 \times 10^{23}\), and (f) \(3.43\) have just three significant figures.

Key Concepts

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a more concise and manageable form. This is particularly helpful in fields like science and engineering, where numbers can often span several orders of magnitude.
In scientific notation, numbers are written as the product of two factors: a coefficient and a power of ten. For example, the number 602,000,000,000,000,000,000,000 can be rewritten as \(6.02 \times 10^{23}\). Here:

• The coefficient is 6.02.
• The power of ten, \(10^{23}\), indicates how many places the decimal has moved.

The coefficient in scientific notation provides the significant figures of the number. In our example, the coefficient 6.02 has three significant figures (6, 0, and 2). This notation helps simplify the expression of numbers and ensures that the significant figures are clearly displayed.
Whenever you're working with scientific notation, focus on the coefficient to determine the number of significant figures, as this will include all non-zero digits and any zeroes between them.

Rounding Rules

Rounding is a crucial skill in mathematics, helping simplify numbers without significantly altering their value. To round a number, you need to know which digit is the last one you want to keep, also known as the rounding digit.
Here are some common rules for rounding:

• If the digit immediately to the right of your rounding digit is less than 5, you simply keep the rounding digit and remove any digits to its right.
• If that digit is 5 or greater, you increase the rounding digit by one and remove the digits to its right.

For example, if you're rounding the number 6.276 to two decimal places, you notice the third digit after the decimal is 6. Since 6 is greater than 5, you round up, resulting in 6.28.
Remember, rounding can affect the significant figures of a number, particularly in scientific measurements, so always be mindful of how you are rounding.

Decimal Points

Decimal points play an essential role in defining significant figures in any number. A decimal point can affect the trailing and leading zeroes, influencing how significant figures are counted.
When there is a decimal point present:

• All non-zero numbers are always significant.
• Zeroes in the middle of non-zero digits are significant (e.g., in 100.4, all four digits are significant).
• Leading zeroes are not significant (e.g., 0.003 has one significant figure, 3).
• Trailing zeroes after a decimal point are significant (e.g., in 2.300, all four numbers are significant).

Without a decimal point, trailing zeroes are not considered significant unless specified by scientific or other notations. Understanding the role of decimal points helps in correctly identifying the number of significant figures and ensures precision in calculations and communication.