Problem 38 - Chemistry: The Central Science

Question

Indicate the number of significant figures in each of the following measured quantities: (a) \(3.774 \mathrm{~km}\), (b) \(205 \mathrm{~m}^{2}\), (c) \(1.700 \mathrm{~cm}\), (d) \(350.00 \mathrm{~K}\), (e) \(307.080 \mathrm{~g}\), (f) \(1.3 \times 10^{3} \mathrm{~m} / \mathrm{s}\).

Step-by-Step Solution

Answer

(a) 4 significant figures, (b) 2 significant figures, (c) 4 significant figures, (d) 5 significant figures, (e) 6 significant figures, (f) 2 significant figures.

Step 1: (a) 3.774 km

The number written in decimal notation has 4 digits, and all of them are non-zero. Therefore, it has 4 significant figures.

Step 2: (b) 205 m^2

The number is a whole number with three digits, including a trailing zero. The zeros at the end of whole numbers are not considered significant, so this number has 2 significant figures.

Step 3: (c) 1.700 cm

The trailing zeros here are considered significant because they are after the decimal point. Therefore, this number has 4 significant figures.

Step 4: (d) 350.00 K

The zeros at the end are considered significant because they are after the decimal point, so this number has 5 significant figures.

Step 5: (e) 307.080 g

All digits in this number are non-zero, and all zeros are between significant digits or trailing digits after the decimal point. Therefore, this number has 6 significant figures.

Step 6: (f) \(1.3 \times 10^{3} \mathrm{~m} / \mathrm{s}\)

The number is in scientific notation with two significant digits in the mantissa (1.3), so it has 2 significant figures.

Key Concepts

Decimal Notation

Decimal notation is a number representation system where numbers are composed of digits arranged in a specific order, with each digit representing a value based on its position and the use of a decimal point. The decimal point separates whole numbers from fractional parts.
A number like 3.774 is a great example of decimal notation. Here, the position of each digit matters:

• **3** is in the 'ones' place.
• **7** is in the 'tenths' place, right after the decimal point.
• **7** in the 'hundredths' place.
• **4** in the 'thousandths' place.

In decimal notation, every digit is significant, which means they all contribute to the precision of the number. For example, in the number 3.774, all four digits are significant figures.

Whole Numbers

Whole numbers are numbers without fractions or decimals, simply counting numbers starting from zero upwards. Examples include 0, 1, 2, 3, and so on.
When dealing with whole numbers like 205, it’s essential to distinguish significant figures.

• Here, the '2' and '5' are significant because they are non-zero digits.
• The '0' in the middle is also significant as it's between non-zero digits, giving us a total of three significant figures. But if there is a trailing zero at the end of whole numbers with no decimal point, it is usually not considered significant.

Trailing Zeros

Trailing zeros are the zeros that appear at the end of a number after the last non-zero digit. Their significance depends on the presence of a decimal point.
For instance, in 1.700, the trailing zeros are after the decimal, and they are considered significant because they imply precision.

• They show that the number has been measured to that level of accuracy.
• Therefore, 1.700 has four significant figures.

However, in a number like 2000 (without a decimal point), the trailing zeros are not automatically significant.
This concept can be confusing, but remembering that a decimal point often implies significance helps clear it up.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers compactly. It expresses numbers in the form \( a \times 10^n \), where "\( a \)" is a number greater than or equal to 1 and less than 10, and "\( n \)" is an integer.
An example of scientific notation is 1.3 × 10³, which simplifies the representation of 1300.

• The "1.3" part, known as the mantissa, dictates the number of significant figures, which is 2 in this case.
• This notation is particularly useful for expressing the magnitude of numbers while also making clear the precision of the mantissa.

This technique avoids ambiguity about trailing zeros, helping you ensure clarity about how many digits are significant.