Question

How many significant figures are there in each of the following values? a. \(6.07 \times 10^{-15}\) b. \(0.003840\) c. \(17.00\) d. \(8 \times 10^{8}\) e. \(463.8052\) f. 300 g. 301 h. 300 .

Short answer

a. 2 significant figures b. 4 significant figures c. 4 significant figures d. 1 significant figure e. 7 significant figures f. 1 significant figure g. 3 significant figures h. 2 significant figures

Step-by-step solution

a. Counting significant figures in 6.07 x 10^{-15}

For scientific notation, like the given number \(6.07 \times 10^{-15}\), only the digits in the coefficient are considered significant. In this case, the coefficient is 6.07. To count the significant figures, we just count the digits in the coefficient, which are 6 and 7. Therefore, there are 2 significant figures in the value.

b. Counting significant figures in 0.003840

In the number 0.003840, the leading zeros (before the first non-zero digit) are not significant. We start counting significant figures from the first non-zero digit, 3. The next digits are 8, 4, and 0. Since the zero follows a non-zero digit and is at the end, it is significant. So, there are 4 significant figures in the value.

c. Counting significant figures in 17.00

The number is 17.00. All non-zero digits are significant, so we count 1 and 7. The two trailing zeros after the decimal point are significant as well. Therefore, the number has 4 significant figures.

d. Counting significant figures in 8 x 10^{8}

In scientific notation, like the given number \(8 \times 10^{8}\), only the digits in the coefficient are considered significant. In this case, the coefficient is 8, which is a single digit. So, there's only 1 significant figure in the value.

e. Counting significant figures in 463.8052

For the number 463.8052, all non-zero digits are significant. We count 4, 6, 3, 8, 0, 5, and 2. The zero in this case is significant because it is between non-zero digits. There are 7 significant figures in the value.

f. Counting significant figures in 300

For the number 300, there are two trailing zeros after the significant non-zero digit 3. Since there's no decimal point and the trailing zeros are at the end of the number, they aren't considered significant. Thus, there's only 1 significant figure in the value.

g. Counting significant figures in 301

For the number 301, there are no leading or trailing zeros, only non-zero digits. We count 3, 0, and 1. The zero is significant as it is between non-zero digits. There are 3 significant figures in the value.

h. Counting significant figures in 300.

The number is 300., which means it has a decimal point at the end. All non-zero digits are significant, so we count 3. The trailing zero at the end of the number becomes significant since it follows a non-zero digit and there is a decimal point. Therefore, there are 2 significant figures in the value.

Key concepts

Understanding Scientific Notation

When dealing with extremely large or very small numbers in chemistry, 'scientific notation' is a convenient way to express them. It involves writing numbers as the product of two parts: a coefficient and a power of ten. For instance, the number \(6.07 \times 10^{-15}\) is in scientific notation, where \(6.07\) is the coefficient, and \(10^{-15}\) indicates that the coefficient should be multiplied by ten to the power of negative fifteen.

This format is particularly useful because it not only simplifies large figures, but it also clearly highlights the 'significant figures' of the number, which are the digits we are confident about. The coefficient \(6.07\) contains these significant figures and in this context, 'counting significant figures' becomes straightforward since we only consider the digits in the coefficient, excluding the exponential part. The purpose of scientific notation is to express the quantity in a more manageable form without losing the information about its precision and significance.

Counting Significant Figures

The 'counting significant figures' technique is vital for students to master as it determines the precision of a measurement or calculation. Significant figures include all non-zero digits, any zeros between them, and any trailing zeros that are after a decimal point.

Let's take the number \(0.003840\) as an example. Here, we disregard any leading zeros --they serve only as placeholders and do not indicate precision. The count begins from the first non-zero digit, and all subsequent digits, including the trailing zero, are significant because they represent measured or estimated figures. A common mistake is to overlook trailing zeros, but they can be significant as in \(17.00\), indicating that the measurement is accurate up to the hundredths place. On the other hand, in a number like \(300\) without a decimal point, the trailing zeros are not considered significant because they could be merely placeholders, not precise measurements.

Trailing Zeros Significance

The significance of 'trailing zeros' often confuses students, but understanding their role is crucial for accurate scientific reporting. A trailing zero is a zero that comes after all other digits. Whether it is significant or not depends on its location and any decimal points present. For example, in \(17.00\), the zeros indicate that the measurement is accurate to the hundredths place, and are therefore significant.

However, in a number like \(300\), without an explicit decimal point, the trailing zeros are not considered significant because they could simply be holding the place value open. Change that to \(300.\), with a decimal point, and the situation is different: now the zeros indicate a measured level of precision and are counted as significant. When learning to count significant figures, it’s important for students to remember that a decimal point makes a world of difference in interpreting the precision of the trailing zeros.