Question

State the number of significant digits in each of the following: (a) \(1.2 \times 10^{0} g\) g (b) \(4.50 \times 10^{1} \mathrm{~g}\) (c) \(5.02 \times 10^{-1} \mathrm{~g}\) (d) \(100 \times 10^{-2} \mathrm{~g}\)

Short answer

(a) 2 significant digits, (b) 3 significant digits, (c) 3 significant digits, (d) 1 significant digit.

Step-by-step solution

Analyzing (a)

For (a), the number is expressed as \(1.2 \times 10^{0} g\). The significant digits in this expression are the actual digits presented in the coefficient, \(1.2\). Therefore, the significant digits are '1' and '2'. The total count is 2 significant digits.

Analyzing (b)

For (b), the number is \(4.50 \times 10^{1} \mathrm{~g}\). The coefficient \(4.50\) has three digits: '4', '5', and '0'. All these digits are significant. Therefore, the count is 3 significant digits.

Analyzing (c)

For (c), we have \(5.02 \times 10^{-1} \mathrm{~g}\). The number \(5.02\) includes the digits '5', '0', and '2', all of which are significant. There are no trailing zeros following a decimal point so all visible digits are significant. The total is 3 significant digits.

Analyzing (d)

For (d), the expression is \(100 \times 10^{-2} \mathrm{~g}\). Here, only the individual digits, '1', '0', and '0', are considered. Since there is no decimal point explicitly shown, only the '1' is significant in common scientific practice, leading to a total of 1 significant digit.

Key concepts

Scientific Notation

Scientific notation is a method used in science to express very large or very small numbers. It helps simplify such figures by reducing the number of digits to manageable forms. Instead of writing out all the digits of a large number, scientific notation lets you move the decimal point. This movement is then expressed as a power of ten.
For instance, in the number \(1.2 \times 10^{0} \text{ g}\), "1.2" is the coefficient, and the "\(10^{0}\)" part indicates the decimal hasn’t moved, since anything raised to the power of zero is one. Similarly, a coefficient like "4.50" in \(4.50 \times 10^{1} \text{ g}\) tells us how many significant digits the number has before considering the magnitude.
When numbers are written in this format, all the digits in the coefficient count towards the total number of significant figures, which is crucial in scientific measurements for ensuring precision.

Measurement Precision

Measurement precision refers to how detailed and exact a measurement is. In scientific terms, precision is communicated through significant figures, which are the numbers in a measurement that contribute to its accuracy. These significant figures include all known digits plus one estimated digit.
For instance, when evaluating \(5.02 \times 10^{-1} \text{ g}\), each digit '5', '0', and '2' holds significance, giving us a precise measurement with three significant figures. Identifying significant figures is essential because they express our confidence in the measurement taken.
Understanding significant figures can involve the following concepts:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.
• Trailing zeros are significant only if they follow a decimal point.

By knowing how many of these figures are significant in a measure, we ensure measurements are accurate and that their precision aligns accurately with the instrument used.

Chemistry Fundamentals

In chemistry, precise measurements are key to experiments and calculations. Chemistry often involves complex balances and reactions where measurement accuracy relates directly to the quality of results.
Significant figures play a vital role in chemistry since they define the precision of a measurement. This can dictate experimental reproducibility and outcome accuracy. For example, in our exercise, the number \(100 \times 10^{-2} \text{ g}\) shows a restricted precision when not accompanied by a decimal point, offering just one significant figure.
Other fundamentals include the use of scientific notation to convey precise narratives without overwhelming detail and different terminologies that standardize and communicate measurements universally. Accurate usage of these basics ensures that experimental data truly reflects the underlying chemical quantitative relationships.