Problem 191 - Introductory Chemistry Atoms First

Question

Do the following calculations and express each answer to the correct number of significant figures. (All values are measurements.) (a) \(\frac{5.03+7.2}{0.003}\) (b) \(\frac{8.93 \times 0.054}{1.32}\) (c) \((6.23 \times 0.042)+9.86\)

Step-by-Step Solution

Answer

(a) $\approx 4000$ (b) $\approx 0.37$ (c) $\approx 10.12$

Step 1: (a) Operation Calculation#a_tag_content# Perform the calculations given in part (a): \[\frac{5.03+7.2}{0.003}.\]

(a) Evaluate Addition#a_tag_content# First, add the two numbers in the numerator: \(5.03 + 7.2 = 12.23\).

Step 2: (a) Evaluate Division#a_tag_content# Next, divide the sum by the denominator: \[\frac{12.23}{0.003} = 4076.666666\ldots\]

(a) Determine Correct Significant Figures#a_tag_content# In the calculation, the measurement \(7.2\) has the fewest decimal places (1) in the addition operation, and \(0.003\) has the fewest significant figures (1) in the division operation. Therefore, the final answer should have one significant figure: \[\frac{5.03+7.2}{0.003} \approx 4000\]

Step 3: (b) Operation Calculation #b_tag_content# Perform the calculations given in part (b): \[\frac{8.93 \times 0.054}{1.32}\].

(b) Evaluate Multiplication #b_tag_content# First, multiply the two numbers in the numerator: \(8.93 \times 0.054 = 0.48222\).

Step 4: (b) Evaluate Division #b_tag_content# Next, divide the product by the denominator: \[\frac{0.48222}{1.32} = 0.36531818181\ldots\]

(b) Determine Correct Significant Figures #b_tag_content# In the calculation, the measurement with the fewest significant figures is \(0.054\) with two significant figures. Therefore, the final answer should have two significant figures: \[\frac{8.93 \times 0.054}{1.32} \approx 0.37\]

Step 5: (c) Operation Calculation #c_tag_content# Perform the calculations given in part (c): \[(6.23 \times 0.042) + 9.86\].

(c) Evaluate Multiplication #c_tag_content# First, multiply the two numbers: \(6.23 \times 0.042 = 0.26166\).

Step 6: (c) Evaluate Addition #c_tag_content# Next, add the product to the second number: \(0.26166 + 9.86 = 10.12166\).

(c) Determine Correct Significant Figures #c_tag_content# In the calculation, the measurement with the fewest decimal places is \(0.042\) with two decimal places in the multiplication operation, and \(9.86\) has the fewest decimal places (two) in the addition operation. Therefore, the final answer should have two decimal places: \[(6.23 \times 0.042) + 9.86 \approx 10.12\]

Key Concepts

Scientific Notation

When working with very large or very small numbers in chemistry, it is often convenient to use scientific notation. This is a way of expressing numbers as a product of two factors: a coefficient and a power of 10. The coefficient must be a number between 1 and 10, and the power of 10 renders the number's scale.

For example, in expressing the number 0.0045 in scientific notation, we would write it as \(4.5 \times 10^{-3}\). This is particularly useful in chemistry for dealing with numbers that represent quantities such as Avogadro's number (\(6.022 \times 10^{23}\)) or the mass of an electron (approximately \(9.11 \times 10^{-31}\) kilograms). Scientific notation allows chemists to easily convey the magnitude of quantities and perform calculations with them, especially when using a calculator or a computer.

Precision in Chemistry

In chemistry, precision refers to how closely individual measurements agree with each other. It is related to the concept of significant figures, which are the digits in a measurement that carry meaning contributing to its precision. This is crucial because it affects how results of experiments are interpreted and reported.

If we measure a substance and find its mass to be 12.3 grams, each of the three digits is significant; they tell us the mass is more than 12 grams but less than 13 grams, and about three tenths of a gram beyond 12. The number of significant figures in a calculation reflects the precision of the result. Less precise measurements affect the final calculated result and limit the number of significant figures that can reliably be reported.

Measurement Uncertainty

Measurement uncertainty is an integral concept in chemistry, which acknowledges that no measurement can be exact. Every measurement is subject to some degree of error or uncertainty, which arises from limitations in the measurement instruments, the observer, and the measured substance's nature.

For instance, a balance may give a reading with a small uncertainty range, indicating that the actual mass of a substance might be slightly more or less than the displayed value. To communicate the amount of uncertainty, significant figures are used. The rule is that the last digit in any reported measurement is assumed to be uncertain. Understanding this concept helps maintain clarity and honesty in scientific communication and limits overstatement of precision.

Arithmetic Operations in Chemistry

Calculations in chemistry must be carried out with attention to the precision of the measurements involved. When performing arithmetic operations, the result should be reported with the correct number of significant figures to ensure that the precision is not overestimated.

For addition and subtraction, the result should have the same number of decimal places as the measurement with the fewest decimal places. In multiplication and division, the number of significant figures in the result should match the measurement with the fewest significant figures. This ensures that the final answer reflects the most precise information available from the measurements and helps maintain the integrity of the calculated results. Keeping these rules in mind is key when performing any arithmetic involving measured quantities in chemistry.