Question

Evaluate each of the following and write the answer to the appropriate number of significant figures. a. \((2.0944+0.0003233+12.22) /(7.001)\) b. \(\left(1.42 \times 10^{2}+1.021 \times 10^{3}\right) /\left(3.1 \times 10^{-1}\right)\) c. \(\left(9.762 \times 10^{-3}\right) /\left(1.43 \times 10^{2}+4.51 \times 10^{1}\right)\) d. \(\left(6.1982 \times 10^{-4}\right)^{2}\)

Short answer

a. \(2.0441\) b. \(3.75 \times 10^{3}\) c. \(0.6828 \times 10^{-4}\) d. \(38.413 \times 10^{-8}\)

Step-by-step solution

Add the numbers: \(2.0944 + 0.0003233 + 12.22 = 14.3147233\) #b. Performing the division# Now, we need to divide this sum by the given denominator: \(\frac{14.3147233}{7.001}\)

Divide the numbers: \(14.3147233 ÷ 7.001 = 2.0441046\) #c. Applying the significant figure rules for division# The least precise measurement has 4 significant figures. Therefore, our answer should have 4 significant figures as well.

Round the result: \(2.0441\) #a. Calculate the sum of both numbers in scientific notation# \(1.42 \times 10^{2} + 1.021 \times 10^{3}\)

Add the numbers: \(1.42 \times 10^{2} + 1.021 \times 10^{3} = 1.42 \times 10^{2} + 10.21 \times 10^{2} = 11.63 \times 10^{2}\) #b. Performing the division in scientific notation# Now, we need to divide this sum by the given value in scientific notation: \(\frac{11.63 \times 10^{2}}{3.1 \times 10^{-1}}\)

Divide the numbers: \(\frac{11.63}{3.1} \times \frac{10^{2}}{10^{-1}} = 3.75 \times 10^{3}\) #c. Divide the numerator value in scientific notation# \( \frac{9.762 \times 10^{-3}}{1.43 \times 10^{2}+4.51 \times 10^{1}} \)

Add the denominator's values: \(1.43 \times 10^{2} + 4.51 \times 10^{1} = 14.3 \times 10^{1}\) #d. Performing the division in scientific notation# \(\frac{9.762 \times 10^{-3}}{14.3 \times 10^{1}}\)

Divide the numbers: \(\frac{9.762}{14.3} \times \frac{10^{-3}}{10^{1}} = 0.6828 \times 10^{-4}\) #e. Square the given value in scientific notation# \(\left(6.1982 \times 10^{-4}\right)^{2}\)

Square the number: \((6.1982)^{2} \times (10^{-4})^{2} = 38.413 \times 10^{-8}\) Final Results: a. \(2.0441\) b. \(3.75 \times 10^{3}\) c. \(0.6828 \times 10^{-4}\) d. \(38.413 \times 10^{-8}\)

Key concepts

Scientific Notation

In the world of science and mathematics, especially in chemistry, scientists deal with very large or very small numbers frequently. Here is where scientific notation becomes incredibly handy as it simplifies these numbers to be easily readable and manageable. Scientific notation presents numbers as a product of a coefficient and a power of ten. For example, instead of writing 1,000 as it is, in scientific notation it would be written as \(1 \times 10^{3}\).

• The coefficient is the number between 1 and 10.
• The exponent shows how many times the coefficient must be multiplied by 10.
• It's a universal method of ensuring precision in numerical expressions.

In the exercise, numbers like \(1.42 \times 10^{2}\) and \(1.021 \times 10^{3}\) are converted into scientific notation to facilitate easy calculations.
This system is especially important in chemistry, where precise values are critical for reactions and experiments.

Precision in Measurement

Precision in measurement refers to how close multiple measurements of a quantity are to each other. In our everyday life, precision ensures the reliability and consistency of measurements. In scientific fields, precise measurements are crucial for credible and repeatable results.
Significant figures are the digits in a number that carry meaningful contributions to its measurement accuracy. They show the precision of the measurement based on the instrument used.

• Numbers from 1-9 are always significant.
• Zeroes between significant digits are also significant.
• Leading zeros are never significant.
• Trailing zeros are significant if there is a decimal point.

While solving problems, the number of significant figures in the answer is dictated by the measurement with the least precision involved in the calculation steps. This ensures that the answer does not imply more precision than is justified by the figures given in the problem. Each step must respect this to produce reliable results.

Mathematical Operations in Chemistry

In chemistry, various mathematical operations such as addition, subtraction, multiplication, and division are used to formulate solutions, balance equations, and perform unit conversions. Each type of operation has its own approach to managing significant figures and must be handled according to the rules of significant figures so as not to distort measurement precision.

• For addition and subtraction: The result should match the least number of decimal places of the numbers involved.
• For multiplication and division: The number of significant figures in the result should match the number of significant figures in the measurement with the fewest significant figures.

In the exercise, when dividing in scientific notation or carrying out operations on values in this format, it’s essential to apply these rules correctly to deliver accurate and meaningful results. Whether adding or multiplying values, carrying out these operations skillfully will highlight the precision of your final answer, essential for chemical equations and reactions in practical applications.