Question

Perform the following mathematical operations, and express the result to the correct number of significant figures. a. \(6.022 \times 10^{23} \times 1.05 \times 10^{2}\) b. \(\frac{6.6262 \times 10^{-34} \times 2.998 \times 10^{8}}{2.54 \times 10^{-9}}\) c. \(1.285 \times 10^{-2}+1.24 \times 10^{-3}+1.879 \times 10^{-1}\) d. \(\frac{(1.00866-1.00728)}{6.02205 \times 10^{23}}\) e. \(\frac{9.875 \times 10^{2}-9.795 \times 10^{2}}{9.875 \times 10^{2}} \times 100(100 \text { is exact) }\) f. \(\frac{9.42 \times 10^{2}+8.234 \times 10^{2}+1.625 \times 10^{3}}{3}(3 \text { is exact) }\)

Short answer

a. \(6.32 \times 10^{25}\) b. \(7.826 \times 10^{-25}\) c. \(2.09 \times 10^{-1}\) d. \(2.300 \times 10^{-27}\) e. \(8.11\%\) f. \(9.79 \times 10^2\)

Step-by-step solution

Perform the multiplication using given values

To solve this problem, multiply the given values: \(6.022 \times 10^{23} \times 1.05 \times 10^{2}\)

Apply scientific notation rules and round the result

Multiply the values and apply the rules of scientific notation. Then round the result to the correct number of significant figures, which is three in this case: \(6.32 \times 10^{25}\) #b. Division#

Perform the division using given values

To solve this problem, divide the given values: \(\frac{6.6262 \times 10^{-34} \times 2.998 \times 10^{8}}{2.54 \times 10^{-9}}\)

Apply scientific notation rules and round the result

Divide the values, apply the rules of scientific notation, and round the result to the correct number of significant figures, which is four in this case: \(7.826 \times 10^{-25}\) #c. Addition#

Perform the addition using given values

To solve this problem, add the given values together: \(1.285 \times 10^{-2}+1.24 \times 10^{-3}+1.879 \times 10^{-1}\)

Align common exponents and round the result

Align the numbers by their common exponent, then add them and round the result to the correct number of significant figures, which is three in this case: \(2.09 \times 10^{-1}\) #d. Division#

Perform the division using given values

To solve this problem, divide the given values: \(\frac{(1.00866-1.00728)}{6.02205 \times 10^{23}}\)

Apply scientific notation rules and round the result

Perform the subtraction and division, and then round the result to the correct number of significant figures, which is four in this case: \(2.300 \times 10^{-27}\) #e. Division and Multiplication#

Perform the calculations using given values

To solve this problem, perform the calculations as given: \(\frac{9.875 \times 10^{2}-9.795 \times 10^{2}}{9.875 \times 10^{2}} \times 100\)

Apply scientific notation rules and round the result

Perform the subtraction, division, and multiplication, and then round the result to the correct number of significant figures, which is three in this case: \(8.11\%\) #f. Division and Addition#

Perform the calculations using given values

To solve this problem, perform the calculations as given: \(\frac{9.42 \times 10^{2}+8.234 \times 10^{2}+1.625 \times 10^{3}}{3}\)

Apply scientific notation rules and round the result

Perform the addition and division, and then round the result to the correct number of significant figures, which is three in this case: \(9.79 \times 10^2\)

Key concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a concise format. It is especially handy in fields like chemistry where we often deal with atomic and molecular scales. In scientific notation, numbers are written as the product of two parts: a coefficient and a power of ten. For example, instead of writing 602,200,000,000,000,000,000,000, we can use scientific notation: \(6.022 \times 10^{23}\).
This notation helps simplify the reading and writing of numbers.
Let's break it down:

• The coefficient is a number greater than or equal to 1 and less than 10.
• The exponent indicates how many times the coefficient must be multiplied (positive exponent) or divided (negative exponent) by 10.

When performing calculations with scientific notation, it's important to apply the math operations on both the coefficients and exponents, ensuring the results remain in the correct form. We're often required to adjust the astronomical numbers in chemistry calculations, so scientific notation keeps everything manageable.

Mathematical Operations in Chemistry

In chemistry, mathematical operations such as multiplication, division, addition, and subtraction are common, often carried out on numbers in scientific notation. Understanding how these operations work helps to precisely calculate concentrations, moles, or reaction rates.

Multiplication and Division

When multiplying or dividing numbers in scientific notation, you handle the coefficients and exponents separately.

• **Multiplication:** Multiply the coefficients, add the exponents.
• **Division:** Divide the coefficients, subtract the exponents.

Addition and Subtraction

Adding or subtracting numbers in scientific notation requires aligning the numbers to the same exponent. You adjust one or more of the numbers such that their exponents match, then perform the operation on the coefficients. Mathematical accuracy is pivotal in chemical calculations to ensure correct outcomes in experiments and formulations. Students should practice these operations consistently to become proficient in handling complex chemical arithmetic while maintaining accuracy with significant figures.

Rounding Rules

Rounding is a critical skill in scientific calculations, ensuring the results reflect the precision of the measured data. While working with significant figures, the rules for rounding ensure that the number of significant digits is appropriate for the context. First, identify the number of significant figures required for the result. This is typically determined by the measurement with the least number of significant figures in the calculation.
Here are some general rules for rounding:

• **If the digit to be dropped is less than 5**, leave the last retained digit unchanged.
• **If the digit to be dropped is 5 or greater**, increase the last retained digit by one.

These rules apply regardless of whether you are multiplying, dividing, adding, or subtracting. Keeping track of significant figures maintains the integrity and reliability of your computed results, a practice that is essential in scientific disciplines like chemistry. It's crucial that students practice rounding correctly to reflect precise measurements and calculations.