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. \(\frac{(1.00866-1.00728)}{6.02205 \times 10^{2.3}}\) e. \(\frac{9.875 \times 10^{2}-9.795 \times 10^{2}}{9.875 \times 10^{2}} \times 100(100\) is exact) f. \(\frac{9.42 \times 10^{2}+8.234 \times 10^{2}+1.625 \times 10^{3}}{3}(3\) is exact)

Short answer

The short answers are: a. \(6.32 \times 10^{25}\) b. \(1.98 \times 10^{-24}\) c. \(2.0 \times 10^{-1}\) d. \(1.9544 \times 10^{-6}\) e. \(0.81\) f. \(1.13 \times 10^3\)

Step-by-step solution

Part a: Multiplying two numbers

To solve this problem, we must multiply the two numbers and then round the result to the correct number of significant figures. We start by multiplying: \(6.022 \times 10^{23} \times 1.05 \times 10^{2} = (6.022 \times 1.05) \times (10^{23} \times 10^2) = 6.3231 \times 10^{25}\) Now, we will round the result to the correct number of significant figures. Since we are multiplying, we should use the least number of significant figures in the given numbers, which is 3. So, the final answer is: \(6.32 \times 10^{25}\)

Part b: Dividing two numbers

To solve this problem, we must calculate the division and then round the result to the correct number of significant figures. We start by dividing: \(\frac{6.6262 \times 10^{-34} \times 2.998 \times 10^{8}}{2.54 \times 10^{-9}} = \frac{(6.6262 \times 2.998) \times (10^{-34} \times 10^8)}{2.54 \times 10^{-9}} = 19.8204 \times 10^{-26}\) Now, we will round the result to the correct number of significant figures. Since we are dividing, we should use the least number of significant figures in the given numbers, which is 3. So, the final answer is: \(1.98 \times 10^{-24}\)

Part c: Adding three numbers

To solve this problem, we must add the three numbers together and then round the result to the correct number of decimal places. We start by adding: \(1.285 \times 10^{-2} + 1.24 \times 10^{-3} + 1.879 \times 10^{-1} = 0.01285 + 0.00124 + 0.1879 = 0.20199\) Now, we will round the result to the correct number of decimal places. Since we are adding, we should look at the least number of decimal places in the given numbers, which is 2. So, the final answer is: \(2.0 \times 10^{-1}\)

Part d: Division and subtraction

In this problem, we must subtract the two numbers and then divide the result by another number. After that, we need to round the result to the correct number of significant figures. We start by subtracting and dividing: \(\frac{(1.00866 - 1.00728)}{6.02205 \times 10^{2.3}} = \frac{0.00138}{6.02205 \times 10^{2.3}} = 1.9544 \times 10^{-6}\) Now, we will round the result to the correct number of significant figures. Since we are subtracting and then dividing, the final result should have the same number of significant figures as the least of the input values, which is 5. So, the final answer is: \(1.9544 \times 10^{-6}\)

Part e: Division and subtraction

In this problem, we must subtract the two numbers, divide the result by another number and then multiply by 100. We start by subtracting the numbers, and dividing by another number: \(\frac{9.875 \times 10^{2} - 9.795 \times 10^{2}}{9.875 \times 10^{2}} = \frac{0.080 \times 10^{2}}{9.875 \times 10^{2}} = 8.0973 \times 10^{-3}\) Now, we multiply the result by 100 to get: \(8.0973 \times 10^{-3} \times 100 = 0.809\nonumber\) We will round the result to the correct number of significant figures, which is 2 in this case. So, the final answer is: \(0.81\)

Part f: Division and addition

In this problem, we must add three numbers together, then divide the result by an exact number (3). We start by adding the numbers and dividing by 3: \(\frac{9.42 \times 10^{2} + 8.234 \times 10^{2} + 1.625 \times 10^{3}}{3} = \frac{(9.42 + 8.234 + 16.25) \times 10^2}{3} = 11.30133 \times 10^2\) We will round the result to the correct number of significant figures. Since we are adding, we should use the least number of significant figures in the given numbers, which is 3. So, the final answer is: \(1.13 \times 10^3\)

Key concepts

Understanding Scientific Notation

When dealing with very large or very small numbers in chemistry, scientific notation is an invaluable tool. It simplifies these numbers into a format that is easier to read, write, and use in calculations. The general form of scientific notation is \( a \times 10^{n} \), where \( a \) is a number greater than or equal to 1 and less than 10, and \( n \) is an integer which represents the number of decimal places the decimal needs to move to convert the number into \( a \) form. For instance, take the number \( 6.022 \times 10^{23} \), this indicates the decimal point has moved 23 places to the right.

Using scientific notation eases the process of performing mathematical operations since you can deal with the coefficient \( a \) and the power of ten separately. After the operations, recombining them gives the final result in scientific notation. Remember, it's important not to round off any numbers until the end of the calculation to maintain precision. This is especially true in chemistry where accurate measurements are crucial.

Mathematical Operations in Chemistry

Mathematical operations in chemistry, such as addition, subtraction, multiplication, and division, follow specific rules when dealing with significant figures. When multiplying or dividing numbers, the result must be rounded to the same number of significant figures as the quantity with the least number of significant figures used in the calculation. For example, when multiplying \( 6.022 \times 10^{23} \) and \( 1.05 \times 10^{2} \), the least number of significant figures in either number is three, which dictates the number of significant figures in the answer, \( 6.32 \times 10^{25} \).

For addition and subtraction, the result should be rounded to the same number of decimal places as the least precise measurement. It's essential to line up the numbers by their decimal points and then perform the operation. After obtaining the raw result, rounding it off to the correct number of decimal places yields the final answer. This process ensures that no false precision is implied in the result.

Calculating Significant Figures

Understanding how to calculate significant figures is fundamental in chemistry to ensure results are accurately reported. Significant figures are the digits in a number that carry meaning contributing to its precision. This includes all non-zero numbers, zeros between significant digits, and zeros which are at the end of a number and to the right of the decimal point. When provided with a number like \( 1.285 \times 10^{-2} \) (4 significant figures), and asked to add it to \( 1.24 \times 10^{-3} \) (3 significant figures) and \( 1.879 \times 10^{-1} \) (4 significant figures), you need to look at the least precise term, which in this case is \( 1.24 \times 10^{-3} \) with only 3 significant figures after the decimal.

Thus, when you add the numbers together, you round the final answer to 2 decimal places, giving a result of \( 2.0 \times 10^{-1} \). It's crucial to remember not to round off until the final result is calculated. By respecting the significant figures rule, you ensure that the precision of your results matches the least precise measurement used in your calculations.