Problem 75 - Chemistry and Chemical Reactivity

Question

Carry out the following operations. Provide the answer with the correct number of significant figures. (a) (1.52)\(\left(6.21 \times 10^{-3}\right)\) (b) \(\left(6.21 \times 10^{3}\right)-\left(5.23 \times 10^{2}\right)\) (c) \(\left(6.21 \times 10^{3}\right) \div\left(5.23 \times 10^{2}\right)\)

Step-by-Step Solution

Answer

(a) 0.00944 (b) 5690 (c) 11.9

Step 1: Multiplication Operation

Multiply the numbers in part (a): \( 1.52 \times 6.21 \times 10^{-3} = 1.52 \times 0.00621 \).Carry out the multiplication: \( 1.52 \times 0.00621 = 0.0094392 \).Determine the number of significant figures: The number with the fewest significant figures is \(6.21\) with 3 significant figures. Round \(0.0094392\) to 3 significant figures to get \(0.00944\).

Step 2: Subtraction Operation

Subtract the numbers in part (b):\( 6.21 \times 10^3 - 5.23 \times 10^2 \).Convert both numbers to the same power of ten: \( 6210 - 523 = 5687 \).Adjust for significant figures: The least precise measurement is \(523\), which has 3 significant figures. Therefore, round \(5687\) to 3 significant figures to get \(5690\).

Step 3: Division Operation

Divide the numbers in part (c):\( \frac{6.21 \times 10^3}{5.23 \times 10^2} \).Perform the division: \( \frac{6210}{523} = 11.87 \).Round to the correct number of significant figures: The number with the fewest significant figures, \(5.23\), has 3 significant figures, so round \(11.87\) to 3 significant figures to get \(11.9\).

Key Concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It's commonly used in science and engineering to handle extremely large or tiny numbers in a manageable way.
To express a number in scientific notation, you'll write it as the product of two parts:

• A number that is greater than or equal to 1 but less than 10.
• A power of ten, which shows how many times the base number should be multiplied by 10.

For example, in the number \[6.21 \times 10^3,\]6.21 is the base or coefficient, and \[10^3\]represents the multiplier. This notation is particularly useful when carrying out calculations involving multiplication and division, as it allows you to separate the coefficients from the powers of ten.
Simply manage the coefficients separately and apply the rules of exponents.

Multiplication and Division

When dealing with multiplication and division in scientific notation, knowing how to handle both the coefficients and the powers of ten is essential.
For multiplication, you multiply the coefficients and then add the exponents of the powers of ten. This maintains the structure of the scientific notation.

• Example: \((a \times 10^{m}) \times (b \times 10^{n}) = (a \times b) \times 10^{m+n}\)

For division, divide the coefficients and subtract the exponents of the powers of ten. This ensures you maintain proper scientific notation form.

• Example: \(\frac{a \times 10^{m}}{b \times 10^{n}} = \frac{a}{b} \times 10^{m-n}\)

Keep in mind the rules for significant figures: The result in both multiplication and division must be expressed with the same number of significant figures as the number with the least significant figures in the operation. This rule helps maintain precision in your results.

Precision in Measurements

Precision in measurements refers to how detailed a measurement is. Significant figures are used to communicate this precision, as they affect the rounding of your final answers.
Significant figures include all non-zero digits, zeroes between non-zero digits, and any zero after the decimal point in a decimal number. Zeroes at the start or those trailing a number in whole numbers without a decimal are not significant.

• If you multiply or divide numbers, the number of significant figures in the product or quotient is determined by the original number with the fewest significant figures. This ensures you don't overstate the precision of your results.
• In addition and subtraction, the precision of the result is determined by the number with the largest uncertainty or the fewest decimal places.

To illustrate, if we consider a measurement of \(6.21\),which comprises 3 significant figures, you would round your results accordingly to match this level of precision. Practicing awareness of significant figures ensures your results are appropriately accurate.