Problem 99 - Chemistry Matter and Change

Question

Complete these problems in scientific notation. Round to the correct number of significant figures. a. \(\left(5.31 \times 10^{-2} \mathrm{cm}\right) \times\left(2.46 \times 10^{5} \mathrm{cm}\right)\) b. \(\left(3.78 \times 10^{3} \mathrm{m}\right) \times\left(7.21 \times 10^{2} \mathrm{m}\right)\) c. \(\left(8.12 \times 10^{-3} \mathrm{m}\right) \times\left(1.14 \times 10^{-5} \mathrm{m}\right)\) d. \(\left(9.33 \times 10^{4} \mathrm{mm}\right) \div\left(3.0 \times 10^{2} \mathrm{mm}\right)\) e. \(\left(4.42 \times 10^{-3} \mathrm{kg}\right) \div\left(2.0 \times 10^{2} \mathrm{kg}\right)\) f. \(\left(6.42 \times 10^{-2} \mathrm{g}\right) \div\left(3.21 \times 10^{-3} \mathrm{g}\right)\)

Step-by-Step Solution

Answer

a) \(1.31 \times 10^{4} \mathrm{cm}^2\); b) \(2.73 \times 10^{6} \mathrm{m}^2\); c) \(9.26 \times 10^{-8} \mathrm{m}^2\); d) \(3.1 \times 10^{2} \mathrm{mm}\); e) \(2.2 \times 10^{-5}\); f) \(2.0 \times 10^{1}\).

Step 1: Multiply the Coefficients for Part a

To solve \( (5.31 \times 10^{-2} \mathrm{cm}) \times (2.46 \times 10^{5} \mathrm{cm}) \), first multiply the coefficients \(5.31\) and \(2.46\). This gives us the product \(13.0626\).

Step 2: Add the Exponents for Part a

Add the exponents of the powers of 10 together: \(-2 + 5 = 3\). So the result is \(13.0626 \times 10^{3} \mathrm{cm}^2\).

Step 3: Round and Convert to Scientific Notation for Part a

The result \(13.0626 \times 10^{3}\) is rounded to three significant figures to get \(1.31 \times 10^{4} \mathrm{cm}^2\).

Step 4: Multiply the Coefficients for Part b

For \( (3.78 \times 10^{3} \mathrm{m}) \times (7.21 \times 10^{2} \mathrm{m}) \), multiply \(3.78\) and \(7.21\) to get \(27.2538\).

Step 5: Add the Exponents for Part b

Add the exponents of the powers of 10: \(3 + 2 = 5\). So the product is \(27.2538 \times 10^{5} \mathrm{m}^2\).

Step 6: Round and Convert to Scientific Notation for Part b

Round \(27.2538\) to three significant figures to get \(2.73 \times 10^{6} \mathrm{m}^2\).

Step 7: Multiply the Coefficients for Part c

To calculate \( (8.12 \times 10^{-3} \mathrm{m}) \times (1.14 \times 10^{-5} \mathrm{m}) \), multiply \(8.12\) and \(1.14\) to get \(9.2568\).

Step 8: Add the Exponents for Part c

Add the exponents: \(-3 + (-5) = -8\). So \(9.2568 \times 10^{-8} \mathrm{m}^2\) is the result.

Step 9: Round and Convert to Scientific Notation for Part c

Round this to three significant figures: \(9.26 \times 10^{-8} \mathrm{m}^2\).

Step 10: Divide the Coefficients for Part d

For \( (9.33 \times 10^{4} \mathrm{mm}) \div (3.0 \times 10^{2} \mathrm{mm}) \), divide \(9.33\) by \(3.0\) to get \(3.11\).

Step 11: Subtract the Exponents for Part d

Subtract the exponents: \(4 - 2 = 2\). So the result is \(3.11 \times 10^{2} \mathrm{mm}\).

Step 12: Round to Two Significant Figures for Part d

The result is already in the format \(3.1 \times 10^{2} \mathrm{mm}\) since the initial measurement \(3.0\) had two significant figures.

Step 13: Divide the Coefficients for Part e

Calculate \( (4.42 \times 10^{-3} \mathrm{kg}) \div (2.0 \times 10^{2} \mathrm{kg}) \) by dividing \(4.42\) by \(2.0\) to get \(2.21\).

Step 14: Subtract the Exponents for Part e

Subtract the exponents: \(-3 - 2 = -5\). So the result is \(2.21 \times 10^{-5}\).

Step 15: Round to Two Significant Figures for Part e

Since the divisor \(2.0\) has two significant figures, round \(2.21\) to \(2.2\), giving \(2.2 \times 10^{-5}\).

Step 16: Divide the Coefficients for Part f

For \( (6.42 \times 10^{-2} \mathrm{g}) \div (3.21 \times 10^{-3} \mathrm{g}) \), divide \(6.42\) by \(3.21\) to get \(2.0\).

Step 17: Subtract the Exponents for Part f

Subtract the exponents: \(-2 - (-3) = 1\). So the result is \(2.0 \times 10^{1}\).

Step 18: Final Result for Part f

The division already respects two significant figures, thus the final result is \(2.0 \times 10^{1}\).

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. Understanding how to determine and use significant figures is crucial in scientific notation and ensures that calculated results are as accurate as the measurements involved allow. Here’s what to keep in mind:

• The number of significant figures includes all non-zero digits and any zeros between them. For example, in the number 5.31, all three digits are significant.
• Leading zeros are not significant. For example, in 0.052, only 5 and 2 are significant.
• Trailing zeros in a number containing a decimal point are significant. In the number 4.50, all three digits are significant.

When performing calculations like multiplication or division in scientific notation, the result should have the same number of significant figures as the number with the smallest count of significant figures from the inputs. This helps ensure that you're not implying the result is more precise than your data supports. For instance, if you multiply 2.46 (three significant figures) by 5.31 (also three significant figures), your result should be rounded to three significant figures too.

Multiplication of Exponents

Multiplying numbers in scientific notation involves both the coefficients and the exponents of the powers of ten. The rule for exponents in multiplication is simple: **add the exponents**. Here's how you handle it:

• Multiply the numerical coefficients. For example, with \(5.31 \times 2.46\), you get \(13.0626\).
• Add the exponents of ten. For instance, if we have \(-2 + 5\), the result is \(10^{3}\).

So, \( (5.31 \times 10^{-2}) \times (2.46 \times 10^{5}) \) results in \(13.0626 \times 10^{3}\), which you would then round appropriately. This process streamlines complex calculations by simplifying the manipulation of large or small numbers often found in scientific data.

Division of Exponents

Dividing numbers in scientific notation follows a process that mirrors multiplication, though the way we handle exponents differs. For division, we need to **subtract the exponents**:

• Divide the coefficients. For \(9.33 \div 3.0\), this gives \(3.11\).
• Subtract the exponents of the powers of ten. For \10^{4} \div 10^{2}\, you compute \(4 - 2 = 2\).

The final result of \( (9.33 \times 10^{4}) \div (3.0 \times 10^{2}) \) becomes \(3.11 \times 10^{2}\). Make sure to verify the significant figures: if the smallest number of significant figures in the inputs is two, the final answer should also have two, potentially leading to further rounding.

Rounding Numbers

Rounding is essential in maintaining the integrity of your calculations, ensuring results don't falsely imply a level of precision that's not present in the data. The general rules for rounding closely align with the significant figures:

• Identify significant figures to retain. If your data points have various significant figures, your result should match the smallest count.
• If a digit immediately after your last significant figure is 5 or greater, round up. For example, rounding \(27.2538\) to three significant figures results in \(27.3\).
• If the digit is less than 5, do not change the last significant figure. For \(13.0626\), rounding to \1.31\ keeps the first significant figures intact.
Applying these principles ensures that all results from operations in scientific notation are consistently accurate and meaningful.