Problem 4 - Analytical Chemistry

Question

Report results for the following calculations to the correct number of significant figures. a. \(4.591+0.2309+67.1=\) b. \(313-273.15=\) c. \(712 \times 8.6=\) d. \(1.43 / 0.026=\) e. \((8.314 \times 298) / 96485=\) f. \(\log \left(6.53 \times 10^{-5}\right)=\) g. \(10^{-7.14}=\) h. \(\left(6.51 \times 10^{-5}\right) \times\left(8.14 \times 10^{-9}\right)=\)

Step-by-Step Solution

Answer

a. 71.9; b. 39.85; c. 6100; d. 55.0; e. 0.02568; f. -4.185; g. 7.24 × 10^{-8}; h. 5.30 × 10^{-13}.

Step 1: Addition of decimal numbers

For the addition of numbers like in (a) \(4.591 + 0.2309 + 67.1\), you should align the numbers by their decimal points and consider significant figures. The least number of decimal places in the original numbers is 1 (in 67.1). Therefore, the answer should have 1 decimal place: \ \(4.591 + 0.2309 + 67.1 = 71.9219\) rounds to \(71.9\).

Step 2: Subtraction to the correct decimal place

In example (b), \(313 - 273.15\), focus on the lowest number of decimal places in the operands. Since \(273.15\) has two decimal places, the answer should have two as well: \ \(313 - 273.15 = 39.85\).

Step 3: Multiplication with significant figures

For (c) \(712 \times 8.6\), count the significant figures in each number. \(712\) has 3 significant figures and \(8.6\) has 2, so the answer should have 2 significant figures: \ \(712 \times 8.6 = 6123.2\) rounds to \(6100\).

Step 4: Division with significant figures

In (d) \(1.43 / 0.026\), each input has a different number of significant figures. \(1.43\) has 3 significant figures, and \(0.026\) has 2; the result should have 2 significant figures: \ \(1.43 / 0.026 \approx 55.0\).

Step 5: Calculating expressions with constants

For (e) \((8.314 \times 298) / 96485\), identify the number of significant figures. \(8.314\) and \(298\) both have 4 significant figures, while \(96485\) has 5, so the result should maintain 4: \ \((8.314 \times 298) / 96485 \approx 0.02568\).

Step 6: Logarithm calculation

For (f) \(\log(6.53 \times 10^{-5})\), the significant figures come from the mantissa given by \(6.53\), which has 3 significant figures. Consequently, the answer is:\ \(\log(6.53 \times 10^{-5}) \approx -4.185\).

Step 7: Exponential calculation

In (g) \(10^{-7.14}\), the number of decimal places in the exponent dictates that precision. Treat \(-7.14\) as having two significant figures when expressing its result \ \(10^{-7.14} \approx 7.24 \times 10^{-8}\).

Step 8: Multiplication with scientific notation

In calculation (h), \((6.51 \times 10^{-5}) \times(8.14 \times 10^{-9})\), the numbers have 3 significant figures each. Thus, the outcome should have 3 significant figures: \ \((6.51 \times 10^{-5}) \times(8.14 \times 10^{-9}) = 5.29734 \times 10^{-13}\) rounds to \(5.30 \times 10^{-13}\).

Key Concepts

Addition and Subtraction Rules

When performing addition or subtraction, the number of decimal places is key for determining significant figures. For instance, in the equation \(4.591 + 0.2309 + 67.1\), align the decimal points first. The term with the least decimal places, \(67.1\) in this case, has one decimal place. Thus, your result should be rounded to one decimal place. After computing the sum, \(71.9219\), it's rounded down to \(71.9\).

Similarly, in subtraction, ensure the result follows the operand with the least decimal places. In the calculation \(313 - 273.15\), \(273.15\) has two decimal places, so the answer retains two decimal places, resulting in \(39.85\). These rules provide precision by keeping track of decimal place consistency.

Multiplication and Division Rules

For multiplication and division, the critical focus is on the number of significant figures, not decimal places. In the multiplication example \(712 \times 8.6\), each number has a different count of significant figures—3 for \(712\) and 2 for \(8.6\). Therefore, the final answer must be limited to 2 significant figures, which rounds \(6123.2\) to \(6100\).

With division, as shown by \(1.43 / 0.026\), use the number with the fewest significant figures to decide the count for your answer. Here, \(0.026\) has 2 significant figures, dictating that the result should be rounded accordingly to \(55.0\), maintaining two significant figures.

• Always use the operand with the least significant figures.
• This ensures results are not reported with more precision than the data supports.

Scientific Notation

Scientific notation is a powerful tool for handling very large or very small numbers. It simplifies expressions like \(6.51 \times 10^{-5} \times 8.14 \times 10^{-9}\) by focusing on significant figures. Both numbers in this expression have 3 significant figures, leading the result \(5.29734 \times 10^{-13}\) to be rounded to \(5.30 \times 10^{-13}\).

When calculating powers of ten, such as in \(10^{-7.14}\), the exponent guides significant figures. Treat the exponent's decimal places as the rule:

• This exponent, \(-7.14\), has two decimal places.
• The expression rounds to \(7.24 \times 10^{-8}\).

Scientific notation streamlines complex calculations, maintaining clarity and precision with significant figures in mind.

Logarithmic Calculations

Logarithmic calculations require careful attention to significant figures, particularly focusing on the mantissa of the number. For \(\log(6.53 \times 10^{-5})\), the leading decimal, \(6.53\), contains 3 significant figures. Thus, the result of the logarithmic operation should mirror this precision: approximately \(-4.185\).

This differs in handling compared to multiplication or division, since precision is derived from the characteristics of the logarithmic input itself.

• Logarithms require special consideration of the initial number's significant figures.
• The result reflects the accuracy of the mantissa used in the calculation.

Meticulous attention to significant figures here ensures the integrity of calculations involving logarithmic functions.