Problem 80 - Chemistry and Chemical Reactivity

Question

Carry out the following calculation, and report the answer to the correct number of significant figures. $$ (1.68)\left[\frac{23.56-2.3}{1.248 \times 10^{3}}\right] $$

Step-by-Step Solution

Answer

The answer is 0.0287, rounded to three significant figures.

Step 1: Perform the Subtraction in the Numerator

In the expression \(23.56 - 2.3\), subtract the numbers while keeping in mind significant figures rules. The number 2.3 has two significant figures, so the result should also have two significant figures. Calculation: \(23.56 - 2.3 = 21.26\) However, since we must adhere to the least number of decimal places, which is one decimal place from 2.3, the result should be \(21.3\).

Step 2: Divide the Result by the Denominator

Now take the result from the numerator and divide by the denominator: \(\frac{21.3}{1.248 \times 10^3}\)Perform the division: \(21.3 \div 1248 = 0.0170673076923077\)The denominator (1.248) limits the significant figures here, which has four significant figures, so the answer should be rounded to four significant figures: \(0.01707\)

Step 3: Multiply by the Number Outside the Brackets

Now, multiply the bracket result by 1.68. Calculation: \(1.68 \times 0.01707 = 0.0286776\)The number 1.68 has three significant figures, so round the result to three significant figures: \(0.0287\).

Step 4: Present the Final Result with Correct Significant Figures

The final computation after multiplication should be presented with the correct significant figures as determined in the previous steps:\[0.0287\]

Key Concepts

Subtraction with Significant Figures

When performing subtraction, significant figures play a crucial role. The guiding rule for subtraction is that the result should have the same number of decimal places as the measurement with the fewest decimal places. This helps in maintaining the precision of the least precise value in the calculation.

In this exercise, we subtracted 2.3 from 23.56. Even though the computation gives us 21.26, we must round to reflect the number with the least decimal places, which is one from 2.3. Therefore, the result is correctly reported as 21.3. An easy way to remember this rule is: "Stick to the lowest decimal places, just like glue!"

• Always check the number of decimal places, not the number of significant figures in subtraction.
• Round your result to have the same decimal precision as the least precise measurement.

Properly handling significant figures in subtraction ensures accuracy and consistency across calculations.

Division with Scientific Notation

When dividing numbers, especially those involving scientific notation, significant figures again become essential. The number of significant figures in the result is determined by the original number with the least significant figures.

In the provided solution, after subtracting, the quotient \(\frac{21.3}{1.248 \times 10^3}\) provides a valuable insight. The denominator (1.248) has four significant figures. Thus, despite the detailed decimal answer being 0.0170673, you should only keep four significant figures, rounding it to 0.01707.

• Count the significant figures in each element involved in the division.
• Match the result's significant figures to the component with the fewest significant figures.
• Using scientific notation correctly can help in making calculations more manageable and less error-prone.

Scientific notation not only helps in dealing with very large or small numbers but also keeps significant figures in check in a busy calculation.

Multiplication with Significant Figures

In multiplication, similar to division, the significant figures in the result should match those of the input number with the least significant figures. This ensures that the precision of your answer is aligned with the measured data.

In the final step of the exercise, we multiply 1.68 by 0.01707. Here, 1.68, having three significant figures, determines that our answer must be rounded to also reflect three significant figures. Thus, the raw result of 0.0286776 is appropriately rounded to 0.0287.

• Observe the significant figures in all numbers being multiplied.
• Limit the significant figures in your final result to match the smallest significant figures in your inputs.
• Always perform rounding only at the very end of your calculation to avoid cumulative errors.

By applying these principles, one can consistently deliver accurate results where precision mirrors the least precise measurement used.