Problem 72 - Chemistry The Science in Context

Question

Perform each of the following calculations, and express the answer with the correct number of significant figures (only the highlighted values are exact): a. \([(12 \times 60.0)+55.3] /\left(5.000 \times 10^{3}\right)=\) b. \((2.00 \times 183.9) /\left[\left(6.02 \times 10^{23}\right) \times\left(1.61 \times 10^{-8}\right)^{3}\right]=\) c. \(0.8161 /\left[2.205 \times(2.54)^{3}\right]=\) d. \((9.00 \times 60.0)+(50.0 \times 60.0)+\left(3.00 \times 10^{1}\right)=\)

Step-by-Step Solution

Answer

Question: Determine the correct number of significant figures for each of the following calculations: a. \([(12 \times 60.0)+55.3] /\left(5.000 \times 10^{3}\right)=\) b. \((2.00 \times 183.9) /\left[\left(6.02 \times 10^{23}\right) \times\left(1.61 \times 10^{-8}\right)^{3}\right]=\) c. \(0.8161 /\left[2.205 \times(2.54)^{3}\right]=\) d. \((9.00 \times 60.0)+(50.0 \times 60.0)+\left(3.00 \times 10^{1}\right)=\) Answer: a. \(0.15506\) b. \(147\) c. \(0.02254\) d. \(3570\)

Step 1: 1. Multiplication

Perform the multiplication: \(12 \times 60.0 = 720.\) In a multiplication, the result has the same number of significant figures as the value with the least significant figures. Here both values have two significant figures (the given and the highlighted), so the result should also have two: \(720.\)

Step 2: 2. Addition

Perform the addition: \(720 + 55.3 = 775.3\) In an addition, the result has the same number of decimal places as the value with the least decimal places. Here, \(720\) has no decimal places and \(55.3\) has one decimal place, so the result should have one decimal place: \(775.3\)

Step 3: 3. Division

Perform the division: \(775.3 /\left(5.000 \times 10^{3}\right) = 0.15506\) In a division, the result has the same number of significant figures as the value with the least significant figures. Here, both values have five significant figures, so the result should also have five: \(0.15506\) Answer a: \(0.15506\) b. \((2.00 \times 183.9) /\left[\left(6.02 \times 10^{23}\right) \times\left(1.61 \times 10^{-8}\right)^{3}\right]=\)

Step 4: 1. Multiplication

Perform the multiplication: \(2.00 \times 183.9 = 367.8\) The result should have three significant figures: \(367.8\)

Step 5: 2. Cubing and multiplication

Perform the operations in the denominator: \((1.61 \times 10^{-8})^{3} = 4.1713441 \times 10^{-24}\) and \((6.02 \times 10^{23}) \times (4.1713441 \times 10^{-24}) = 2.51119736\) The result should have three significant figures: \(2.51\)

Step 6: 3. Division

Perform the division: \(367.8 / 2.51 = 146.532\) The result should have three significant figures: \(147\) Answer b: \(147\) c. \(0.8161 /\left[2.205 \times(2.54)^{3}\right]=\)

Step 7: 1. Cubing

Cube the value: \((2.54)^{3} = 16.387064\)

Step 8: 2. Multiplication

Perform the multiplication: \(2.205 \times 16.387064 = 36.1739\)

Step 9: 3. Division

Perform the division: \(0.8161/36.1739 = 0.022544\) The result should have four significant figures: \(0.02254\) Answer c: \(0.02254\) d. \((9.00 \times 60.0)+(50.0 \times 60.0)+\left(3.00 \times 10^{1}\right)=\)

Step 10: 1. Multiplication

Perform the multiplications: \(9.00 \times 60.0 = 540\), \(50.0 \times 60.0 = 3000\), \(3.00 \times 10^{1} = 30\) The results should have two significant figures, since all multiplications are between 2-significant figure values\(: 540, 3000,\) and \(30\)

Step 11: 2. Addition

Perform the additions: \(540 + 3000 + 30 = 3570\) Answer d: \(3570\)

Key Concepts

Significant Figures in Multiplication

When you're dealing with multiplication, significant figures are crucial in determining the precision of your final answer. Essentially, the rule is that your product should have the same number of significant figures as the factor with the fewest significant figures in the calculation. For example, consider the multiplication of 12 and 60.0. Here, both values have two significant figures. So, the result (720) is expressed with these two significant figures.
This concept prevents overstating the precision of your result. By maintaining the balance of significant figures, you're ensuring that your calculations remain accurate and meaningful. Always look for the factor with the smallest number of significant figures to decide how precise your final product should be.

Significant Figures in Addition

When performing addition (or subtraction), the key to finding the correct number of significant figures is to look at the decimal places, not the significant figures. To express the answer properly, it must have the same number of decimal places as the number in the calculation with the least decimal places.
For instance, if you are adding 720 and 55.3, you notice that 720 has no decimal places, and 55.3 has one decimal place. Thus, the sum should have one decimal place, resulting in 775.3.
This rule ensures that you aren't assuming more precision in your results than provided by your initial numbers. By focusing on supporting proper decimal placement, you keep your calculations accurate and relevant without assuming extra precision.

Significant Figures in Division

In division, similar to multiplication, the result should have the same number of significant figures as the value in the calculation that has the smallest number of significant figures. It's all about keeping your answer as precise as your least precise number.
Let's consider dividing 775.3 by 5,000 (5.000 x 10³). Both numbers involved in this case have five significant figures, which means the result of the division (0.15506) should also be rounded to five significant figures.
This rule helps maintain the integrity of your calculations, ensuring that you don't falsely indicate a greater level of precision than your initial data allows.

Scientific Notation and Significant Figures

Scientific notation is a method to express very large or very small numbers elegantly and clearly, often used in conjunction with significant figures. When you express a number in scientific notation, you only retain significant figures that truly convey the precision of your measurement.
For example, the calculation \[(6.02 \times 10^{23}) \times (1.61 \times 10^{-8})^3\] demonstrates how large and small values are managed easily using scientific notation. It also ensures clarity when counting significant figures. The outcome is expressed compactly without losing information.
Through scientific notation, significant figures are easily identifiable, and numbers remain manageable for calculations, helping emphasize precision without complicating the maths.