Problem 79 - Chemistry: An Atoms-Focused Approach

Question

Perform each of the following calculations and express the answer with the correct number of significant figures: a. \(3.15 \times 2255 / 7.7=\) b. \(\left(6.7399 \times 10^{-18}\right) \times\left(1.0135 \times 10^{3}\right) /(52.67+0.144)=\) c. \((4.7+58.69) /\left(6.022 \times 10^{23} \times 6.864\right)=\) d. \((76.2-60.0) /\left[43.53 \times\left(9.988 \times 10^{4}\right)\right]=\)

Step-by-Step Solution

Answer

Question: Calculate the final value of each expression, considering the significant figures: a. \(3.15 \times 2255 / 7.7=\) b. \(\left(6.7399 \times 10^{-18}\right) \times\left(1.0135 \times 10^{3}\right) /(52.67+0.144)=\) c. \((4.7+58.69) /\left(6.022 \times 10^{23} \times 6.864\right)=\) d. \((76.2-60.0) /\left[43.53 \times\left(9.988 \times 10^{4}\right)\right]=\) Answer: a. 110 b. \(1.29 \times 10^{-16}\) c. \(1.5 \times 10^{-23}\) d. \(3.73 \times 10^{-6}\)

Step 1: Determine the significant figures

We need to find the significant figures in each number involved in the calculation to determine the final answer having the correct number of significant figures. 3.15 = 3 significant figures 2255 = 4 significant figures 7.7 = 2 significant figures

Step 2: Perform the calculation

We perform the given calculation: \(3.15 \times 2255 / 7.7 = 869.675324675 / 7.7 = 112.933272727\)

Step 3: Determine the least significant figures

We will now round our answer to the least number of significant figures among our given numbers, which is 2. The final answer will have 2 significant figures. \(112.933272727 \approx 110\) Answer: 110 #b. \(\left(6.7399 \times 10^{-18}\right) \times\left(1.0135 \times 10^{3}\right) /(52.67+0.144)=\) #

Step 4: Determine the significant figures

Identify the significant figures in each number: 6.7399 \(\times\) 10^{-18} = 5 significant figures 1.0135 \(\times\) 10^3 = 5 significant figures 52.67 = 4 significant figures 0.144 = 3 significant figures

Step 5: Perform the calculation

We perform the given calculation: \(\left(6.7399 \times 10^{-18}\right) \times\left(1.0135 \times 10^{3}\right) /(52.67+0.144) = 6.83332225 \times 10^{-15} / 52.814\)

Step 6: Determine the least significant figures

We round our answer to the least number of significant figures, which is 3. \(6.83332225 \times 10^{-15} / 52.814 \approx 1.29 \times 10^{-16}\) Answer: \(1.29 \times 10^{-16}\) #c. \((4.7+58.69) /\left(6.022 \times 10^{23} \times 6.864\right)=\)#

Step 7: Determine the significant figures

Identify the significant figures in each number: 4.7 = 2 significant figures 58.69 = 4 significant figures 6.022 \(\times\) 10^23 = 4 significant figures 6.864 = 4 significant figures

Step 8: Perform the calculation

We perform the given calculation: \((4.7+58.69) /\left(6.022 \times 10^{23} \times 6.864\right) = 63.39 / \left(4.1365848 \times 10^{24}\right)\)

Step 9: Determine the least significant figures

We round our answer to the least number of significant figures, which is 2. \(63.39 / \left(4.1365848 \times 10^{24}\right) \approx 1.5 \times 10^{-23}\) Answer: \(1.5 \times 10^{-23}\) #d. \((76.2-60.0) /\left[43.53 \times\left(9.988 \times 10^{4}\right)\right]=\)#

Step 10: Determine the significant figures

Identify the significant figures in each number: 76.2 = 3 significant figures 60.0 = 3 significant figures 43.53 = 4 significant figures 9.988 \(\times\) 10^4 = 4 significant figures

Step 11: Perform the calculation

We perform the given calculation: \((76.2-60.0) /\left[43.53 \times\left(9.988 \times 10^{4}\right)\right] = 16.2 / \left(4.3468394 \times 10^{6}\right)\)

Step 12: Determine the least significant figures

We round our answer to the least number of significant figures, which is 3. \(16.2 / \left(4.3468394 \times 10^{6}\right) \approx 3.73 \times 10^{-6}\) Answer: \(3.73 \times 10^{-6}\)

Key Concepts

Scientific Notation

Scientific notation is a useful way to handle very large or very small numbers, making them easy to read, write, and calculate with. It's essentially a way to express numbers as a product of a coefficient and a power of 10.
For example, the number 6,739,900,000,000,000,000 can be written more compactly in scientific notation as \(6.7399 \times 10^{18}\). Similarly, very small numbers like 0.000000000067399 can be expressed as \(6.7399 \times 10^{-11}\).
When using scientific notation, the coefficient must be a number between 1 and 10, but it can include decimal places. The power of 10 indicates how many places to move the decimal point:

• A positive exponent moves the decimal to the right, making the number larger.
• A negative exponent moves the decimal to the left, making the number smaller.

In calculations like multiplication or division, you multiply or divide the coefficients as usual while adding or subtracting the powers of 10 according to the operation. Always keep track of the significant figures to ensure your final answer reflects the correct precision.

Rounding Numbers

Rounding numbers is an essential part of working with significant figures in numerical calculations. When we perform operations like addition, subtraction, multiplication, or division, the precision of our result is limited by the number with the least significant figures.
Significant figures refer to the digits in a number that contribute to its precision, including all digits except:

• Leading zeros, which are the zeros at the start of a number.
• Trailing zeros only when they follow a decimal point.

To round a number to a certain number of significant figures, look at the digit right after the last significant figure you want to keep:

• If it's 5 or more, round up.
• If it's less than 5, leave the last desired figure as it is.

For example, if your calculation yields 112.933272727 but you need two significant figures, consider only the first two significant digits (11), then assess the next digit (2). Since it's less than 5, the number rounds down to 110.
Accurate rounding is crucial when reporting results, especially in scientific contexts where maintaining the precision of measurements is important.

Numerical Calculations

Numerical calculations involving significant figures demand both precision and accuracy. In practice, this means carefully attending to both the values you calculate and how you report them, maintaining proper numerical integrity.
The process generally involves:

• Performing operations in the correct order (parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right).
• Keeping track of significant figures at each step to ensure accurate results. The number of significant figures in the final result is dictated by the factor in the calculation with the fewest significant figures.

For example, when calculating \((3.15 \times 2255 \div 7.7)\), the significant figures of each number (3, 4, and 2 respectively) lead us to express the final answer with two significant figures. This is because 7.7 has the least number of significant figures. Hence, following through the calculations properly and managing significant figures ensures that the results realistically reflect the precision of the measures involved.