Problem 56 - Chemistry Principles and Practice

Question

Perform the indicated calculations, and express the answer to the correct number of significant figures. Use scientific notation where appropriate. (a) \(1.88 \times 36.305\) (b) \(1.04 \times 3.114 / 42\) (c) \(28.5+4.43+0.073\) (d) \(3.10 \times 10^{2}-5.1 \times 10^{1}\)

Step-by-Step Solution

Answer

(a) 68.2, (b) 0.0771, (c) 33.0, (d) 2.59 x 10^2.

Step 1: Calculation for Part (a)

To calculate \(1.88 \times 36.305\), multiply the two numbers. \[1.88 \times 36.305 = 68.2404\] Since \(1.88\) has the least number of significant figures (3), the final answer should also have 3 significant figures. Hence, round to \(68.2\).

Step 2: Calculation for Part (b)

For \(1.04 \times 3.114 / 42\), first multiply and then divide the numbers: \[(1.04 \times 3.114) = 3.23856\] After which, divide by 42: \[\frac{3.23856}{42} = 0.0771095238\] The factor with the least significant figures is \(1.04\) (3), so round to 3 significant figures, resulting in \(0.0771\).

Step 3: Calculation for Part (c)

Sum the numbers: \[28.5 + 4.43 + 0.073 = 33.003\] Here, the number with the least decimal places is \(28.5\) (1 decimal place), so round the result to one decimal place, giving \(33.0\).

Step 4: Calculation for Part (d)

Subtract the numbers in scientific notation: \(3.10 \times 10^{2} - 5.1 \times 10^{1}\). First, convert \(5.1 \times 10^{1}\) to \(51\), then perform the subtraction: \[310 - 51 = 259\] Since \(5.1\) has the least number of decimal places after subtraction (1 decimal place), the result can remain as-is or be expressed as scientific notation with one decimal place: \(2.59 \times 10^2\).

Key Concepts

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form, making it easier to manage calculations involving these numbers. This is especially useful in scientific calculations where such numbers are common.
For example, instead of writing out a lengthy number like 310,000, we can express it in scientific notation as \(3.10 \times 10^{5}\). Here, 3.10 is called the coefficient, and the power of ten (\(10^{5}\)) represents the number of places the decimal point has been moved to the right. Similarly, a tiny number like 0.00052 can be written as \(5.2 \times 10^{-4}\), indicating the decimal point has been moved four places to the left.

• Coefficient: A number usually between 1 and 10, showing the significant figures.
• Exponent: Indicates how many places the decimal point was shifted.

Mathematical Calculations

Mathematical calculations often involve a sequence of operations including addition, subtraction, multiplication, and division. Each operation follows a specific order of precedence, which is critical to obtaining the correct result.

• Perform operations within parentheses first.
• Next, handle exponents and radicals.
• Then, execute multiplication and division from left to right.
• Finally, perform addition and subtraction from left to right.

Maintaining proper significant figures during calculations is essential. For instance, when multiplying or dividing numbers, the final answer should have the same number of significant figures as the number with the least significant figures from the original values.
However, when adding or subtracting, the result should match the least number of decimal places of any number in the operation.

Rounding Rules

Rounding is a process used to reduce the number of digits in a number while keeping its value similar to the original number. It is an important concept in mathematical calculations to express the answer in a suitable form with the appropriate number of significant figures.
Here are some basic rounding rules:

• Identify the digit at the place you need to round to.
• Look at the digit immediately to the right (the next smallest place value).
• If that digit is less than 5, round down by leaving the identified place value unchanged.
• If the digit is 5 or more, round up by increasing the identified place value by one.

For example, rounding 68.2404 to three significant figures gives us 68.2 because the fourth digit (0) is less than 5. Always choose the number of significant figures based on the number with the least significant figures in your calculation, as shown in the exercise solutions.