Problem 126 - Chemistry Matter and Change

Question

Complete the following problems in scientific notation. Round off to the correct number of significant figures. (Chapter 2\()\) $$\begin{array}{l}{\text { a. }\left(5.31 \times 10^{-2} \mathrm{cm}\right) \times\left(2.46 \times 10^{5} \mathrm{cm}\right)} \\ {\text { b. }\left(6.42 \times 10^{-2} \mathrm{g}\right) \div\left(3.21 \times 10^{-3} \mathrm{g}\right)} \\ {\text { c. }\left(9.87 \times 10^{4} \mathrm{g}\right)-\left(6.2 \times 10^{3} \mathrm{g}\right)}\end{array}$$

Step-by-Step Solution

Answer

a. \(1.31 \times 10^4 \text{ cm}^2\), b. \(2.00 \times 10^1\), c. \(9.25 \times 10^4 \text{ g}\).

Step 1: Understanding Scientific Notation

Scientific notation is a method of expressing numbers as a product of a coefficient and a power of ten. For instance, \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer. It helps in handling very large or very small numbers efficiently.

Step 1: Multiplication Solution for a

Given the expression: \((5.31 \times 10^{-2} \text{ cm}) \times (2.46 \times 10^5 \text{ cm})\). Multiply the coefficients: \(5.31 \times 2.46 = 13.0626\). Sum the exponents: \(-2 + 5 = 3\). Thus, \(13.0626 \times 10^3\), which in scientific notation is \(1.30626 \times 10^4\). Round to three significant figures: \(1.31 \times 10^4 \text{ cm}^2\).

Step 2: Division Solution for b

Given the expression: \((6.42 \times 10^{-2} \text{ g}) \div (3.21 \times 10^{-3} \text{ g})\). Divide the coefficients: \(6.42 \div 3.21 = 2.\). Subtract the exponents: \(-2 - (-3) = 1\). Thus, \(2 \times 10^1\), which rounds to three significant figures: \(2.00 \times 10^1 \text{ (dimensionless)}\).

Step 3: Subtraction Solution for c

Given the expression: \((9.87 \times 10^4 \text{ g}) - (6.2 \times 10^3 \text{ g})\). Convert \(6.2 \times 10^3\) to \(0.62 \times 10^4\) for compatibility: \(9.87 \times 10^4 - 0.62 \times 10^4 = 9.25 \times 10^4\). It remains in the form of scientific notation, and rounds to three significant figures: \(9.25 \times 10^4 \text{ g}\).

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. When measuring or expressing data, it is crucial to use the appropriate number of significant figures to accurately reflect the precision of your instruments or calculations. This ensures consistency and clarity in scientific communication.
Here are some general rules to remember:

• All non-zero digits are significant. For example, in the number 123.45, all numbers are significant because they are non-zero.
• Any zeros between significant digits are also significant. For example, in 105.0, zeros are significant.
• Leading zeros are not significant. In 0.0023, only the digits 2 and 3 are significant.
• Trailing zeros in a decimal number are significant – like in 50.00, all digits are significant because they follow a decimal point.

When rounding off calculations, match the number of significant figures to the number used in the given measurements. This helps maintain precision consistency throughout your work.

Multiplication in Scientific Notation

Multiplying numbers in scientific notation involves two main steps: multiplying the coefficients and adding the exponents of the ten’s powers. This approach simplifies calculations, especially when dealing with very large or very small numbers.
Let's break down the process:

• Step 1: Multiply the coefficients. For example, when you multiply 5.31 by 2.46, the result is 13.0626.
• Step 2: Add the exponents. If the exponents are -2 and 5, you would add them to get 3.
• Step 3: Combine the results: you would get 13.0626 × 10^3.

To express in proper scientific notation, adjust to have only one non-zero digit to the left of the decimal point, resulting in 1.30626 × 10^4. Finally, apply significant figures for your result, which typically matches the lowest number of significant figures involved in your initial data, leading to the answer of 1.31 × 10^4.

Division in Scientific Notation

Dividing numbers in scientific notation is quite similar to multiplying, but it involves subtracting the exponents instead. This technique is very effective for simplifying division operations.
Here is a step-by-step approach:

• Step 1: Divide the coefficients of the numbers. For instance, dividing 6.42 by 3.21 gives 2.
• Step 2: Subtract the exponents. Given -2 and -3 as the exponents, subtract -3 from -2 to get 1.
• Step 3: Put it together. Having a result like 2 × 10^1 clearly illustrates the division.

To maintain the number of significant figures, ensure the final answer reflects the precision of the original figures, which helps preserve accuracy. For the given example, this means showing the result as 2.00 × 10^1.

Addition and Subtraction in Scientific Notation

When adding or subtracting numbers in scientific notation, it is essential to express them with the same exponent for consistency. This alignment allows for easier arithmetic operations.
Here's how to go about it:

• Step 1: Adjust the numbers such that they both have the same exponent. For example, convert 6.2 × 10^3 to 0.62 × 10^4.
• Step 2: Perform the addition or subtraction of the coefficients as you would for regular numbers. With the modified numbers 9.87 × 10^4 and 0.62 × 10^4, subtract to get 9.25 × 10^4.

Ensure that the result, if needed, is in proper scientific notation and adjust the number of significant figures based on the least precise measurement in the calculation. This leads to the result being accurately reflected in the number of significant digits, for instance, 9.25 × 10^4 for the given problem.