Problem 80 - Introductory Chemistry Atoms First

Question

Do these calculations using a scientific calculator and report your answers in scientific notation: (a) \(\left(3.33 \times 10^{4} \mathrm{~km}\right)+\left(2.22 \times 10^{5} \mathrm{~km}\right)\) (b) \(\left(2.444 \times 10^{9} \mathrm{~J}\right) \div\left(2.444 \times 10^{-9} \mathrm{~J}\right)\) (c) \(\left(2.34 \times 10^{2} \mathrm{~m}\right)-\left(2.34 \times 10^{1} \mathrm{~m}\right)\) (d) \(\left(4.00 \times 10^{4} \mathrm{~L}\right)+\left(6.00 \times 10^{-1} \mathrm{~L}\right)\)

Step-by-Step Solution

Answer

(a) \(2.553 \times 10^5\) km (b) \(1 \times 10^{18}\) (c) \(2.106 \times 10^2\) m (d) \(4.006 \times 10^4\) L

Step 1: Align the exponents

To add the numbers, we need to have the same exponent. Here, we will rewrite \(3.33 \times 10^4\) as \(0.333 \times 10^5\).

Step 2: Add the numbers with the same exponent

Now that we have the same exponent, we can add the mantissas together: \(0.333 + 2.22 = 2.553\).

Step 3: Write the result in scientific notation

The final result is \(2.553 \times 10^5\) km. (b) $\left(2.444 \times 10^{9} \mathrm{~J}\right) \div\left(2.444 \times 10^{-9} \mathrm{~J}\right)$

Step 1: Divide the mantissas

We need to divide the mantissas: \(\frac{2.444}{2.444} = 1\).

Step 2: Subtract the exponents

After dividing the mantissas, we need to subtract the exponents: \(9 - (-9) = 18\).

Step 3: Write the result in scientific notation

The final result is \(1 \times 10^{18}\). (c) $\left(2.34 \times 10^{2} \mathrm{~m}\right)-\left(2.34 \times 10^{1} \mathrm{~m}\right)$

Step 1: Align the exponents

To subtract the numbers, we need to have the same exponent. Here, we will rewrite \(2.34 \times 10^1\) as \(0.234 \times 10^2\).

Step 2: Subtract the numbers with the same exponent

Now that we have the same exponent, we can subtract the mantissas: \(2.34 - 0.234 = 2.106\).

Step 3: Write the result in scientific notation

The final result is \(2.106 \times 10^2\) m. (d) $\left(4.00 \times 10^{4} \mathrm{~L}\right)+\left(6.00 \times 10^{-1} \mathrm{~L}\right)$

Step 1: Align the exponents

To add the numbers, we need to have the same exponent. Here, we will rewrite \(6.00 \times 10^{-1}\) as \(0.006 \times 10^4\).

Step 2: Add the numbers with the same exponent

Now that we have the same exponent, we can add the mantissas together: \(4.00 + 0.006 = 4.006\).

Step 3: Write the result in scientific notation

The final result is \(4.006 \times 10^4\) L.

Key Concepts

Scientific Notation

Scientific notation is a convenient way of expressing very large or very small numbers. This method simplifies complex numbers by using powers of ten. In scientific notation, a number is written as the product of two parts: a decimal number (called the mantissa) and an exponent of ten. For example, the number 7,500 can be written as \(7.5 \times 10^3\). The mantissa provides the significant digits, while the exponent indicates the order of magnitude or how many places the decimal point has been moved.

• The mantissa is typically a number between 1 and 10.
• The exponent shows how many places the decimal point is moved.

The primary benefit of using scientific notation is to easily perform arithmetic operations on large numbers. This format is especially useful in scientific fields such as chemistry and physics, where extremely large or small numbers are frequently encountered.

Mantissa and Exponent Manipulation

Understanding how to manipulate the mantissa and the exponent correctly is crucial when working with scientific notation. When performing arithmetic operations, the mantissas are adjusted based on the operation, while the exponents must be aligned:

• Aligning Exponents: Before adding or subtracting, make sure the exponents are the same. Adjust the mantissa accordingly. This might mean shifting the decimal point in the mantissa so that the exponents match.
• Adjusting Mantissas: Once the exponents are aligned, add or subtract the mantissas as you would in regular decimal form.

This manipulation ensures that our computations remain consistent and gives us results in an understandable format. As seen in our example problems, aligning exponents is key for both addition and subtraction in scientific notation.

Arithmetic Operations with Exponents

Performing arithmetic operations with exponents involves some unique rules:

• Adding/Subtracting: Align the exponents before performing operations on the mantissas, as explained earlier. The operation then proceeds as normal based on the mantissas.
• Multiplying: For multiplication, simply multiply the mantissas together and add the exponents. For example, \((2 \times 10^3) \times (3 \times 10^2)\) becomes \(6 \times 10^{(3+2)} = 6 \times 10^5\).
• Dividing: Similar to multiplication, divide the mantissas and subtract the exponents. For instance, dividing \( (8 \times 10^5) \div (2 \times 10^2)\) results in \(4 \times 10^{(5-2)} = 4 \times 10^3\).

These methods make complex calculations simpler and are part of what makes scientific notation so useful for scientific computations.

Problem-Solving Steps in Chemistry

Problem-solving in chemistry often requires working with very precise and sometimes cumbersome amounts. Using scientific notation allows chemists to simplify calculations. It is essential to proceed systematically through these problems by following a structured approach:

• Identify the Given Information: Start by determining what is given and what you need to find. Write the quantities in scientific notation if they aren't already.
• Perform Necessary Calculations: Use knowledge of mantle and exponent manipulation to align exponents where necessary and perform arithmetic operations.
• Check Units and Significant Figures: Ensure that the units are correct and that the number of significant figures is maintained throughout the calculation.
• Write the Final Result Clearly: After completing calculations, express your result in scientific notation to maintain clarity and accuracy.

This framework helps students tackle questions systematically, ensuring accuracy and clarity, crucial in scientific study.