Problem 15 - General Chemistry: Principles and Modern Applications

Question

Express each value in exponential form. Where appropriate, include units in your answer. (a) speed of sound (sea level): 34,000 centimeters per second (b) equatorial radius of Earth: 6378 kilometers (c) the distance between the two hydrogen atoms in the hydrogen molecule: 74 trillionths of a meter (d) \(\frac{\left(2.2 \times 10^{3}\right)+\left(4.7 \times 10^{2}\right)}{5.8 \times 10^{-3}}=\)

Step-by-Step Solution

Answer

(a) \(3.4 \times 10^{4}\) cm/sec, (b) \(6.378 \times 10^{3}\) km, (c) \(7.4 \times 10^{-11}\) m, (d) \(4.6 \times 10^{5}\).

Step 1: Express the Speed of Sound in Exponential Form

To express the speed of sound in exponential form, identify the power of ten that the number 34,000 can be represented by. It can be seen that 34,000 = 3.4 x 10^4. Hence, the speed of sound at sea level is \(3.4 \times 10^{4}\) centimeters per second.

Step 2: Express the Equatorial Radius of Earth in Exponential Form

To express 6378 kilometers in exponential form, observe that 6378 = 6.378 x 10^3. Hence, the equatorial radius of Earth is \(6.378 \times 10^{3}\) kilometers.

Step 3: Express the Distance between Hydrogen Atoms in Exponential Form

To express 74 trillionths of a meter in exponential form, note that a trillionth is equivalent to \(10^{-12}\). Therefore, 74 trillionths = 74 x \(10^{-12}\), which simplifies to \(7.4 \times 10^{-11}\). Hence, the distance between the hydrogen atoms in a hydrogen molecule is \(7.4 \times 10^{-11}\) meters.

Step 4: Evaluate the Given Expression

Observe that the result of the expression will depend on the rules of operations, i.e., the order of operations: parentheses first, then multiplication and division (from left to right). \n To simplify the expression, begin by solving the operation in the parenthesis: \(2.2 \times 10^{3} + 4.7 \times 10^{2} = 2.2 \times 1000 + 4.7 \times 100 = 2200 + 470 = 2670\). \n Then divide this result by \(5.8 \times 10^{-3}\): \(\frac{2670}{5.8 \times 10^{-3}}\), which simplifies to \(4.6 \times 10^{5}\).

Key Concepts

Speed of Sound

The speed of sound is a fascinating concept in physics. At sea level, the speed of sound is known to be around 34,000 centimeters per second. When expressing this number in exponential notation, we break down 34,000 to its core components.
Using exponential notation simplifies the number to 3.4 multiplied by a power of 10 that represents the number of zeros followed by 3.4, which is 10,000, as in 10 to the power of 4. Thus, the speed of sound becomes:

• \(3.4 \times 10^{4} \text{ cm/s}\)

Exponential notation is very useful when dealing with very large or very small numbers, making them easier to read and compare. It's important to understand that this form maintains the value of the number.

Equatorial Radius of Earth

The Earth's equatorial radius is a key measurement in understanding our planet's size and shape. It is approximately 6,378 kilometers. Expressing it in exponential form involves identifying the place value of each digit.
To simplify, 6,378 can be written as:

• \(6.378 \times 10^{3} \text{ kilometers}\)

This breaks down the number into 6.378 followed by 3 zeros, as indicated by the power of 3. Exponential notation makes it easier to perform calculations with large measurements like the ones involved in astronomical or geographical data.

Distance between Hydrogen Atoms

In the world of chemistry, the distance between hydrogen atoms in a hydrogen molecule is incredibly small. It's measured in trillionths of a meter, specifically 74 trillionths.
A trillionth is expressed as a power of ten: \(10^{-12}\). Therefore, 74 trillionths is:

• \(7.4 \times 10^{-11} \text{ meters}\)

Writing numbers like these in exponential notation is crucial for scientists, as it allows for easier manipulation and comparison of tiny distances at an atomic level.

Order of Operations

The order of operations is crucial in mathematics for solving expressions accurately. It's often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

For example, consider the expression \(\frac{(2.2 \times 10^{3})+(4.7 \times 10^{2})}{5.8 \times 10^{-3}}\).
Start by solving the parentheses:

• \(2.2 \times 10^{3} + 4.7 \times 10^{2} = 2200 + 470 = 2670\)

Next, divide by the denominator:

• \(\frac{2670}{5.8 \times 10^{-3}}\) simplifies to \(4.6 \times 10^{5}\)

Understanding and applying these steps ensures consistency and accuracy in solving mathematical problems.