Problem 66 - Chemistry: Atoms First

Question

The distance between the centers of the two oxygen atoms in an oxygen molecule is \(1.21 \times 10^{-8} \mathrm{cm}\). What is this distance in inches?

Step-by-Step Solution

Answer

The distance between the centers of the two oxygen atoms in an oxygen molecule is approximately \(4.76 \times 10^{-10}\) inches.

Step 1: Identify Conversion Factor

The conversion factor from centimeters to inches is 1 inch = 2.54 cm. This means that to convert a length from centimeters to inches, you divide the number of centimeters by 2.54.

Step 2: Set Up Conversion Equation

To find the distance in inches, use the conversion factor to set up the equation: \[\begin{equation} \text{Distance in inches} = \text{Distance in cm} \times \left(\frac{1\text{ inch}}{2.54\text{ cm}}\right). \end{equation}\]

Step 3: Convert Measurement

Insert the given distance into the equation and calculate: \[\begin{equation} \text{Distance in inches} = 1.21 \times 10^{-8} \text{ cm} \times \left(\frac{1\text{ inch}}{2.54\text{ cm}}\right). \end{equation}\]Perform the multiplication to obtain the distance in inches.

Step 4: Calculate the Answer

Perform the calculation to get: \[\begin{equation} \text{Distance in inches} = 1.21 \times 10^{-8} \times \frac{1}{2.54} \approx 4.76 \times 10^{-10} \text{ inches}. \end{equation}\]

Key Concepts

Metric to Imperial Conversion

Understanding how to convert measurements between different systems is an essential part of chemistry and many other scientific disciplines. The metric system is commonly used worldwide and is based on units like meters, liters, and grams. However, in the United States, the imperial system is also prevalent, using units such as inches, gallons, and pounds.

To convert units from metric to imperial, or vice versa, you need to know the correct conversion factors. For example, when converting from centimeters to inches, the conversion factor is that 1 inch equals 2.54 centimeters. This means that if you have a measurement in centimeters, you can convert it to inches by dividing the number of centimeters by 2.54.

It's important to note that conversion factors are specific to each type of measurement. Hence, while you could convert lengths using the centimeters to inches conversion factor, converting volumes or masses would require different factors, such as liters to gallons or grams to pounds.

Oxygen Molecule Bond Length

The oxygen molecule, composed of two oxygen atoms, is a vital component of the air we breathe. In an oxygen molecule, the two atoms are held together by a covalent bond. The length of this bond is the distance between the nuclei of the two atoms. For an oxygen molecule (\text{O}_2), this bond length is approximately 1.21 angstroms, or \(1.21 \times 10^{-10}\) meters. In chemistry, knowing the bond length is crucial because it can affect the physical and chemical properties of the molecule.

To put this into perspective, the bond lengths in molecules are usually in the range of 1 to 2 angstroms, which is extremely small and hence measured in units suitable for atomic scales. Understanding bond lengths is key when discussing molecular shapes, sizes, and how they interact with one another. It also plays an important role in understanding the energy levels associated with certain molecular bonds.

Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact format. It's particularly useful in fields like chemistry where reactions can occur on the atomic scale, which involves incredibly small numbers, and in astronomy, where the distances are immensely vast.

A number in scientific notation is written as the product of two numbers: a coefficient and a power of 10. The coefficient is typically a number that is greater than or equal to 1, but less than 10, while the power of 10 shows how many times 10 should be multiplied or divided to achieve the original number. For example, the bond length in an oxygen molecule can be written as \(1.21 \times 10^{-10}\) meters in scientific notation.

This format significantly simplifies calculations and comparisons of magnitude. Without scientific notation, working with such numbers would be cumbersome and error-prone, making it an indispensable tool in any field that deals with quantities outside of the human scale.