Problem 1 - Chemistry and Global Awareness

Question

What is the wavelength (in \(\mathrm{mm}\) ) of light if its frequency is \(1.55 \times 10^{10} \mathrm{~s}^{-1}\) ?

Step-by-Step Solution

Answer

The wavelength is 19.35 mm.

Step 1: Identify the Formula

To find the wavelength of light, we use the formula that relates wavelength \( \lambda \), frequency \( f \), and the speed of light \( c \): \[\lambda = \frac{c}{f}\]The speed of light \( c \) in vacuum is approximately \( 3 \times 10^8 \text{ m/s} \).

Step 2: Insert the Known Values

Substitute the given frequency \( f = 1.55 \times 10^{10} \text{ s}^{-1} \) and the speed of light \( c = 3 \times 10^8 \text{ m/s} \) into the formula:\[\lambda = \frac{3 \times 10^8 \text{ m/s}}{1.55 \times 10^{10} \text{ s}^{-1}}\]

Step 3: Solve for Wavelength in Meters

Perform the division to find the wavelength in meters:\[\lambda = \frac{3 \times 10^8}{1.55 \times 10^{10}} = 0.01935 \text{ m}\]

Step 4: Convert Wavelength to Millimeters

Since 1 meter equals 1000 millimeters, convert the wavelength from meters to millimeters:\[\lambda = 0.01935 \text{ m} \times 1000 \text{ mm/m} = 19.35 \text{ mm}\]

Key Concepts

Speed of Light

The speed of light, symbolized by the letter 'c', is a fundamental constant in physics. It is the speed at which light travels in a vacuum, which is approximately \(3 \times 10^8 \text{ m/s}\). This means that in one second, light can travel about 300 million meters. This constant is crucial because it establishes the relationship between the wavelength and frequency of light. Understanding this concept allows us to calculate other properties of light once we know either its frequency or wavelength.
This constant is not just vital in the study of light but also in the broader realm of physics, as it plays a pivotal role in Einstein's Theory of Relativity. It underpins our comprehension of the universe and forms the baseline for the speed limit of the universe itself.

Conversion of Units

Unit conversion is an essential mathematical operation, especially when dealing with scientific formulas. In the context of wavelength calculation, it becomes necessary to convert units, such as from meters to millimeters. This often occurs because not all systems of measurement are directly compatible with how we wish to express our results or compare them to other values.
Understandably, when converting from meters to millimeters, we multiply by 1000 because there are 1000 millimeters in a meter. This conversion ensures that our answer is in a more useful or desired format for reporting or comparison.

• Remember that unit consistency is critical in calculations to prevent errors.
• Keep a handy conversion table or tool when working with different units of measure frequently.

By getting accustomed to unit conversion, you'll find it easier to handle many types of calculations across various scientific fields.

Frequency of Light

Frequency, represented by the letter 'f', refers to how often the waves of light pass a certain point in a given time frame, usually one second. It is measured in hertz (Hz), which is equivalent to \(\text{s}^{-1}\). In our original exercise, the frequency was given as \(1.55 \times 10^{10} \text{ s}^{-1}\), illustrating that this is the number of cycles occurring per second.
This parameter helps distinguish different types of light or electromagnetic waves. For instance, light with a higher frequency appears more towards the blue end of the visible spectrum, while light with a lower frequency appears more to the red.

• Frequency and wavelength are inversely related—when one increases, the other decreases.
• This relationship is key in fields like telecommunications, where different frequencies can carry varying types of signals.

By mastering the concept of frequency, you can better predict and understand how light and other waves behave in different environments.