Problem 6 - Chemistry: The Central Science

Question

The familiar phenomenon of a rainbow results from the diffraction of sunlight through raindrops. (a) Does the wavelength of light increase or decrease as we proceed outward from the innermost band of the rainbow? (b) Does the frequency of light increase or decrease as we proceed outward? (c) Suppose that instead of sunlight, the visible light from a hydrogen discharge tube (Figure 6.10) was used as the light source. What do you think the resulting "hydrogen discharge rainbow" would look like? [Section 6.3]

Step-by-Step Solution

Answer

(a) The wavelength of light increases as we proceed outward from the innermost band of the rainbow. (b) The frequency of light decreases as we proceed outward from the innermost band of the rainbow. (c) The hydrogen discharge rainbow would display discrete spectral lines at the specific wavelengths emitted by the hydrogen discharge tube, instead of a smooth transition of colors found in a traditional rainbow.

Step 1: Understanding the rainbow formation

A rainbow is formed when sunlight passes through raindrops in the atmosphere. The sunlight is both refracted (bent) and reflected (bounced off surfaces) within the raindrops, and this separates the light into its constituent colors, which form a circular arc. The separation of light into different colors is due to the fact that different wavelengths of light are refracted by different amounts. Shorter wavelengths (blue and violet) will be refracted more than longer wavelengths (red and orange).

Step 2: Wavelength changes from inner to outer bands

As we proceed outward from the innermost band of the rainbow, we see the colors changing from violet to blue, green, yellow, orange, and finally red. This change in color corresponds to a change in the wavelength of light. Since the red light has a longer wavelength than the violet light, the wavelength of light increases as we proceed outward from the innermost band of the rainbow. Answer (a): The wavelength of light increases as we proceed outward from the innermost band of the rainbow.

Step 3: Frequency changes from inner to outer bands

Now that we know that the wavelength increases as we proceed outward from the innermost band of the rainbow, we can determine the change in frequency. Recall that the speed of light (c) is equal to the product of its wavelength (λ) and frequency (f): \(c = λf\) Considering that the speed of light is constant, if the wavelength increases, the frequency must decrease to maintain this relationship. Therefore, as we proceed outward from the innermost band of the rainbow, the frequency of light decreases. Answer (b): The frequency of light decreases as we proceed outward from the innermost band of the rainbow.

Step 4: Hydrogen discharge rainbow

Instead of sunlight, if we used the visible light from a hydrogen discharge tube as the light source, the resulting "hydrogen discharge rainbow" would have a different appearance. A hydrogen discharge tube emits light only at specific wavelengths, which correspond to the allowed energy transitions of the hydrogen atom. These wavelengths would be fewer and more distinct compared to the continuous spectrum of sunlight. So, instead of a smooth transition of colors like the one seen in traditional rainbows, the hydrogen discharge rainbow would display very specific, sharp, and bright spectral lines at the discrete wavelengths emitted by the hydrogen discharge tube. Answer (c): The hydrogen discharge rainbow would display discrete spectral lines at the specific wavelengths emitted by the hydrogen discharge tube, instead of a smooth transition of colors found in a traditional rainbow.

Key Concepts

Wavelength

Wavelength is a fundamental concept in understanding light and its behavior. Imagine light as a wave, similar to ocean waves. The wavelength is the distance between consecutive peaks or crests of this wave.

Light waves have different wavelengths, which determines their color. In a rainbow, red light has the longest wavelength, while violet has the shortest.

• Red: Longest wavelength
• Violet: Shortest wavelength

When light passes through a raindrop, it separates into different colors because each color bends by a different amount. This is called refraction. Shorter wavelengths bend more, and longer wavelengths bend less. As you move outward in a rainbow, the wavelength increases, causing a change from violet to red.

Frequency

Frequency is directly related to how often the waves pass a certain point in a second. While wavelength measures the distance between waves, frequency measures how many crests pass a point in a given time.

The important thing to remember is the relationship between wavelength and frequency, which is tied to the speed of light, denoted as "c." The equation is:

\[ c = \lambda f \]

• \(c\) is the speed of light
• \(\lambda\) is the wavelength
• \(f\) is the frequency

As you move out from the innermost part of the rainbow, the wavelength increases, which means the frequency must decrease to keep the speed of light constant. Longer wavelengths correspond to lower frequencies which are why the outer bands of a rainbow, like red, have lower frequencies than the inner bands, like violet.

Hydrogen Discharge Tube

A hydrogen discharge tube might look like a regular tube, but what happens inside is fascinating. It's used in laboratories to study the light emitted by hydrogen gas.

When you apply a high voltage to the hydrogen gas in the tube, the electrons get excited to higher energy levels. As these electrons fall back to lower energy levels, they emit light at specific wavelengths, unique to hydrogen.

This produces what's called a "line spectrum" because the emitted light consists of distinct lines rather than a continuous range of colors. These lines are specific to hydrogen and can tell scientists a lot about its properties.

Spectral Lines

Spectral lines are the distinct lines of color, or specific wavelengths, seen in the spectrum of light emitted or absorbed by an element, like hydrogen.

Each element has a unique pattern of spectral lines, much like a fingerprint. These lines occur because electrons in the atom jump between energy levels.

• When an electron absorbs energy, it moves to a higher energy level, showing up as an absorption line.
• When it falls back, it emits energy and creates an emission line.

For hydrogen, these spectral lines in the visible range are called the Balmer series. This phenomenon helps us determine the composition of stars and other celestial bodies, as each element has its distinct spectral lines.