Problem 90 - Chemistry The Molecular Nature of Matter

Question

Use the Rydberg equation to calculate the wavelength in nanometers of the spectral line of hydrogen for which \(n_{2}=5\) and \(n_{1}=2\). (Report your answer using three significant figures.) In what region of the electromagnetic spectrum would this line be found? Would we be expected to see the light corresponding to this spectral line? Explain your answer.

Step-by-Step Solution

Answer

The wavelength is approximately \(434.0 nm\) and the spectral line would be visible in the violet region of the electromagnetic spectrum.

Step 1: Understanding the Rydberg Equation

The Rydberg equation for a hydrogen atom is given by the formula \(\frac{1}{\lambda} = R_{H} \left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)\), where \(\lambda\) is the wavelength, \(R_{H}\) is the Rydberg constant \((1.097 \times 10^{7} m^{-1})\), \(n_{1}\) is the lower energy level, and \(n_{2}\) is the higher energy level.

Step 2: Substituting Known Values

Substitute the given values \(n_{2}=5\) and \(n_{1}=2\) and the Rydberg constant into the Rydberg equation: \(\frac{1}{\lambda} = 1.097 \times 10^{7} \left(\frac{1}{2^{2}}-\frac{1}{5^{2}}\right)\).

Step 3: Calculating Wavelength

Calculate the inverse wavelength \(\frac{1}{\lambda}\) by simplifying inside the parenthesis: \(\frac{1}{\lambda} = 1.097 \times 10^{7} \left(\frac{1}{4}-\frac{1}{25}\right)\). This simplifies to \(\frac{1}{\lambda} = 1.097 \times 10^{7} \left(\frac{21}{100}\right)\), which then simplifies to \(\frac{1}{\lambda} = 1.097 \times 10^{7} \times 0.21\). Thus \(\frac{1}{\lambda} = 2.3037 \times 10^{6} m^{-1}\). To find the wavelength \(\lambda\), take the reciprocal of this number.

Step 4: Converting to Nanometers

The reciprocal of the inverse wavelength gives the wavelength in meters. To convert this to nanometers (nm), use the conversion \(1 m = 10^{9} nm\). Therefore, the wavelength in nanometers is \(\lambda = \frac{1}{2.3037 \times 10^{6} m^{-1}} \times 10^{9} nm\), which is approximately \(434.0 nm\) when rounded to three significant figures.

Step 5: Identifying the Region of the Electromagnetic Spectrum

The calculated wavelength of \(434.0 nm\) falls within the visible range of the electromagnetic spectrum, which is approximately from \(380 nm\) to \(750 nm\). Therefore, the spectral line would be in the visible region.

Step 6: Visibility of the Spectral Line

Since the wavelength \(434.0 nm\) is within the visible light range, we would be expected to see the light corresponding to this spectral line with the naked eye, appearing as a violet hue.

Key Concepts

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all wavelengths of electromagnetic radiation, ranging from extremely short gamma rays to incredibly long radio waves. It includes, in order of increasing wavelength, gamma rays, X-rays, ultraviolet light, visible light, infrared radiation, microwaves, and radio waves. Each type of radiation in the spectrum has unique properties and interacts with matter in distinct ways.

Visible light, which human eyes can detect, is a small portion of this spectrum, situated between ultraviolet light and infrared radiation. When scientists examine the hydrogen spectrum, they often use the electromagnetic spectrum to identify the region in which a specific wavelength falls. For example, in the exercise, by using the Rydberg equation, a wavelength within the visible light range was identified, indicating that it is a wavelength we can observe without any specialized equipment.

Spectral Lines

Spectral lines are unique patterns of light that are emitted or absorbed by substances, such as hydrogen, when the electrons transition between energy levels. These lines serve as fingerprints for elements, allowing scientists to identify their presence in distant stars or other celestial objects. The patterns result from quantized energy levels within an atom, meaning electrons can only exist at specific energy levels and the energy difference between these levels corresponds to a photon of a specific wavelength.

When an electron makes a transition from a higher energy level to a lower one, it releases energy in the form of light, and this light has a wavelength that can be calculated using the Rydberg formula. The resulting spectral line can be mapped onto the electromagnetic spectrum to understand its characteristics and detectability.

Wavelength Calculation

Wavelength calculation is a fundamental aspect of understanding the characteristics of light emitted or absorbed by an atom. The Rydberg equation provides a method to calculate the specific wavelength of light involved in an electronic transition of a hydrogen atom. By inputting the initial and final energy levels of the electron's transition into the Rydberg formula, the inverse of the wavelength can be determined.

The actual wavelength can then be obtained by taking the reciprocal of this value. In this exercise, the detailed steps involved calculating an inverse wavelength and then converting it to the actual wavelength in meters, which was further converted to nanometers (nm) for practical use. The process requires attention to detail and proper handling of units to ensure accurate results.

Visible Light

Visible light is the portion of the electromagnetic spectrum that is detectable by the human eye. It ranges approximately from 380 nm to 750 nm, and each wavelength within this range corresponds to a different color, from violet at the shortest wavelengths to red at the longest. This feature of light is why we see the world in a rainbow of colors.

In the exercise, the calculated wavelength of the spectral line for hydrogen is 434.0 nm, which falls within the range of visible light. This tells us that the spectral line produces a color visible to humans, specifically within the blue-violet range, which would appear as a violet hue to the naked eye. Understanding visible light and its relation to wavelength helps explain why certain emissions from atoms are noticeable without the need for instrumentation and others are not.