Problem 83 - Chemistry The Central Science In Si Units

Question

Certain elements emit light of a specific wavelength when they are burned or heated in a non-luminous flame. Historically, chemists used such emission wavelengths to determine whether specific elements were present in a sample. Some characteristic wavelengths for a few of the elements are given in the following table: $$\begin{array}{llll} \hline \mathrm{Ag} & 328.1 \mathrm{nm} & \mathrm{Fe} & 372.0 \mathrm{nm} \\ \mathrm{Au} & 267.6 \mathrm{nm} & \mathrm{K} & 404.7 \mathrm{nm} \\ \mathrm{Ba} & 455.4 \mathrm{nm} & \mathrm{Mg} & 285.2 \mathrm{nm} \\ \mathrm{Ca} & 422.7 \mathrm{nm} & \mathrm{Na} & 589.6 \mathrm{nm} \\ \mathrm{Cu} & 324.8 \mathrm{nm} & \mathrm{Ni} & 341.5 \mathrm{nm} \\ \hline \end{array}$$ (a) Determine which of these emissions occur in the ultraviolet part of the spectrum. (b) Which emission has the highest frequency and which one has the lowest frequency? (c) When burned, a sample of an unknown substance is found to emit light of frequency \(6.58 \times 10^{14} \mathrm{~s}^{-1} .\) Which of these elements is probably in the sample?

Step-by-Step Solution

Answer

(a) UV emissions: Ag, Au, Fe, Mg, Ni. (b) Highest frequency: Au; lowest: Na. (c) Element: Ba.

Step 1: Identify Ultraviolet Wavelengths

The ultraviolet (UV) part of the electromagnetic spectrum ranges from approximately 10 nm to 400 nm. We need to identify which of the given wavelengths are less than 400 nm to be considered in the UV range. From the table: \(\mathrm{Ag} (328.1 \mathrm{nm})\), \(\mathrm{Au} (267.6 \mathrm{nm})\), \(\mathrm{Fe} (372.0 \mathrm{nm})\), \(\mathrm{K} (404.7 \mathrm{nm})\), \(\mathrm{Mg} (285.2 \mathrm{nm})\), and \(\mathrm{Ni} (341.5 \mathrm{nm})\) are all investigated. Excluding \(\mathrm{K} (404.7 \mathrm{nm})\) as it is above 400 nm, the rest occur in the UV range.

Step 2: Determine Highest and Lowest Frequency

Frequency \(u\) and wavelength \(\lambda\) are related by the formula \(u = \frac{c}{\lambda}\), where \(c\) is the speed of light, \(\approx 3.00 \times 10^{8} \mathrm{~m/s}\). The highest frequency corresponds to the shortest wavelength, and the lowest frequency corresponds to the longest wavelength. From the table, \(\mathrm{Au} (267.6 \mathrm{nm})\) has the shortest wavelength, and \(\mathrm{Na} (589.6 \mathrm{nm})\) has the longest, hence the highest and lowest frequencies respectively.

Step 3: Find Element from Emission Frequency

To find which element corresponds to the given frequency \(6.58 \times 10^{14} \mathrm{~s}^{-1}\), convert each given wavelength to frequency using \(u = \frac{c}{\lambda} \). Matching the calculated frequency to \(6.58 \times 10^{14} \mathrm{~s}^{-1}\), the corresponding wavelength can be calculated as \( \lambda = \frac{c}{u} = \frac{3.00 \times 10^8 \mathrm{~m/s}}{6.58 \times 10^{14} \mathrm{~s}^{-1}} = 455.3 \mathrm{nm}\), which is very close to \(\mathrm{Ba} (455.4 \mathrm{nm})\). Hence, Ba is likely in the sample.

Key Concepts

Ultraviolet Spectrum

The ultraviolet (UV) spectrum is a portion of the electromagnetic spectrum that spans wavelengths from approximately 10 nm to 400 nm. This range includes shorter wavelengths than visible light, making UV light invisible to the human eye. Nevertheless, many chemical elements exhibit characteristic emission lines in this spectral region.
When elements are excited, either through thermal energy as in a flame or electrical energy in a discharge tube, they emit light at specific wavelengths. These wavelengths can be used to identify the presence of certain elements within a sample. Ultraviolet emission spectroscopy has thus become a valuable tool in elemental analysis due to its accuracy in detecting and distinguishing between different elements.
In the problem, you'll notice certain elements like Silver (Ag), Gold (Au), Iron (Fe), Magnesium (Mg), and Nickel (Ni) have emissions within this UV wavelength range. This information is crucial for determining the presence of these elements in a sample when analyzed through UV emission spectroscopy.

Frequency and Wavelength Relationship

Understanding the relationship between frequency and wavelength is key to interpreting spectra. The frequency (\(u\)) and wavelength (\(\lambda\)) of light are inversely related through the equation: \[u = \frac{c}{\lambda}\] where \(c\) is the speed of light, approximately \(3.00 \times 10^8\) m/s.
- Shorter wavelengths correspond to higher frequencies.- Longer wavelengths correspond to lower frequencies.
This relationship is fundamental when analyzing emission spectra. By measuring the wavelength of light emitted from a sample, you can calculate its frequency, which is crucial in identifying elements. For instance, the element with the shortest wavelength \((267.6\text{ nm})\) in the exercise is Gold (Au), which consequently, due to the inverse relationship, has the highest frequency. Conversely, Sodium (Na) with the longest wavelength \((589.6\text{ nm})\) has the lowest frequency.
Being able to convert between wavelength and frequency enables students and scientists alike to connect observed spectral data to elemental composition efficiently.

Elemental Analysis

Elemental analysis using emission spectroscopy is a precise method for determining what elements are present in a material. When a substance is heated or electrically excited, its atoms emit light at specific and unique wavelengths corresponding to the element’s electron transitions.
This emitted light can be captured and analyzed, breaking it down into its specific wavelength components using a spectroscope. By comparing observed wavelengths to known emission lines of elements, you can determine which elements are present in the sample.
For example, in the exercise, the unknown sample emitted light at a frequency of \(6.58 \times 10^{14} \text{ s}^{-1}\). By converting this frequency into a wavelength, it was matched with the known emission wavelength of Barium (Ba) at \(455.4\text{ nm}\), suggesting that Barium is present in the sample. This method of elemental analysis is both cost-effective and efficient, making it widely used in environmental monitoring, forensic analysis, and quality control processes.

Electromagnetic Spectrum

The electromagnetic spectrum is the entire range of electromagnetic radiation, from the longest wavelengths like radio waves to the shortest, such as gamma rays. It encompasses various types of light, including:

• Radio waves
• Microwaves
• Infrared
• Visible light
• Ultraviolet (UV)
• X-rays
• Gamma rays

Each type has different properties and applications across science and technology.
In the context of emission spectroscopy, we often focus on the visible and UV parts of the spectrum. These are the regions where most elements emit light that can be detected using basic laboratory equipment. While visible light is directly perceivable by our eyes, ultraviolet light, with its shorter wavelengths, offers insights invisible to humans yet crucial in scientific measurements.
The electromagnetic spectrum is foundational to fields like spectroscopy, telecommunications, and medical imaging, demonstrating the wide applicability and importance of understanding electromagnetic radiation's various forms and effects.