Problem 105 - Chemistry Central Science

Question

Moseley established the concept of atomic number by studying X-rays emitted by the elements. The X-rays emitted by some of the elements have the following wavelengths: \begin{tabular}{ll} \hline Element & Wavelength (A) \\ \hline \(\mathrm{Ne}\) & \(14.610\) \\ \(\mathrm{Ca}\) & \(3.358\) \\ \(\mathrm{Zn}\) & \(1.435\) \\ \(\mathrm{Zr}\) & \(0.786\) \\ \(\mathrm{Sn}\) & \(0.491\) \\ \hline \end{tabular} (a) Calculate the frequency, \(\nu\), of the \(X\) -rays emitted by each of the elements, in Hz. (b) Using graph paper (or

Step-by-Step Solution

Answer

The frequencies of the X-rays emitted by the given elements are approximately: Ne: \(2.05 \times 10^{18}\) Hz Ca: \(8.93 \times 10^{18}\) Hz Zn: \(2.09 \times 10^{19}\) Hz Zr: \(3.82 \times 10^{19}\) Hz Sn: \(6.11 \times 10^{19}\) Hz To plot the graph, use the atomic numbers of the elements as the x-axis and the calculated frequencies as the y-axis.

Step 1: Convert Wavelengths from Angstroms (A) to Meters (m)

To convert the given wavelengths in Angstroms to meters, use the conversion factor: 1 Angstrom = \(1 \times 10^{-10}\) meters. The converted wavelength for each element is: Ne: \(14.610 \times 10^{-10}\) m Ca: \(3.358 \times 10^{-10}\) m Zn: \(1.435 \times 10^{-10}\) m Zr: \(0.786 \times 10^{-10}\) m Sn: \(0.491 \times 10^{-10}\) m

Step 2: Calculate Frequency using Speed of Light Formula

The speed of light, \(c\), is given by the formula: \(c = \nu \lambda\), where \(\nu\) is the frequency, \(\lambda\) is the wavelength, and \(c\) is the speed of light (\(3 \times 10^8\) m/s). To find the frequency for each element, rearrange the formula for frequency and substitute the converted wavelength values: \(\nu = \frac{c}{\lambda}\) Ne: \(\nu = \frac{3 \times 10^8}{14.610 \times 10^{-10}}\) Hz Ca: \(\nu = \frac{3 \times 10^8}{3.358 \times 10^{-10}}\) Hz Zn: \(\nu = \frac{3 \times 10^8}{1.435 \times 10^{-10}}\) Hz Zr: \(\nu = \frac{3 \times 10^8}{0.786 \times 10^{-10}}\) Hz Sn: \(\nu = \frac{3 \times 10^8}{0.491 \times 10^{-10}}\) Hz Calculate the frequency for each element: Ne: \(\nu \approx 2.05 \times 10^{18}\) Hz Ca: \(\nu \approx 8.93 \times 10^{18}\) Hz Zn: \(\nu \approx 2.09 \times 10^{19}\) Hz Zr: \(\nu \approx 3.82 \times 10^{19}\) Hz Sn: \(\nu \approx 6.11 \times 10^{19}\) Hz

Step 3: Understanding Plotting the Graph

Since we cannot draw on this platform, we will help you understand the process of plotting the graph. On the graph paper, the x-axis will represent the elements' atomic number and the y-axis will represent the frequency of the X-rays. The atomic numbers for the elements are: Ne (10), Ca (20), Zn (30), Zr (40), Sn (50). Plot the points (10, \(2.05 \times 10^{18}\)), (20, \(8.93 \times 10^{18}\)), (30, \(2.09 \times 10^{19}\)), (40, \(3.82 \times 10^{19}\)), and (50, \(6.11 \times 10^{19}\)) on the graph paper. Then draw a line that best fits these points. This graph would visually represent the relationship between the atomic number and the frequency of the X-rays emitted by each element.

Key Concepts

X-ray Wavelength Calculation

Understanding how to calculate the wavelength of X-rays is vital for interpreting data in fields such as chemistry and physics. To begin, X-rays are a form of electromagnetic radiation with wavelengths typically ranging from 0.01 to 10 nanometers. While working with X-ray data, you'll often find the wavelength provided in units called Angstroms (\r{A}), where 1 \r{A} equals to \(1 \times 10^{-10}\) meters.

When converting from Angstroms to meters, it's as simple as applying the conversion factor. For example, if we have an X-ray wavelength of \(14.610\) \r{A}, simply multiply by \(1 \times 10^{-10}\) to get the metric equivalent: \(14.610 \times 10^{-10}\) meters. This first step is crucial because it transitions the information into a standardized unit, making further calculations, such as finding frequency, more straightforward.

Speed of Light Formula

The speed of light is a fundamental constant in physics, represented by the symbol \(c\) and approximately valued at \(3 \times 10^8\) meters per second. This velocity represents the fastest speed at which energy, matter, or information can travel in a vacuum. It plays a pivotal role in numerous formulas and equations, especially in the realm of electromagnetism.

For example, the relationship between frequency (\(u\)), wavelength (\(\r{A}\)), and the speed of light (\(c\)) is expressed by the equation \(c = u \times \r{A}\). To find the frequency of an electromagnetic wave, you can rearrange the formula to \(u = \frac{c}{\r{A}}\). This inverse relationship between frequency and wavelength demonstrates that as the wavelength of light decreases, its frequency increases and vice versa.

Frequency and Wavelength Relationship

Traffic lights through their colors showcase the relationship between frequency and wavelength. When we move from red to violet, we actually observe a decrease in wavelength and an increase in frequency, much like the spectrum of X-rays or any other electromagnetic radiation. The relationship is captured by the equation \(u = \frac{c}{\r{A}}\), indicating that frequency (\(u\)) is inversely proportional to wavelength (\(\r{A}\)), with a constant speed of light (\(c\)).

This mathematical connection is crucial when it comes to X-rays. Given the speed of light, one can calculate the frequency by reorganizing this relationship, which provides the key to unlocking various atomic properties. Understanding this equation allows for the determination of unknown variables provided the others are known, facilitating deeper insights into the nature of electromagnetic radiation.

Moseley's Law

Discovering the atomic number's significance was credited to Henry Moseley through his groundbreaking work known as Moseley's Law. This scientific law established a linear relationship between the square root of the frequency of the X-rays emitted by an element and its atomic number. The equation for Moseley's Law is represented as \(u = K(Z - \beta)^2\), where \(u\) is the frequency, \(K\) is a proportionality constant, \(Z\) is the atomic number, and \(\beta\) is the screening constant.

This relationship enables us to predict the frequency of the X-rays an element can emit based upon its position in the periodic table. Moseley's work not only provided a method to categorize elements but also reinforced the concept of atomic number as a fundamental property, which was a significant leap forward in the field of atomic physics. Additionally, Moseley's Law aids in the identification of elements and the understanding of their electronic structure.