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: $$ \begin{array}{lc} \hline \text { Element } & \text { Wavelength } \\ \hline \mathrm{Ne} & 14.610 \\ \mathrm{Ca} & 3.358 \\ \mathrm{Zn} & 1.435 \\ \mathrm{Zr} & 0.786 \\ \mathrm{Sn} & 0.491 \\ \hline \end{array} $$ (a) Calculate the frequency, \(\nu\), of the \(\mathrm{X}\) rays emitted by each of the elements, in Hz. (b) Plot the square root of \(\nu\) versus the atomic number of the element. What do you observe about the plot? (c) Explain how the plot in part (b) allowed Moseley to predict the existence of undiscovered elements. (d) Use the result from part (b) to predict the \(\mathrm{X}\)-ray wavelength emitted by iron. (e) A particular element emits X rays with a wavelength of \(0.980 \AA\). What element do you think it is?

Short answer

The relationship between the square root of frequency and the atomic number of the elements is linear. Moseley used this linear relationship to predict the existence of undiscovered elements by identifying gaps in the sequence. By analyzing the given data, we can predict the X-ray wavelength emitted by iron and identify an element based on its X-ray wavelength.

Step-by-step solution

Calculate the frequency of X-rays

First, we need to calculate the frequency of X-rays emitted by each element. The formula to convert wavelength to frequency is given by: \(ν = \dfrac{c}{λ}\), where \(ν\) is the frequency, \(c\) is the speed of light (\(3.0 \times 10^8 m/s\)), and \(λ\) is the wavelength of the X-ray. Now, for each element, we can calculate the frequency using the given wavelength.

Calculate the square root of frequency

After finding the frequency for each element, we have to calculate the square root of these frequencies. Using a calculator, find the square root of each calculated frequency.

Plot the relationship

Create a graph, with the atomic number of the elements on the x-axis and the square root of the frequency on the y-axis. Plot the values of the square root of frequency for each element and analyze the relationship.

Analyze the relationship

Observe the plotted graph to analyze the relationship between the square root of frequency and the atomic number. You will notice that as the atomic number increases, the square root of frequency also increases.

Explain the plot

The plot in part (b) shows a linear relationship between the square root of frequency and the atomic number of the elements. Moseley was able to predict the existence of undiscovered elements because if there were a missing element in the sequence, there would be a gap in the plot. By analyzing the plot, Moseley was able to predict where undiscovered elements would fit in the sequence.

Predict an X-ray wavelength

In part (d), we are asked to predict the X-ray wavelength emitted by iron. First, find the atomic number of iron (which is 26). Now, use the relationship between the square root of frequency and the atomic number derived from the plot to predict the square root of the frequency for iron. Next, to find the frequency of iron, square the value obtained for the square root of frequency. Finally, use the formula \(λ = \dfrac{c}{ν}\) to calculate the wavelength of X-ray emission for iron.

Identify the element

In part (e), we are given the X-ray wavelength for a particular element (0.980 Angstroms). First, convert the wavelength to meters. Then, using the formula \(ν = \dfrac{c}{λ}\), calculate the frequency for the given wavelength. Next, find the square root of the frequency and use the relationship between the square root of frequency and the atomic number found in the plot to predict the atomic number of the element. Lastly, identify the element corresponding to the predicted atomic number.

Key concepts

Atomic Number

The atomic number is a fundamental concept in chemistry and physics, representing the number of protons found in the nucleus of an atom. It is a unique identifier for elements, differentiating one element from another. Moseley, through his pioneering work on X-ray spectroscopy, established the theoretical and empirical significance of the atomic number.

• Each element has a unique atomic number, denoting the number of protons and thus positively charged units in its core.
• This number also directly influences the elemental properties, solidsation state, and chemical behavior.
• Moseley's Law demonstrated that the square root of the frequency of X-ray emissions from an element is directly proportional to its atomic number.

Thus, the atomic number has become crucial in arranging elements in the periodic table. Before Moseley’s work, elements were primarily ordered by atomic weight, which led to inconsistencies.

X-ray Spectroscopy

X-ray spectroscopy is a technique used to study the properties of elements through the analysis of their X-ray emissions. This method played a vital role in Moseley’s discoveries.

• When elements are hit with high energy, they emit X-rays at characteristic wavelengths.
• X-ray spectroscopy measures these wavelengths to determine an element’s specific structure and characteristics.
• Moseley noted that the emitted X-ray frequency from each element varied systematically with the atomic number.

This technique provided a new way to study the inner structure of atoms, leading to more precise and accurate predictions of new elements. Moseley's findings also further validated the utility of the atomic number beyond ordering elements.

Frequency Calculation

Frequency calculation is essential for understanding X-ray emissions according to Moseley's Law. This calculation involves converting the measured wavelength of X-rays into frequency, using the well-known equation:
\[ν = \dfrac{c}{λ}\]where:

• \(ν\) = frequency of the X-rays
  
• \(c\) = speed of light, approximately \(3.0 \times 10^8\) meters per second
  
• \(λ\) = wavelength of the emitted X-ray
  

This calculation is a crucial step as understanding the frequency helps in visualizing the relationship seen in Moseley’s plot of square root frequency versus atomic number. This mathematical process assists in determining elemental properties and inferences about undiscovered elements.

Undiscovered Elements Prediction

Moseley’s Law significantly contributed to the prediction of undiscovered elements. By using the relationship between X-ray frequency and atomic number, gaps in the periodic table were identified.

• Plotting the atomic number against the square root of frequency showed a linear relationship.
• This clear and consistent relationship suggested the presence of known anomalies, predicting where elements should exist yet had not been discovered.
• If there was a missing or unexpected gap, Moseley could hypothesize that an unknown element filled this space.

Moseley’s work allowed chemists and physicists to refine the periodic table, driving the search for new elements, and enhancing our understanding of atomic structure.