Uncovering the Shortest Wavelength of Be3+ Lines

In summary, the shortest wavelength for the series of lines in the triply ionized beryllium spectrum is calculated using the equation: 1/λ = Rz^2(1/n1^2 - 1/n2^2), where R is the rydberg constant, z is the atomic number, and n1 is the final shell it jumps to from n2.
  • #1
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Homework Statement



Doubly ionized lithium Li2+ (Z = 3) and triply ionized beryllium Be3+ (Z = 4) each emit a line spectrum. For a certain series of lines in the lithium spectrum, the shortest wavelength is 1979.8 nm. For the same series of lines in the beryllium spectrum, what is the shortest wavelength?

Homework Equations





The Attempt at a Solution



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  • #2
Hehe... give it a shot man. Tell you what, the equation you need is:

[tex]\frac{1}{\lambda}=Rz^2(\frac{1}{n_1^2}-\frac{1}{n_2^2})[/tex]

where R is the rydberg constant, z is the atomic number, and n2 is the shell it jumps from to n1 (the final shell)
 
  • #3


I would approach this problem by first understanding the concept of atomic spectra and the significance of different energy levels in atoms. I would then use the Rydberg formula, which relates the wavelengths of spectral lines to the energy levels of an atom, to calculate the shortest wavelength for Be3+. The Rydberg formula is given by:

1/λ = R(Z^2/n^2), where λ is the wavelength, R is the Rydberg constant, Z is the atomic number, and n is the energy level.

Since we are dealing with Be3+, which has an atomic number of 4, we can substitute these values into the formula and solve for λ.

1/λ = R(4^2/n^2)

Since we are looking for the shortest wavelength, we can assume that n= ∞, which represents the highest energy level. This simplifies the equation to:

1/λ = R(4^2/∞^2)

Solving for λ, we get:

λ = 1/R(4^2/∞^2)

Substituting in the value for the Rydberg constant (1.097x10^-7 m^-1), we get:

λ = 1/(1.097x10^-7)(4^2/∞^2)

λ = 9.91x10^-8 m

Therefore, the shortest wavelength for the Be3+ spectrum is approximately 99.1 nm. This is significantly shorter than the shortest wavelength for Li2+ (1979.8 nm), which makes sense since Be3+ has a higher atomic number and therefore a greater number of energy levels. I would also note that this value is an approximation since we assumed n=∞, but it gives us a good estimate of the shortest wavelength for the given series of lines in the Be3+ spectrum.
 

1. What is the significance of uncovering the shortest wavelength of Be3+ lines?

The shortest wavelength of Be3+ lines is important because it can provide valuable information about the energy levels and electronic structure of beryllium ions. This can help scientists better understand the properties and behavior of these ions in different environments.

2. How do scientists go about uncovering the shortest wavelength of Be3+ lines?

Scientists use a variety of techniques, such as spectroscopy, to study the emission of light from beryllium ions. By analyzing the wavelengths of light emitted, they can determine the energy levels and transitions of the ions, including the shortest wavelength of Be3+ lines.

3. What applications can benefit from knowing the shortest wavelength of Be3+ lines?

The knowledge of the shortest wavelength of Be3+ lines can have various applications in fields such as astrophysics, plasma physics, and materials science. It can help in the analysis and interpretation of astronomical data, understanding the behavior of plasma in fusion experiments, and designing new materials with specific electronic properties.

4. Is the shortest wavelength of Be3+ lines a constant value?

No, the shortest wavelength of Be3+ lines can vary depending on the specific conditions of the beryllium ions, such as temperature and pressure. It can also be affected by the presence of other elements or molecules in the surrounding environment.

5. What other factors can impact the shortest wavelength of Be3+ lines?

In addition to external conditions, the electronic structure and energy levels of beryllium ions can also affect the shortest wavelength of Be3+ lines. Changes in the ionization state or the number of electrons can alter the energy levels and thus the wavelengths of light emitted by the ions.

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