Uncovering the Shortest Wavelength of Be3+ Lines

[MrDMD83]

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

none

 

[chaoseverlasting]

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

$$\frac{1}{\lambda}=Rz^2(\frac{1}{n_1^2}-\frac{1}{n_2^2})$$

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

 

[m2003]

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.