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Chapter 8: Problem 97

Use appropriate relationships from the chapter to determine the wavelength of the line in the emission spectrum of \(\mathrm{He}^{+}\) produced by an electron transition from \(n=5\) to \(n=2\).

Short Answer

Expert verified
The wavelength of the emission line produced by the electron transition from n=5 to n=2 in the helium ion (\(\mathrm{He}^{+}\)) is as calculated in the steps above, in nanometers.

Step by step solution

01

Identify Variables in the Balmer-Rydberg Equation

The Balmer-Rydberg equation relates the wavelength of light emitted by an electron undergoing a transition to a lower energy level to the principal quantum numbers of the initial and final states. The equation is given by \[ \frac{1}{\lambda} = Z^2R \left( \frac{1}{n_{1}^2} - \frac{1}{n_{2}^2} \right) \] where for \(\mathrm{He}^{+}\), Z=2. Here \(n_{1}\) is the lower energy level (n=2 in this case), \(n_{2}\) is the higher energy level (n=5 in this case), \(\lambda\) is the wavelength we want to find and R is the Rydberg constant (R = 1.097 \times 10^7 m^{-1}).
02

Calculation using the Balmer-Rydberg Equation

Substitute the known values into the equation: \[ \frac{1}{\lambda} = 2^2 \times 1.097 \times 10^7 m^{-1} \left( \frac{1}{2^2} - \frac{1}{5^2} \right) \] Solve this equation for \(\frac{1}{\lambda}\) to find the reciprocal of the wavelength.
03

Convert \(\frac{1}{\lambda}\) to \(\lambda\)

To compute for the wavelength \(\lambda\), take the reciprocal of the answer from Step 2. This will give the wavelength in meters.
04

Converting Wavelength to Nanometers

Generally, wavelengths of light are represented in nanometers (nm) for convenience, since they are usually very small. Thus, convert the wavelength from meters to nanometers by multiplying by \(10^9\). This will give the final answer in nanometers (nm).

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Most popular questions from this chapter

An atom in which just one of the outer-shell electrons is excited to a very high quantum level \(n\) is called a "high Rydberg" atom. In some ways, all these atoms resemble a Bohr hydrogen atom with its electron in a high-numbered orbit. Explain why you might expect this to be the case.

Use the basic rules for electron configurations to indicate the number of (a) unpaired electrons in an atom of \(\mathrm{P} ;\) (b) \(3 d\) electrons in an atom of \(\mathrm{Br} ;\) (c) \(4 p\) electrons in an atom of \(\mathrm{Ge} ;\) (d) \(6 \mathrm{s}\) electrons in an atom of \(\mathrm{Ba}\) (e) \(4 f\) electrons in an atom of Au.

How long does it take light from the sun, 93 million miles away, to reach Earth?

Based on the relationship between electron configurations and the periodic table, give the number of (a) outer-shell electrons in an atom of \(\mathrm{Sb} ;\) (b) electrons in the fourth principal electronic shell of \(\mathrm{Pt} ;\) (c) elements whose atoms have six outer-shell electrons; (d) unpaired electrons in an atom of Te; (e) transition elements in the sixth period.

Electromagnetic radiation can be transmitted through a vacuum or empty space. Can heat be similarly transferred? Explain.
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