In an alternate universe, electron energies in an atom are found to obey the
following equation where n is the principle quantum number:
E = 3.0 x 10^(-19) J * nm * (1/n^2)
..where n is the principle quantum number
A gas discharge lamp, like the lamps used in lab, contains only atoms
from the alternate universe. You turn the lamp on and observe the
spectrum. Two energy transitions that occur within the visible spectrum
(400-700 nm), provide the wavelength of the photon and corresponding
energy levels that produce the photon.
...Where do I even begin?
To begin solving this problem, we need to find the energy levels and wavelengths that correspond to the given visible spectrum range (400-700 nm).
1. Start by setting up an equation using the given formula: E = 3.0 x 10^(-19) J * nm * (1/n^2).
2. Since we are interested in the visible spectrum, the wavelength values should fall within the range of 400-700 nm.
3. Substitute the given wavelengths into the equation and solve for the principle quantum number, n.
4. Calculate the energy levels using the determined values of n.
5. Identify the energy transitions that correspond to the observed spectrum and find the corresponding photon energy levels.
Let's go step by step to find the energy levels and wavelengths:
Step 1: Setting up the equation
E = 3.0 x 10^(-19) J * nm * (1/n^2)
Step 2: Determine the values of n for the given wavelength range
The given range is 400-700 nm, so we want to find the corresponding principle quantum numbers (n) for these values.
For the lower wavelength (400 nm):
400 = 3.0 x 10^(-19) J * nm * (1/n^2)
For the higher wavelength (700 nm):
700 = 3.0 x 10^(-19) J * nm * (1/n^2)
Step 3: Solve for n in each equation
Let's solve for n in each of the above equations separately.
For 400 nm:
Divide both sides by (3.0 x 10^(-19) J * nm):
400 / (3.0 x 10^(-19) J * nm) = 1/n^2
Take the reciprocal of both sides:
(3.0 x 10^(-19) J * nm) / 400 = n^2
n^2 = (3.0 x 10^(-19) J * nm) / 400
Take the square root of both sides to solve for n:
n = sqrt((3.0 x 10^(-19) J * nm) / 400)
Similarly, solve for n in the equation for 700 nm.
Step 4: Calculate the energy levels using the determined values of n
Now that we have calculated the values of n, we can substitute them back into the original equation to calculate the corresponding energy levels (E).
For each value of n, plug it into the equation: E = 3.0 x 10^(-19) J * nm * (1/n^2).
Step 5: Identify the energy transitions within the visible spectrum
Compare the calculated energy levels with the visible spectrum range (400-700 nm) to find the energy transitions that correspond to the observed spectrum.
Remember, the wavelength of a photon is inversely proportional to its energy. So, the energy transition from a higher energy level to a lower energy level would produce photons with wavelengths falling within the visible spectrum range.
By following these steps, you can determine the energy transitions and corresponding energy levels that produce the photons observed in the gas discharge lamp in the alternate universe.