Which electron transition in a hydrogen atom will emit a photon with the longest wavelength?
n = 2 to n = 1
n = 3 to n = 1
n = 1 to n = 2
n = 1 to n = 3
I would assume it would be a short electron transition since energy and wavelength are inversely related. But I am a little confused as if it would be going from to n = 2 to n = 1 or vice versa.
First, your rationale is OK about energy and wavelength. You KNOW the answer can't be c or d because the atom ABSORBS energy to do that. Atoms EMIT energy only when they go from a higher level to a lower level. So that leaves a or b as the only possibilities so you can guess and be right half the time. You can be right all the time by using
delta E = k[1/(n1)^2 - 1/(n2)^2]
n1 is always < n2 in this formula for emission.
For a we have
dE = k(1/1 - 1/4) = 0.75k and we don't worry about the value of k.
For b we have
dE = k(1/1 - 1/9) = approx 0.9k which makes b the more energetic one and that means the wavelength is shorter and your rationale is right on to pick a as the longer wavelength.
To determine which electron transition in a hydrogen atom will emit a photon with the longest wavelength, we need to consider the energy levels associated with each transition. In hydrogen, electrons transition between different energy levels, which are indicated by the principal quantum number, n.
The energy of a photon is directly proportional to its frequency, and wavelength is inversely proportional to frequency. According to the energy equation for photons, E = hf, where E is the energy, h is Planck's constant, and f is the frequency.
Now, we can use the equation c = λf, where c is the speed of light, λ is the wavelength, and f is the frequency, to relate wavelength to frequency.
Combining these two equations, we have E = hc/λ.
For electron transitions in hydrogen, we know that the energy difference between two energy levels is given by ΔE = E2 - E1 = hc/λ, where ΔE is the energy difference and λ is the wavelength.
The longer the wavelength (larger λ), the smaller the value of energy difference (ΔE) and vice versa.
Now, let's compare the possible electron transitions and determine which one will emit a photon with the longest wavelength:
Transition 1: n = 2 to n = 1
Transition 2: n = 3 to n = 1
Transition 3: n = 1 to n = 2
Transition 4: n = 1 to n = 3
The energy difference between energy levels decreases as n increases. Therefore, the transition with the longest wavelength will be the one with the largest value of n for the final energy level.
In this case, n = 3 to n = 1 (Transition 2) will emit a photon with the longest wavelength.
To determine which electron transition in a hydrogen atom will emit a photon with the longest wavelength, we need to understand the relationship between electron transitions and the resulting photon wavelength.
In general, an electron transition from a higher energy level (n = 3, for example) to a lower energy level (n = 2 or n = 1) corresponds to the emission of a photon. The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength.
According to the Bohr model of the hydrogen atom, the energy of an electron in a particular energy level is given by the equation:
E = -13.6 eV / n²
Where E represents the energy of the electron and n represents the principal quantum number. For example, an electron in the n = 1 energy level would have an energy of -13.6 eV.
To calculate the energy difference between two energy levels, we subtract the energy of the initial level (E₁) from the energy of the final level (E₂):
ΔE = E₂ - E₁ = (-13.6 eV / n₂²) - (-13.6 eV / n₁²)
To find the wavelength of the emitted photon, we can use the equation:
λ = c / ν
Where λ represents the wavelength, c represents the speed of light (approximately 3.00 × 10^8 meters per second), and ν represents the frequency.
For a photon, the energy is related to the frequency by the equation:
E = hν
Where E represents the energy, h represents Planck's constant (approximately 6.63 × 10^-34 Joule-seconds), and ν represents the frequency.
Since ΔE = hν, we can rewrite the equation as:
λ = c / (ΔE / h)
Therefore, the wavelength is inversely proportional to the energy difference between two energy levels.
Now, let's consider the given options:
n = 2 to n = 1:
Here, ΔE = (-13.6 eV / 1²) - (-13.6 eV / 2²) = -13.6 eV - (-3.4 eV) = -10.2 eV
n = 3 to n = 1:
Here, ΔE = (-13.6 eV / 1²) - (-13.6 eV / 3²) = -13.6 eV - (-1.51 eV) = -12.1 eV
n = 1 to n = 2:
Here, ΔE = (-13.6 eV / 2²) - (-13.6 eV / 1²) = -3.4 eV - (-13.6 eV) = 10.2 eV
n = 1 to n = 3:
Here, ΔE = (-13.6 eV / 3²) - (-13.6 eV / 1²) = -1.51 eV - (-13.6 eV) = 12.1 eV
From the calculations, we can see that the transition from n = 2 to n = 1 has an energy difference (ΔE) of -10.2 eV, which is the largest absolute value. Since wavelength is inversely proportional to |ΔE|, this transition will emit a photon with the longest wavelength.
Therefore, the correct answer is n = 2 to n = 1.