An electron is in the ground state of an infinite square well. The longest wavelength photon that the electron can absorb is 400 nm.
What is the length of thw well?
The Correct Answer and Explanation is :
To solve for the length of the infinite square well, we need to use the relationship between the energy levels of an electron in such a well and the wavelength of the photon it can absorb. The energy levels for an electron in an infinite square well are given by:
[
E_n = \frac{n^2 h^2}{8mL^2}
]
Where:
- ( E_n ) is the energy of the (n)-th state,
- ( h ) is Planck’s constant ((6.626 \times 10^{-34}) J·s),
- ( m ) is the mass of the electron ((9.109 \times 10^{-31}) kg),
- ( L ) is the length of the well,
- ( n ) is the principal quantum number.
Step 1: Determine the energy difference for the transition
The longest wavelength photon corresponds to the smallest energy transition the electron can make. In the ground state, (n = 1), and the next allowed state is (n = 2). The energy difference between these two states is:
[
\Delta E = E_2 – E_1 = \frac{4h^2}{8mL^2} – \frac{h^2}{8mL^2} = \frac{3h^2}{8mL^2}
]
Step 2: Relating energy to photon wavelength
The energy of a photon is related to its wavelength by:
[
E = \frac{hc}{\lambda}
]
Where:
- ( E ) is the energy of the photon,
- ( h ) is Planck’s constant,
- ( c ) is the speed of light ((3.0 \times 10^8) m/s),
- ( \lambda ) is the wavelength of the photon.
The longest wavelength corresponds to the transition from (n = 1) to (n = 2), so the energy of the photon is:
[
\Delta E = \frac{hc}{\lambda}
]
Given that the longest wavelength photon is 400 nm, we can substitute this value:
[
\frac{3h^2}{8mL^2} = \frac{hc}{\lambda}
]
Step 3: Solve for the length of the well (L)
Substitute ( \lambda = 400 \times 10^{-9} ) m and solve for ( L ):
[
\frac{3h^2}{8mL^2} = \frac{h \cdot c}{400 \times 10^{-9}}
]
Simplify and solve for ( L ):
[
L = \sqrt{\frac{3h}{8mc} \cdot 400 \times 10^{-9}}
]
After plugging in the known values for ( h ), ( m ), and ( c ), the length of the well (L) is approximately 2.6 nm.
Explanation:
The energy levels in an infinite square well depend on the length of the well and the quantum number of the state. The electron can absorb photons corresponding to transitions between these energy levels. The longest wavelength photon corresponds to the smallest energy difference, which is between the ground state ((n = 1)) and the first excited state ((n = 2)). Using the relationship between energy and wavelength, we can calculate the length of the well based on the given wavelength of the absorbed photon.