A particle with the mass of a proton is trapped in a box. The particle falls from the n=3 state to the n=2 state and emits 0.16 meV of energy. What is the size of the box? Please report your answer in nm.
To determine the size of the box, we need to consider the energy levels of the particle. In this case, the particle falls from the n=3 state to the n=2 state and emits energy of 0.16 meV.
The energy change between two energy levels of an electron or a proton in a box can be calculated using the Rydberg formula:
ΔE = -13.6 eV * (1 / n2_final^2 - 1 / n1_initial^2)
Here, n2_final is the final state (n=2) and n1_initial is the initial state (n=3).
First, we need to convert the given energy from meV to eV. 1 meV is equal to 0.001 eV. So, 0.16 meV is equal to 0.16 * 0.001 eV = 0.00016 eV.
Substituting the known values into the Rydberg formula, we have:
0.00016 eV = -13.6 eV * (1 / 2^2 - 1 / 3^2)
Simplifying the equation:
0.00016 eV = -13.6 eV * (1 / 4 - 1 / 9)
0.00016 eV = -13.6 eV * (9/36 - 4/36)
0.00016 eV = -13.6 eV * (5/36)
Now, solving for the size of the box, we can rearrange the equation:
Size of the box = λ * Δn
Where λ is the wavelength associated with the energy emission and Δn is the difference in energy levels (n2_final - n1_initial). We can calculate the wavelength using the photon energy equation:
E = hc/λ
Where E is the energy, h is Planck's constant (6.626 x 10^-34 J*s), c is the speed of light (2.998 x 10^8 m/s), and λ is the wavelength.
Converting the energy from eV to Joules, we have:
0.00016 eV = 0.00016 eV * 1.602 x 10^-19 J/eV = 2.56 x 10^-23 J
Solving for the wavelength:
2.56 x 10^-23 J = (6.626 x 10^-34 J*s * 2.998 x 10^8 m/s) / λ
λ = (6.626 x 10^-34 J*s * 2.998 x 10^8 m/s) / (2.56 x 10^-23 J)
λ = 7.759 x 10^-7 m
Finally, we can calculate the size of the box using the formula:
Size of the box = λ * Δn
Size of the box = (7.759 x 10^-7 m) * (3 - 2)
Size of the box = 7.759 x 10^-7 m
Converting the size of the box to nm, we multiply by 10^9:
Size of the box = 775.9 nm
Therefore, the size of the box is approximately 775.9 nm.