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.
An electron is trapped in a box of length L = 100 pm. There is a single node in the center of the box where the electron cannot exist. What is the energy of the electron, in eV?
(6.632*10^-34)^2*4/(8*9.109*1.602*10^-70)=150.7eV
To find the size of the box in the first question, we need to use the formula for energy difference in an atom or a particle. The formula is given by:
ΔE = -13.6 eV * (1/n_f^2 - 1/n_i^2)
where ΔE is the energy difference, n_f is the final quantum state, and n_i is the initial quantum state.
In this case, the particle falls from the n=3 state to the n=2 state and emits 0.16 meV of energy. First, we need to convert the energy difference from meV to eV. 1 meV is equal to 0.001 eV, so 0.16 meV is equal to 0.16 * 0.001 = 0.00016 eV.
Plugging in the values into the 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)
To find the size of the box, we can rearrange the equation:
Size of the box = (h^2 / (8 * m * E))^1/3
where h is the Planck's constant, m is the mass of the particle (in this case, the mass of a proton), and E is the energy difference.
The value of h is 6.626 x 10^-34 J·s and the mass of a proton is approximately 1.67 x 10^-27 kg.
First, we need to convert the energy difference from eV to J. 1 eV is equal to 1.6 x 10^-19 J. So, 0.00016 eV is equal to 0.00016 * 1.6 x 10^-19 J = 2.56 x 10^-23 J.
Plugging in the values into the equation:
Size of the box = (6.626 x 10^-34 J·s)^2 / (8 * 1.67 x 10^-27 kg * 2.56 x 10^-23 J))^1/3
Calculating the value inside the parentheses:
Value inside the parentheses = (6.626 x 10^-34 x 6.626 x 10^-34) / (8 * 1.67 x 10^-27 * 2.56 x 10^-23))^1/3
Value inside the parentheses = (4.38 x 10^-67 J^2) / (2.04 x 10^-48 kg·J))^1/3
Calculating the cube root:
Value inside the parentheses = (2.148 x 10^-20 kg·J)^1/3
Value inside the parentheses = 2.48 x 10^-7 kg^(1/3)·J^(1/3)
Now we can substitute this value back into the equation:
Size of the box = (2.48 x 10^-7 kg^(1/3)·J^(1/3))^1/3
Calculating the cube root:
Size of the box = 5.07 x 10^-3 kg^(1/9)·J^(1/9)
Finally, we need to convert the answer from units of kg^(1/9)·J^(1/9) to nm. We can use the conversion factor:
1 m = 10^9 nm
Therefore,
Size of the box = 5.07 x 10^-3 kg^(1/9)·J^(1/9) * (1 m / 10^9 nm)
Size of the box = 5.07 x 10^-12 m^(1/9)·J^(1/9) = 5.07 nm
Therefore, the size of the box is approximately 5.07 nm.