Find the energy levels of an electron in a box 1 nm wide. Mass of electron is 9.1 × 10–31 kg. Also findthe energy levels of 10 gm marble in a box 10 cm wide.
Question
Find the energy levels of an electron in a box 1 nm wide. Mass of electron is 9.1 × 10–31 kg. Also findthe energy levels of 10 gm marble in a box 10 cm wide.
Solution
The energy levels of a particle in a box can be calculated using the formula for a particle in a one-dimensional box:
E = n²h² / 8mL²
where:
- E is the energy of the level,
- n is the quantum number (1, 2, 3, ...),
- h is Planck's constant (6.626 x 10^-34 J.s),
- m is the mass of the particle, and
- L is the width of the box.
- For an electron in a box 1 nm wide:
Substitute the given values into the formula:
E = n²(6.626 x 10^-34 J.s)² / 8(9.1 x 10^-31 kg)(1 x 10^-9 m)²
Solving this will give the energy levels for the electron in the box.
- For a 10 gm marble in a box 10 cm wide:
First, convert the mass of the marble to kg (since the SI unit of mass is kg): 10 gm = 0.01 kg
Then, convert the width of the box to m (since the SI unit of length is m): 10 cm = 0.1 m
Substitute these values into the formula:
E = n²(6.626 x 10^-34 J.s)² / 8(0.01 kg)(0.1 m)²
Solving this will give the energy levels for the marble in the box.
Note: The energy levels calculated will be in Joules (J), since this is the SI unit of energy.
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