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The difference in energy levels of n = 2 and n = 1 of a particle -in - a one dimensional box is 9 units of energy. In the same units, what is the difference in energy levels of n = 3 and n =2 for the above system?
- a)4
- b)5
- c)10
- d)15
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
The difference in energy levels of n = 2 and n = 1 of a particle -in -...
For a one-dimensional box, the energy is given by
where n is the quantum number,
m is the mass of the particle,
and L is the box length.
For ground state, n=1. Thus,
Also for the n=2 state, the energy will be
The difference in energy levels of n = 2 and n = 1 will be
Now, the difference in energy levels of n = 2 and n = 1 of a particle -in - a one-dimensional box is 9 units of energy
so,
The difference in energy levels of n = 3 and n =2 will be,
In the same units, the difference in energy levels of n = 3 and n =2 for the above system will be
= 15 units
Hence, the difference in energy levels of n = 3 and n =2 for the above system is 15 units.
where n is the quantum number,
m is the mass of the particle,
and L is the box length.
For ground state, n=1. Thus,
Also for the n=2 state, the energy will be
The difference in energy levels of n = 2 and n = 1 will be
Now, the difference in energy levels of n = 2 and n = 1 of a particle -in - a one-dimensional box is 9 units of energy
so,
The difference in energy levels of n = 3 and n =2 will be,
In the same units, the difference in energy levels of n = 3 and n =2 for the above system will be
= 15 units
Hence, the difference in energy levels of n = 3 and n =2 for the above system is 15 units.
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Community Answer
The difference in energy levels of n = 2 and n = 1 of a particle -in -...
Difference in energy levels of n = 2 and n = 1:
The energy levels of a particle in a one-dimensional box are given by the equation:
E = (n^2 * h^2) / (8mL^2)
where E is the energy level, n is the quantum number, h is the Planck's constant, m is the mass of the particle, and L is the length of the box.
For n = 2, the energy level is:
E2 = (2^2 * h^2) / (8mL^2)
For n = 1, the energy level is:
E1 = (1^2 * h^2) / (8mL^2)
The difference in energy levels between n = 2 and n = 1 is given by:
ΔE = E2 - E1
ΔE = [(2^2 * h^2) / (8mL^2)] - [(1^2 * h^2) / (8mL^2)]
ΔE = [(4 * h^2) / (8mL^2)] - [(1 * h^2) / (8mL^2)]
ΔE = [(4 - 1) * h^2] / (8mL^2)
ΔE = (3 * h^2) / (8mL^2)
Given that ΔE = 9 units of energy, we can write:
(3 * h^2) / (8mL^2) = 9
Difference in energy levels of n = 3 and n = 2:
To find the difference in energy levels between n = 3 and n = 2, we can use the same formula:
For n = 3, the energy level is:
E3 = (3^2 * h^2) / (8mL^2)
For n = 2, the energy level is:
E2 = (2^2 * h^2) / (8mL^2)
The difference in energy levels between n = 3 and n = 2 is given by:
ΔE = E3 - E2
ΔE = [(3^2 * h^2) / (8mL^2)] - [(2^2 * h^2) / (8mL^2)]
ΔE = [(9 * h^2) / (8mL^2)] - [(4 * h^2) / (8mL^2)]
ΔE = [(9 - 4) * h^2] / (8mL^2)
ΔE = (5 * h^2) / (8mL^2)
Since we know that ΔE = 9 units of energy for the n = 2 and n = 1 case, we can substitute this value into the equation:
(5 * h^2) / (8mL^2) = 9
Simplifying the equation, we get:
5 * h^2 = 9 * 8mL^2
5 * h^2 = 72mL^2
Dividing both sides by 72mL^2, we get:
5 * h^2 / 72m
The energy levels of a particle in a one-dimensional box are given by the equation:
E = (n^2 * h^2) / (8mL^2)
where E is the energy level, n is the quantum number, h is the Planck's constant, m is the mass of the particle, and L is the length of the box.
For n = 2, the energy level is:
E2 = (2^2 * h^2) / (8mL^2)
For n = 1, the energy level is:
E1 = (1^2 * h^2) / (8mL^2)
The difference in energy levels between n = 2 and n = 1 is given by:
ΔE = E2 - E1
ΔE = [(2^2 * h^2) / (8mL^2)] - [(1^2 * h^2) / (8mL^2)]
ΔE = [(4 * h^2) / (8mL^2)] - [(1 * h^2) / (8mL^2)]
ΔE = [(4 - 1) * h^2] / (8mL^2)
ΔE = (3 * h^2) / (8mL^2)
Given that ΔE = 9 units of energy, we can write:
(3 * h^2) / (8mL^2) = 9
Difference in energy levels of n = 3 and n = 2:
To find the difference in energy levels between n = 3 and n = 2, we can use the same formula:
For n = 3, the energy level is:
E3 = (3^2 * h^2) / (8mL^2)
For n = 2, the energy level is:
E2 = (2^2 * h^2) / (8mL^2)
The difference in energy levels between n = 3 and n = 2 is given by:
ΔE = E3 - E2
ΔE = [(3^2 * h^2) / (8mL^2)] - [(2^2 * h^2) / (8mL^2)]
ΔE = [(9 * h^2) / (8mL^2)] - [(4 * h^2) / (8mL^2)]
ΔE = [(9 - 4) * h^2] / (8mL^2)
ΔE = (5 * h^2) / (8mL^2)
Since we know that ΔE = 9 units of energy for the n = 2 and n = 1 case, we can substitute this value into the equation:
(5 * h^2) / (8mL^2) = 9
Simplifying the equation, we get:
5 * h^2 = 9 * 8mL^2
5 * h^2 = 72mL^2
Dividing both sides by 72mL^2, we get:
5 * h^2 / 72m
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The difference in energy levels of n = 2 and n = 1 of a particle -in - a one dimensional box is 9 units of energy. In the same units, what is the difference in energy levels of n = 3 and n =2 for the above system?a)4b)5c)10d)15Correct answer is option 'D'. Can you explain this answer?
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