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Calculate E1 for The N 2 Energy Level

Reviewed by Calculator Editorial Team

This calculator computes the E1 energy level for the n=2 state in quantum mechanics. The E1 energy level represents the first excited state of a hydrogen-like atom, where the electron is in the n=2 principal quantum state.

What is E1 in Quantum Mechanics?

The E1 energy level refers to the first excited state of a hydrogen-like atom, where the electron is in the n=2 principal quantum state. In quantum mechanics, energy levels are quantized, meaning they can only take specific discrete values. The E1 energy level is the energy of the electron when it is in the first excited state, just above the ground state (n=1).

This energy level is crucial in understanding atomic structure and transitions between energy states. When an electron transitions from the n=2 state to the n=1 state, it emits a photon with energy equal to the difference between these two levels.

The Formula for E1

The energy of the n=2 state in a hydrogen-like atom can be calculated using the following formula:

En = -R∞hc / n2

Where:

  • En is the energy of the nth state
  • R∞ is the Rydberg constant (1.0973731568539 × 107 m-1)
  • h is Planck's constant (6.62607015 × 10-34 J·s)
  • c is the speed of light (2.99792458 × 108 m/s)
  • n is the principal quantum number (2 for the first excited state)

For the n=2 state, the formula simplifies to:

E1 = -R∞hc / 4

This gives the energy of the electron in the n=2 state relative to the ground state (n=1).

Worked Example

Let's calculate the E1 energy level for the n=2 state using the given formula.

  1. Identify the constants:
    • R∞ = 1.0973731568539 × 107 m-1
    • h = 6.62607015 × 10-34 J·s
    • c = 2.99792458 × 108 m/s
    • n = 2
  2. Plug the values into the formula:

    E1 = - (1.0973731568539 × 107 × 6.62607015 × 10-34 × 2.99792458 × 108) / 4

  3. Calculate the numerator:

    Numerator = 1.0973731568539 × 6.62607015 × 2.99792458 × 10(7-34+8) = 2.17987236 × 10-15 J

  4. Divide by 4:

    E1 = - (2.17987236 × 10-15) / 4 = -5.4496809 × 10-16 J

The E1 energy level for the n=2 state is approximately -5.4496809 × 10-16 joules.

Interpreting the Results

The negative sign indicates that the energy is below the ground state (n=1). The magnitude of the energy shows how much higher the n=2 state is compared to the ground state. This energy difference is crucial for understanding atomic transitions and spectral lines.

In practical terms, this calculation helps in understanding the behavior of electrons in atoms and the energy required for transitions between different energy levels.

Frequently Asked Questions

What is the difference between E1 and the ground state energy?
The E1 energy level (n=2) is higher than the ground state energy (n=1). The difference between these two levels is the energy of the photon emitted when an electron transitions from n=2 to n=1.
Can this formula be used for other atoms besides hydrogen?
Yes, this formula can be generalized for hydrogen-like atoms (atoms with a single electron) by adjusting the Rydberg constant to account for the atomic mass and nuclear charge.
What units are used for the energy calculation?
The energy is calculated in joules (J), which is the SI unit of energy. For other units, you can convert the result accordingly.