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How to Calculate E for N 2 Energy Level

Reviewed by Calculator Editorial Team

Calculating the energy level for n=2 in the hydrogen atom is fundamental to understanding atomic structure. This guide explains the Rydberg formula, provides a calculator, shows worked examples, and answers common questions.

Introduction

The energy levels of electrons in hydrogen atoms are quantized, meaning they can only have specific values. The principal quantum number n determines the energy level, with n=1 being the ground state and higher n values representing excited states.

For n=2, we're looking at the first excited state. Calculating its energy requires the Rydberg formula, which relates energy levels to the Rydberg constant and quantum numbers.

Rydberg Formula

The energy of an electron in a hydrogen atom can be calculated using the Rydberg formula:

E = -R∞hc / n2

Where:

  • E = Energy of the electron (in joules)
  • R∞ = Rydberg constant (109,737.31534 cm-1)
  • h = Planck's constant (6.62607015 × 10-34 J·s)
  • c = Speed of light (299,792,458 m/s)
  • n = Principal quantum number

For n=2, we substitute n=2 into the formula to find the energy of the first excited state.

Calculation Steps

  1. Identify the principal quantum number n (in this case, n=2)
  2. Recall the Rydberg constant (R∞ = 109,737.31534 cm-1)
  3. Convert the Rydberg constant to joules (1 cm-1 = 1.986 × 10-23 J)
  4. Calculate the energy using the formula E = -R∞hc / n2
  5. Convert the result to electron volts (1 eV = 1.60218 × 10-19 J) for practical interpretation

Worked Examples

Example 1: Calculating Energy for n=2

Using the Rydberg formula:

E = -109,737.31534 × 1.986 × 10-23 × 6.62607015 × 10-34 × 299,792,458 / (2)2

Calculating step by step:

  1. Convert R∞ to joules: 109,737.31534 × 1.986 × 10-23 = 2.1798 × 10-18 J
  2. Multiply by hc: 2.1798 × 10-18 × 6.62607015 × 10-34 × 299,792,458 ≈ 4.1356 × 10-18 J
  3. Divide by n2: 4.1356 × 10-18 / 4 ≈ 1.0339 × 10-18 J
  4. Convert to eV: 1.0339 × 10-18 / 1.60218 × 10-19 ≈ -0.6456 eV

The energy for n=2 is approximately -0.6456 eV, which is -1.0545 × 10-18 J.

Example 2: Comparing n=1 and n=2

Quantum Number (n) Energy (eV) Energy (J)
1 -13.6057 -2.1798 × 10-18
2 -3.4014 -5.4473 × 10-19

Note that the energy becomes less negative (more positive) as n increases, meaning the electron is less bound to the nucleus.

FAQ

What is the difference between n=1 and n=2 energy levels?

The n=1 level is the ground state with the lowest energy (-13.6057 eV). The n=2 level is the first excited state with higher energy (-3.4014 eV). The difference represents the energy needed to excite an electron from n=1 to n=2.

Why is the energy negative?

Negative energy indicates that the electron is bound to the nucleus. The more negative the value, the more tightly bound the electron is.

Can this formula be used for other atoms?

The Rydberg formula is specific to hydrogen. For other atoms, more complex quantum mechanical models are needed that account for multiple electrons and nuclear charge.