Problem 6 - Chemistry in 30 days

Question

The amount of energy needed to excite a hydrogen atom from its ground state to the second excited state is (a) \(12.1 \mathrm{eV}\) (b) \(10.2 \mathrm{eV}\) (c) \(13.6 \mathrm{eV}\) (d) \(34 \mathrm{eV}\)

Step-by-Step Solution

Answer

The energy needed is 12.1 eV, so option (a) is correct.

Step 1: Identify the Initial and Final States

In this problem, the hydrogen atom is excited from the ground state (n=1) to the second excited state. In hydrogen, the ground state corresponds to the principal quantum number n=1, and the second excited state corresponds to n=3.

Step 2: Use the Energy Level Formula for Hydrogen

The energy levels of a hydrogen atom can be calculated using the formula: \[ E_n = -13.6 \left( \frac{1}{n^2} \right) \, \text{eV} \]where \( E_n \) is the energy of the n-th level.

Step 3: Calculate the Energy for Initial State

Substitute n=1 into the formula to find the energy of the ground state:\[ E_1 = -13.6 \left( \frac{1}{1^2} \right) = -13.6 \, \text{eV} \].

Step 4: Calculate the Energy for Final State

Substitute n=3 into the formula to find the energy of the second excited state:\[ E_3 = -13.6 \left( \frac{1}{3^2} \right) = -13.6 \left( \frac{1}{9} \right) = -1.51 \, \text{eV} \].

Step 5: Find the Energy Difference

The energy needed to excite the hydrogen atom is the energy difference between the states:\[ \Delta E = E_3 - E_1 = -1.51 \, \text{eV} - (-13.6 \, \text{eV}) = 12.1 \, \text{eV} \].

Step 6: Select the Correct Option

The calculated energy difference is 12.1 eV. Therefore, the correct option is (a) \(12.1 \mathrm{eV}\).

Key Concepts

Quantum Numbers

Quantum numbers are crucial for understanding the structure and behavior of atoms, like hydrogen. They define the unique properties and possible energy states of electrons. There are four types of quantum numbers:

• The **principal quantum number** ( ): Determines the energy level of the electron. It can take any integer value starting from 1. For example, in this exercise, the ground state is =1 and the second excited state is =3.
• The **azimuthal quantum number** (l): Describes the shape of the electron's orbital and ranges from 0 up to -1.
• The **magnetic quantum number** (m): Represents the orientation of the orbital in space and varies from -l to +l.
• The **spin quantum number** (s): Indicates the spin of the electron, either +1/2 or -1/2.

These numbers are like an address for the electron, telling us where it is and what it is doing.

Energy Level Formula

The energy levels of a hydrogen atom are calculated using a specific formula that tells us how much energy is in each level. It all comes down to the principal quantum number .The formula is:\[ E_n = -13.6 \left( \frac{1}{n^2} \right) \, \text{eV} \]Here, \( E_n \) is the energy for a certain level . The constant -13.6 is crucial as it is the energy in electronvolts of the atom's most stable, or ground, state.The negative sign indicates that the energy is bound, meaning it's lower than the energy of an electron at rest at an infinite distance from the nucleus. As increases, the energy becomes less negative, indicating higher energy levels.

Energy Difference Calculation

To find the energy needed to move an electron from one energy level to another, we calculate the energy difference.Here's how:1. **Find the energy of the initial state** using \( n = 1 \): \[ E_1 = -13.6 \left( \frac{1}{1^2} \right) = -13.6 \, \text{eV} \] 2. **Find the energy of the final state** using \( n = 3 \): \[ E_3 = -13.6 \left( \frac{1}{3^2} \right) = -1.51 \, \text{eV} \] 3. **Subtract the initial energy from the final energy** to get the energy difference: \[ \Delta E = E_3 - E_1 = -1.51 \, \text{eV} - (-13.6 \, \text{eV}) = 12.1 \, \text{eV} \] This difference, 12.1 eV, is the energy needed to excite the atom from the ground state to the second excited state.

Excited States

Excited states occur when an electron in an atom absorbs energy and moves to a higher energy level than its ground state.

• In the hydrogen atom, the ground state is the lowest energy state, where =1.
• Higher energy levels where >1 are referred to as excited states. So, if an electron in a hydrogen atom is at =3, this is the second excited state.
• A move from =1 to =3 requires energy, which, in this case, is 12.1 eV.

Such transitions are temporary, as electrons tend to return to lower energy states, sometimes releasing energy in the form of light. Understanding excited states helps in fields like spectroscopy and quantum chemistry.