Question

To move an electron in a hydrogen atom from the ground state to the second excited state, $12.084\mathrm{eV}$ is required. IIow much energy is required to cause one mole of hydrogen atoms to undergo this transition? (1) $984\mathrm{kJ}$ (2) $1036\mathrm{kJ}$ (3) $1166\mathrm{kJ}$ (4) $1312\mathrm{kJ}$

Short answer

1036 kJ (option 2)

Step-by-step solution

– Identify Energy per Transition

It takes 12.084 eV to move an electron in a hydrogen atom from the ground state to the second excited state.

– Convert eV to Joules

Use the conversion factor between electron volts and joules: 1 eV = 1.60218 × 10^{-19} J. Therefore, 12.084 eV = 12.084 × 1.60218 × 10^{-19} J.

– Calculate the Energy for One Mole

One mole of hydrogen atoms contains Avogadro's number of atoms, which is approximately 6.022 × 10^{23} atoms/mol. Multiply the energy for one atom by Avogadro's number: (12.084 × 1.60218 × 10^{-19} J) × 6.022 × 10^{23} atoms/mol.

– Convert to Kilojoules

Convert the resultant energy in joules (J) to kilojoules (kJ) by dividing by 1000: $$\frac{(12.084\times 1.60218\times {10}^{-19}\text{J})\times 6.022\times {10}^{23}}{1000}$$.

– Calculate and Compare with Options

Perform the calculation to get the energy in kJ and compare with the given options.

Key concepts

Energy Transitions in Hydrogen Atoms

An energy transition is when an electron in an atom moves from one energy level to another. In a hydrogen atom, electrons can absorb energy to jump to a higher energy level, known as an excited state. For instance, moving an electron from the ground state (the lowest energy level, n=1) to the second excited state (n=3) involves absorbing a specific amount of energy. This energy is fixed and unique to each transition.

Electron Volts to Joules Conversion

Energy in atoms is often measured in electron volts (eV). However, for larger calculations, it’s convenient to convert this into joules (J). One electron volt is equivalent to approximately 1.60218 × 10^{-19} joules. For example, if moving an electron from the ground state to an excited state requires 12.084 eV, we find the energy in joules by multiplying: 12.084 eV × 1.60218 × 10^{-19} J/eV.

Avogadro's Number

Avogadro’s number (6.022 × 10^{23}) is hugely important in chemistry. It tells us how many particles, like atoms, are in one mole of a substance. If you know the energy required for one electron to transition states, you can find the energy for an entire mole of electrons by multiplying the single-electron energy by Avogadro’s number. In our exercise, we first converted 12.084 eV to joules, then multiplied by Avogadro's number to find the energy for one mole of hydrogen atoms.

Understanding Ground State

The ground state of an atom is its lowest energy state, where the electron is in its most stable position (n=1 for hydrogen). When an electron is in the ground state, it has the lowest possible energy and is closest to the nucleus. To move to a higher energy level, the electron must absorb a specific amount of energy. This is what we calculated in our exercise when moving the electron from the ground state to an excited state.

Excited State Explained

When an electron absorbs energy, it moves to a higher energy level, known as an excited state. For hydrogen, just above the ground state (n=1), we have the first excited state (n=2), second excited state (n=3), and so on. The energy required to move an electron between these levels is specific and quantized. In our problem, moving the electron to the second excited state required 12.084 eV. In general, excited states are less stable, and electrons will eventually return to the ground state, often emitting energy as light.