To determine the amount of energy required for an electron to jump from the [tex]\( n=1 \)[/tex] energy level to the [tex]\( n=3 \)[/tex] energy level in a hydrogen atom, we need to follow these steps:
1. Identify the given constants and conversions:
- Planck's constant, [tex]\( h = 6.626 \times 10^{-19} \, \text{J} \cdot \text{s} \)[/tex].
- Conversion factor, [tex]\( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \)[/tex].
2. Apply the Bohr model to determine energy levels:
According to the Bohr model, the energy [tex]\( E_n \)[/tex] of an electron in the [tex]\( n \)[/tex]-th energy level is given by:
[tex]\[
E_n = - \frac{13.6 \, \text{eV}}{n^2}
\][/tex]
For [tex]\( n=1 \)[/tex]:
[tex]\[
E_1 = - \frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV}
\][/tex]
For [tex]\( n=3 \)[/tex]:
[tex]\[
E_3 = - \frac{13.6 \, \text{eV}}{3^2} = - \frac{13.6}{9} \, \text{eV} = -1.511111111111111 \, \text{eV}
\][/tex]
3. Convert these energy levels from electron volts (eV) to joules (J):
[tex]\[
E_1 \, (\text{in J}) = E_1 \times 1.6 \times 10^{-19} \, \text{J/eV} = -13.6 \times 1.6 \times 10^{-19} \, \text{J} = -2.176 \times 10^{-18} \, \text{J}
\][/tex]
[tex]\[
E_3 \, (\text{in J}) = E_3 \times 1.6 \times 10^{-19} \, \text{J/eV} = -1.511111111111111 \times 1.6 \times 10^{-19} \, \text{J} = -2.4177777777777774 \times 10^{-19} \, \text{J}
\][/tex]
4. Calculate the energy difference required for the electron to transition from [tex]\( n=1 \)[/tex] to [tex]\( n=3 \)[/tex]:
[tex]\[
\Delta E \, (\text{in J}) = E_3 - E_1 = -2.4177777777777774 \times 10^{-19} \, \text{J} - (-2.176 \times 10^{-18} \, \text{J})
\][/tex]
[tex]\[
\Delta E \, (\text{in J}) = 2.176 \times 10^{-18} \, \text{J} - 2.4177777777777774 \times 10^{-19} \, \text{J} = 1.934222222222222 \times 10^{-18} \, \text{J}
\][/tex]
5. Convert the energy difference back to electron volts (eV):
[tex]\[
\Delta E \, (\text{in eV}) = \frac{\Delta E \, (\text{in J})}{1.6 \times 10^{-19} \, \text{J/eV}} = \frac{1.934222222222222 \times 10^{-18} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}}
\][/tex]
[tex]\[
\Delta E \, (\text{in eV}) = 12.088888888888887 \, \text{eV}
\][/tex]
Therefore, the amount of energy needed for an electron to jump from the [tex]\( n=1 \)[/tex] energy level to the [tex]\( n=3 \)[/tex] energy level is approximately [tex]\( 12.09 \, \text{eV} \)[/tex]. None of the options provided in the question (91.8 eV, 108.8 eV, 117.5 eV, 114.8 eV) match the correct value.
