To find the maximum wavelength that a sample of hydrogen atoms can absorb, we can follow these steps:
### Step-by-Step Solution:
1. **Understand the Relationship Between Energy and Wavelength**:
The energy \( E \) of a photon is related to its wavelength \( \lambda \) by the equation:
\[
E = \frac{hc}{\lambda}
\]
where \( h \) is Planck's constant and \( c \) is the speed of light.
2. **Identify the Condition for Maximum Wavelength**:
Since energy is inversely proportional to wavelength, to maximize \( \lambda \), we need to minimize \( E \). The minimum energy corresponds to the transition with the smallest energy difference.
3. **Determine the Relevant Energy Transition**:
For hydrogen, the transition from the second energy level (n=2) to the first energy level (n=1) has the smallest energy difference. We can calculate this energy difference using the formula for the energy levels of hydrogen:
\[
E_n = -\frac{13.6 \, \text{eV}}{n^2}
\]
Thus, for \( n=2 \) and \( n=1 \):
\[
E_2 = -\frac{13.6 \, \text{eV}}{2^2} = -\frac{13.6}{4} = -3.4 \, \text{eV}
\]
\[
E_1 = -\frac{13.6 \, \text{eV}}{1^2} = -13.6 \, \text{eV}
\]
4. **Calculate the Energy Difference**:
The energy difference \( \Delta E \) for the transition from \( n=2 \) to \( n=1 \) is:
\[
\Delta E = E_1 - E_2 = (-13.6) - (-3.4) = -10.2 \, \text{eV}
\]
5. **Use the Energy to Find Wavelength**:
Now, substituting \( \Delta E \) into the energy-wavelength equation:
\[
\Delta E = \frac{hc}{\lambda}
\]
Rearranging gives:
\[
\lambda = \frac{hc}{\Delta E}
\]
6. **Substituting Values**:
Using \( h = 4.135667696 \times 10^{-15} \, \text{eV s} \) and \( c = 3 \times 10^8 \, \text{m/s} \):
\[
\lambda = \frac{(4.135667696 \times 10^{-15} \, \text{eV s})(3 \times 10^8 \, \text{m/s})}{10.2 \, \text{eV}}
\]
Calculating this gives:
\[
\lambda \approx 121.56 \, \text{nm}
\]
7. **Convert to Angstroms**:
Since \( 1 \, \text{nm} = 10 \, \text{Å} \), we convert:
\[
\lambda \approx 1215.6 \, \text{Å}
\]
### Final Answer:
The maximum wavelength that a sample of hydrogen atoms can absorb is approximately **1215.6 Å**.
