A sample of hydrogen (in the form of atoms), is made to absorb white light. `52%` of the hydrogen atoms got ionised. In order to calculate the ionisation energy of hydrogen from its absorption spectrum `("assuming the electrons that got ejected have" KE=0)`, it is possible by measuring the frequency of the:

Options

A. line of shortest wavelength
B. line of longest wavelength
C. line of greatest intensity
D. line of smallest intensity

Answer

To solve the problem of calculating the ionization energy of hydrogen based on the absorption of white light and the ionization of hydrogen atoms, we can follow these steps:

### Step 1: Understand the Concept of Ionization Energy
Ionization energy is the energy required to remove an electron from an atom in its gaseous state. In this case, we are dealing with hydrogen atoms.

### Step 2: Analyze the Given Information
We know that 52% of the hydrogen atoms got ionized when exposed to white light. This indicates that the energy absorbed by these atoms was sufficient to overcome the ionization energy.

### Step 3: Relate Ionization Energy to Frequency
The energy absorbed by the hydrogen atoms can be related to the frequency of the light using the equation:
\[ E = h \cdot \nu \]
where:
- \( E \) is the energy absorbed,
- \( h \) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)),
- \( \nu \) is the frequency of the light.

### Step 4: Consider the Condition of Kinetic Energy
Since it is given that the kinetic energy (KE) of the ejected electrons is zero, the energy absorbed by the hydrogen atoms is equal to the ionization energy (IE) of hydrogen:
\[ IE = E \]

### Step 5: Determine the Frequency of Light
To find the frequency of the light that corresponds to the ionization energy, we need to consider the longest wavelength of light that can still ionize the hydrogen atoms. The longest wavelength corresponds to the minimum energy required to ionize the hydrogen atoms.

### Step 6: Calculate the Ionization Energy
The ionization energy of hydrogen is known to be approximately \( 1312 \, \text{kJ/mol} \). To convert this to energy per atom, we can use Avogadro's number (\(6.022 \times 10^{23} \, \text{atoms/mol}\)):
\[ IE \text{ (per atom)} = \frac{1312 \, \text{kJ/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} \]

### Step 7: Find the Frequency
Now, we can rearrange the equation for energy to solve for frequency:
\[ \nu = \frac{IE}{h} \]

### Step 8: Final Calculation
Substituting the values into the equation will give us the frequency of the light that corresponds to the ionization energy of hydrogen.

### Summary of Steps:
1. Understand ionization energy.
2. Analyze the percentage of ionized atoms.
3. Relate ionization energy to frequency using \( E = h \cdot \nu \).
4. Consider that KE = 0 means energy equals ionization energy.
5. Determine the frequency of light corresponding to the ionization energy.
6. Calculate the ionization energy per atom.
7. Rearrange the equation to find frequency.
8. Perform the final calculation.