Which of the expression given below gives IE of H atom in terms of Rydberg's constant `(R_(H))`?

Options

A. `R_(H).hc`
B. `R_(H)N_(A).hc`
C. `R_(H)(2hc)`
D. `R_(H).c`

Explanation

To find the expression for the ionization energy (IE) of the hydrogen atom in terms of Rydberg's constant (R_H), we can follow these steps:

### Step-by-Step Solution:

1. **Understand Ionization Energy**:
The ionization energy (IE) of an atom is the energy required to remove an electron from the atom in its ground state. For hydrogen, this can be expressed in terms of frequency (ν) or wavelength (λ).

2. **Relate Ionization Energy to Frequency**:
The ionization energy can be expressed using the equation:
\[ IE = h \nu \]
where \( h \) is Planck's constant and \( \nu \) is the frequency of the emitted photon.

3. **Relate Frequency to Wavelength**:
We can also express the ionization energy in terms of wavelength:
\[ IE = \frac{hc}{\lambda} \]
where \( c \) is the speed of light.

4. **Use Rydberg's Formula**:
Rydberg's formula relates the wavelength (λ) to the Rydberg constant (R_H):
\[ \frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]
For hydrogen, \( Z = 1 \) and for ionization, we consider the transition from \( n = 1 \) to \( n = \infty \) (where the electron is removed):
\[ \frac{1}{\lambda} = R_H \]

5. **Substitute into the Ionization Energy Equation**:
Now substituting \( \frac{1}{\lambda} = R_H \) into the ionization energy equation:
\[ IE = hc \cdot R_H \]

6. **Final Expression**:
Therefore, the expression for the ionization energy of the hydrogen atom in terms of Rydberg's constant is:
\[ IE = hc R_H \]