The rydberg formula, for the spectrum of the hydorgen aotm where all terms have their usual menaning is

Options

A. `hupsilon_(if)=(me^(4))/(8epsilon_0^(2)h^2)((1)/(n_f)-(1)/(n_i))`
B. `hupsilon_(if)=(me^(4))/(8epsilon_0^2h^2)((1)/n_(f)^2-(1)/(n_(i)^(2)))`
C. `hupsilon_(if)=(8epsilon_0^2h^2)/(me^(4))((1)/(n_f)-(1)/(n_(i)))`
D. `hupsilon_(if)=(8epsilon_0^2h^2)/(me^(4))((1)/n_(f)^(2)-(1)/(n_(i)^(2)))`

Explanation

To derive the Rydberg formula for the spectrum of the hydrogen atom, we will follow these steps:

### Step 1: Understand the Energy Levels
The energy of an electron in the nth orbit of a hydrogen atom is given by the formula:
\[ E_n = -\frac{M e^4}{8 \epsilon_0^2 h^2 n^2} \]
where:
- \( E_n \) is the energy of the electron in the nth orbit,
- \( M \) is the mass of the electron,
- \( e \) is the charge of the electron,
- \( \epsilon_0 \) is the permittivity of free space,
- \( h \) is Planck's constant,
- \( n \) is the principal quantum number.

### Step 2: Calculate the Energy Difference
When an electron transitions from an initial energy level \( n_i \) to a final energy level \( n_f \), the change in energy (\( \Delta E \)) can be calculated as:
\[ \Delta E = E_{n_f} - E_{n_i} \]
Substituting the energy levels:
\[ \Delta E = \left(-\frac{M e^4}{8 \epsilon_0^2 h^2 n_f^2}\right) - \left(-\frac{M e^4}{8 \epsilon_0^2 h^2 n_i^2}\right) \]
This simplifies to:
\[ \Delta E = \frac{M e^4}{8 \epsilon_0^2 h^2} \left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right) \]

### Step 3: Relate Energy to Frequency
According to Planck's relation, the energy of a photon emitted or absorbed during this transition is also given by:
\[ \Delta E = h \nu \]
where \( \nu \) is the frequency of the emitted or absorbed radiation.

### Step 4: Combine the Equations
Equating the two expressions for \( \Delta E \):
\[ h \nu = \frac{M e^4}{8 \epsilon_0^2 h^2} \left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right) \]
From this, we can express the frequency \( \nu \):
\[ \nu = \frac{M e^4}{8 \epsilon_0^2 h^3} \left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right) \]

### Step 5: Write the Rydberg Formula
The Rydberg formula for the wavelengths of the spectral lines can be derived from the frequency:
\[ \frac{1}{\lambda} = R \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \]
where \( R \) is the Rydberg constant given by:
\[ R = \frac{M e^4}{8 \epsilon_0^2 h^3} \]

### Final Result
Thus, the Rydberg formula for the spectrum of the hydrogen atom is:
\[ \frac{1}{\lambda} = R \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \]

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