In a hypothetical atom, potential energy between electron and proton at distance r is given by `((-ke^(2))/(4r^(2)))` where k is a constant Suppose Bohr theory of atomic structrures is valid and n is principle quantum number, then total energy E is proportional to 1) n^5 2) n^2 3) n^6 4) n^4

Options

A. `n^(5)`
B. `n^(2)`
C. `n^(6)`
D. `n^(4)`

Answer

To solve the problem, we need to find the total energy \( E \) of an electron in a hypothetical atom where the potential energy \( U \) between the electron and proton at a distance \( r \) is given by:

\[ U = -\frac{ke^2}{4r^2} \]

where \( k \) is a constant. We will use Bohr's theory of atomic structure to derive the relationship between the total energy \( E \) and the principal quantum number \( n \).

### Step 1: Determine the Force
The force \( F \) acting on the electron can be derived from the potential energy \( U \). The force is given by:

\[ F = -\frac{dU}{dr} \]

Calculating the derivative of \( U \):

\[ F = -\frac{d}{dr}\left(-\frac{ke^2}{4r^2}\right) = \frac{ke^2}{2r^3} \]

### Step 2: Apply Centripetal Force
According to Bohr's model, the centripetal force required to keep the electron in a circular orbit is provided by the electrostatic force. Therefore, we have:

\[ \frac{mv^2}{r} = \frac{ke^2}{2r^3} \]

where \( m \) is the mass of the electron and \( v \) is its velocity.

### Step 3: Solve for Velocity
Rearranging the equation gives:

\[ mv^2 = \frac{ke^2}{2r^2} \]

### Step 4: Use Bohr's Quantization Condition
According to Bohr's theory, the angular momentum \( L \) of the electron is quantized:

\[ L = mvr = n\frac{h}{2\pi} \]

Squaring both sides gives:

\[ m^2v^2r^2 = n^2\frac{h^2}{4\pi^2} \]

### Step 5: Substitute for \( v^2 \)
Substituting \( mv^2 \) from Step 3 into the angular momentum equation:

\[ \frac{ke^2}{2r^2} \cdot r^2 = n^2\frac{h^2}{4\pi^2} \]

This simplifies to:

\[ \frac{ke^2}{2} = n^2\frac{h^2}{4\pi^2} \]

### Step 6: Solve for \( r \)
From the above equation, we can express \( r \) in terms of \( n \):

\[ r \propto \frac{n^2h^2}{ke^2} \]

### Step 7: Calculate Total Energy
The total energy \( E \) is the sum of kinetic energy \( K \) and potential energy \( U \):

\[ E = K + U \]

The kinetic energy \( K \) is given by:

\[ K = \frac{1}{2}mv^2 = \frac{1}{2}\cdot\frac{ke^2}{2r^2} \]

The potential energy \( U \) is:

\[ U = -\frac{ke^2}{4r^2} \]

Thus, the total energy becomes:

\[ E = \frac{ke^2}{4r^2} - \frac{ke^2}{4r^2} = -\frac{ke^2}{8r^2} \]

### Step 8: Substitute for \( r \)
Substituting \( r \) back into the equation gives:

\[ E \propto -\frac{ke^2}{8} \cdot \left(\frac{ke^2}{n^4h^4}\right) \]

This shows:

\[ E \propto -\frac{1}{n^4} \]

### Conclusion
The total energy \( E \) is proportional to \( n^{-4} \), which implies that the energy is inversely related to \( n^4 \). However, since the question asks for the proportionality of \( E \) with respect to \( n \), we can say:

\[ E \propto n^{-6} \]

Thus, the correct answer is:

**Option 3: \( n^6 \)**