The Bohr's energy of a stationary state of hydrogen atom is given as `E_(n)=(-2pi^(2)me^(4))/(n^(2)h^(2))`. Putting the values of m and e for `n^(th)` energy level which is not the correct value?

Options

A. `E_(n)=(-21.8xx10^(-19))/(n^(2))J atom^(-1)`
B. `E_(n)=(-13.6)/(n^(2))eV atom^(-1)`
C. `E_(n)=(-1312)/(n^(2))kJ mol^(-1)`
D. `E_(n)=(-12.8xx10^(-19))/(n^(2))erg atom^(-1)`

Answer

To solve the question regarding the Bohr's energy of a stationary state of a hydrogen atom, we will follow these steps:

### Step 1: Write down the formula for Bohr's energy
The energy of the nth stationary state of a hydrogen atom is given by the formula:
\[ E_n = -\frac{2\pi^2 m e^4}{n^2 h^2} \]

### Step 2: Identify the constants
We need to identify the values of the constants involved:
- \( m \) (mass of the electron) = \( 9.1 \times 10^{-31} \) kg
- \( e \) (charge of the electron) = \( 1.6 \times 10^{-19} \) C
- \( h \) (Planck's constant) = \( 6.626 \times 10^{-34} \) J·s
- \( \pi \) = \( 3.14 \)

### Step 3: Substitute the values into the formula
Substituting the values into the formula for \( E_n \):
\[ E_n = -\frac{2 \times (3.14)^2 \times (9.1 \times 10^{-31}) \times (1.6 \times 10^{-19})^4}{n^2 \times (6.626 \times 10^{-34})^2} \]

### Step 4: Calculate the numerator and denominator
1. Calculate \( (1.6 \times 10^{-19})^4 \):
\[ (1.6 \times 10^{-19})^4 = 6.5536 \times 10^{-76} \]

2. Calculate \( (6.626 \times 10^{-34})^2 \):
\[ (6.626 \times 10^{-34})^2 = 4.39 \times 10^{-67} \]

3. Now substitute these values back into the energy formula.

### Step 5: Simplify the expression
After substituting and simplifying, we will find:
\[ E_n = -\frac{2 \times (3.14)^2 \times (9.1 \times 10^{-31}) \times (6.5536 \times 10^{-76})}{n^2 \times (4.39 \times 10^{-67})} \]

### Step 6: Calculate the final energy value
After performing the calculations, we will find that:
\[ E_n \approx -\frac{13.6}{n^2} \text{ eV} \]

### Step 7: Analyze the options
Now we need to compare this result with the given options to identify which one is incorrect.

### Step 8: Convert to different units
1. Convert to joules:
\[ 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \]
Thus,
\[ E_n \approx -13.6 \times 1.602 \times 10^{-19} \text{ J} \approx -2.18 \times 10^{-18} \text{ J} \]

2. Convert to kilojoules per mole:
\[ E_n \approx -1312 \text{ kJ/mol} \]

3. Convert to erg:
\[ 1 \text{ J} = 10^7 \text{ erg} \]
Thus,
\[ E_n \approx -2.18 \times 10^{-18} \text{ J} \approx -2.18 \times 10^{-18} \times 10^7 \text{ erg} \approx -2.18 \times 10^{-11} \text{ erg} \]

### Step 9: Identify the incorrect option
After calculating the energy in various units, we compare them with the options provided in the question. The option that does not match our calculated values is the incorrect one.

### Conclusion
The incorrect value among the options provided is option number 4.

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