The first excited state of hydrogen atom is `10.2 eV` above its ground state. The temperature is needed to excite hydrogen atoms to first excited level is

Options

A. `7.9xx 10^(4) K`
B. `3.5xx 10^(4) K`
C. `5.8xx 10^(4) K`
D. `14xx 10^(4) K`

Explanation

To find the temperature needed to excite hydrogen atoms to the first excited state (which is 10.2 eV above the ground state), we can use the relationship between energy and temperature given by the kinetic interpretation of temperature.

### Step-by-Step Solution:

1. **Understand the Energy Requirement**:
The first excited state of the hydrogen atom is 10.2 eV above the ground state. This means that to excite an electron from the ground state (n=1) to the first excited state (n=2), we need to provide 10.2 eV of energy.

2. **Convert Energy from eV to Joules**:
The energy in electron volts can be converted to joules using the conversion factor:
\[ 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \]
Therefore,
\[ 10.2 \text{ eV} = 10.2 \times 1.6 \times 10^{-19} \text{ J} = 1.632 \times 10^{-18} \text{ J} \]

3. **Relate Energy to Temperature**:
According to the kinetic theory of gases, the average kinetic energy of a particle is given by:
\[ \text{K.E.} = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \text{ J/K} \)) and \( T \) is the temperature in Kelvin.

4. **Set Up the Equation**:
We set the kinetic energy equal to the energy required to excite the electron:
\[ \frac{3}{2} k T = 1.632 \times 10^{-18} \text{ J} \]

5. **Solve for Temperature \( T \)**:
Rearranging the equation to solve for \( T \):
\[ T = \frac{2 \times 1.632 \times 10^{-18}}{3 \times 1.38 \times 10^{-23}} \]
Now, calculate \( T \):
\[ T = \frac{3.264 \times 10^{-18}}{4.14 \times 10^{-23}} \approx 7.9 \times 10^{4} \text{ K} \]

6. **Final Answer**:
The temperature needed to excite hydrogen atoms to the first excited level is approximately \( 7.9 \times 10^{4} \text{ K} \).