The de Brogile wavelength of an electron (mass `=1 xx 10^(-30)` kg, charge `= 1.6 xx 10^(-19)C)` with a kinetic energy of 200 eV is (Planck's constant `= 6.6 xx 10^(-34) Js)`

Options

A. `9. 60 xx 10^(-11)` m
B. `8. 25 xx 10^(-11)` m
C. `6.25 xx 10^(-11)` m
D. `5.00 xx 10^(-11)` m

Explanation

To find the de Broglie wavelength of an electron with a given mass and kinetic energy, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where:
- \( h \) is Planck's constant
- \( p \) is the momentum of the particle

### Step 2: Relate momentum to kinetic energy
The momentum (\( p \)) of the electron can be expressed in terms of its kinetic energy (\( K \)):
\[ K = \frac{p^2}{2m} \]
From this, we can express momentum as:
\[ p = \sqrt{2mK} \]

### Step 3: Convert kinetic energy from eV to Joules
Given that the kinetic energy is 200 eV, we need to convert this to Joules. The conversion factor is:
\[ 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \]
Thus,
\[ K = 200 \text{ eV} = 200 \times 1.6 \times 10^{-19} \text{ J} = 3.2 \times 10^{-17} \text{ J} \]

### Step 4: Substitute values into the momentum formula
Now we can substitute the mass of the electron and the kinetic energy into the momentum formula:
- Mass \( m = 1 \times 10^{-30} \text{ kg} \)
- Kinetic energy \( K = 3.2 \times 10^{-17} \text{ J} \)

Calculating momentum:
\[ p = \sqrt{2 \times (1 \times 10^{-30}) \times (3.2 \times 10^{-17})} \]
Calculating the value inside the square root:
\[ = \sqrt{6.4 \times 10^{-47}} = 8.0 \times 10^{-24} \text{ kg m/s} \]

### Step 5: Substitute momentum into the de Broglie wavelength formula
Now substitute \( p \) back into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{p} = \frac{6.6 \times 10^{-34}}{8.0 \times 10^{-24}} \]
Calculating this gives:
\[ \lambda = 8.25 \times 10^{-11} \text{ m} \]

### Final Answer
The de Broglie wavelength of the electron is:
\[ \lambda \approx 8.25 \times 10^{-11} \text{ m} \]