Calculate de - Broglie wavelength of an electron having kinetic energy `2.8xx10^(-23)J` electron having kinetic energy `2.8xx10^(-23)J. (m_(e)=9.1xx10^(-31)kg)`

Options

A. `9.28 xx10^(-4)m`
B. `9.28 xx10^(-7)m`
C. `9.28 xx10^(-8)m`
D. `9.28 xx10^(-10)m`

Answer

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

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

### Step 2: Relate momentum to kinetic energy
Momentum \( p \) can be expressed in terms of kinetic energy \( KE \):
\[ KE = \frac{1}{2} mv^2 \]
From this, we can derive:
\[ p = mv = \sqrt{2m \cdot KE} \]

### Step 3: Substitute momentum in the de Broglie wavelength formula
Substituting the expression for momentum into the de Broglie wavelength formula, we get:
\[ \lambda = \frac{h}{\sqrt{2m \cdot KE}} \]

### Step 4: Insert known values
Now, we can insert the known values into the formula. The values are:
- Planck's constant \( h = 6.626 \times 10^{-34} \, \text{J s} \)
- Mass of the electron \( m = 9.1 \times 10^{-31} \, \text{kg} \)
- Kinetic energy \( KE = 2.8 \times 10^{-23} \, \text{J} \)

Substituting these values into the equation:
\[ \lambda = \frac{6.626 \times 10^{-34}}{\sqrt{2 \cdot 9.1 \times 10^{-31} \cdot 2.8 \times 10^{-23}}} \]

### Step 5: Calculate the denominator
First, calculate the term under the square root:
\[ 2 \cdot 9.1 \times 10^{-31} \cdot 2.8 \times 10^{-23} = 5.096 \times 10^{-53} \]
Now take the square root:
\[ \sqrt{5.096 \times 10^{-53}} \approx 7.13 \times 10^{-27} \]

### Step 6: Calculate the de Broglie wavelength
Now substitute back into the equation for wavelength:
\[ \lambda = \frac{6.626 \times 10^{-34}}{7.13 \times 10^{-27}} \approx 9.28 \times 10^{-8} \, \text{m} \]

### Final Answer
The de Broglie wavelength of the electron is approximately:
\[ \lambda \approx 9.28 \times 10^{-8} \, \text{m} \]

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