Calculate the wavelength associated with a moving electron having kinetic energy of `1.375 xx 10^(-25) J`. (mass of `e = 9.1 xx 10^(-31) kg, h = 6.63 xx 10^(-34) kg m^2 s^(-1))`.

Explanation

To calculate the wavelength associated with a moving electron having a kinetic energy of \(1.375 \times 10^{-25} \, \text{J}\), we will use the de Broglie wavelength formula. Let's go through the steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (\(\lambda\)) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where:
- \(h\) is Planck's constant (\(6.63 \times 10^{-34} \, \text{kg m}^2 \text{s}^{-1}\)),
- \(m\) is the mass of the electron (\(9.1 \times 10^{-31} \, \text{kg}\)),
- \(v\) is the velocity of the electron.

### Step 2: Relate kinetic energy to velocity
The kinetic energy (KE) of the electron is given by:
\[ KE = \frac{1}{2} mv^2 \]
From this, we can express the velocity \(v\) in terms of kinetic energy:
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

### Step 3: Substitute the kinetic energy into the velocity formula
Substituting the given kinetic energy value:
\[ v = \sqrt{\frac{2 \cdot 1.375 \times 10^{-25} \, \text{J}}{9.1 \times 10^{-31} \, \text{kg}}} \]

### Step 4: Calculate the velocity
Calculating the value:
\[ v = \sqrt{\frac{2 \cdot 1.375 \times 10^{-25}}{9.1 \times 10^{-31}}} \]
\[ v = \sqrt{\frac{2.75 \times 10^{-25}}{9.1 \times 10^{-31}}} \]
\[ v = \sqrt{3.02 \times 10^{5}} \approx 5.49 \times 10^{2} \, \text{m/s} \]

### Step 5: Substitute \(v\) back into the de Broglie wavelength formula
Now that we have \(v\), we can substitute it back into the de Broglie wavelength formula:
\[ \lambda = \frac{h}{mv} = \frac{6.63 \times 10^{-34}}{9.1 \times 10^{-31} \cdot 5.49 \times 10^{2}} \]

### Step 6: Calculate the wavelength
Calculating the denominator:
\[ mv = 9.1 \times 10^{-31} \cdot 5.49 \times 10^{2} \approx 5.00 \times 10^{-28} \]
Now substituting this back into the wavelength formula:
\[ \lambda = \frac{6.63 \times 10^{-34}}{5.00 \times 10^{-28}} \approx 1.325 \times 10^{-6} \, \text{m} \]

### Final Answer
The wavelength associated with the moving electron is approximately:
\[ \lambda \approx 1.325 \times 10^{-6} \, \text{m} \text{ or } 1325 \, \text{nm} \]

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