Calculate the wavelength a particle of mass `m = 6.6 xx 10^(-27) kg` moving with kinetic energy `7.425 xx 10^(-13) J ( h = 6.6 xx 10^(-34) kg m^(2) s ^(-1)`

Explanation

To calculate the wavelength of a particle using the de Broglie wave equation, we can follow these steps:

### Step 1: Understand the de Broglie wavelength formula
The de Broglie wavelength (λ) of a particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where:
- \( h \) is Planck's constant (\( 6.6 \times 10^{-34} \, \text{kg m}^2/\text{s} \)),
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.

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

### Step 3: Substitute the expression for velocity into the de Broglie equation
Substituting \( v \) into the de Broglie wavelength formula gives:
\[ \lambda = \frac{h}{m \sqrt{\frac{2 \cdot KE}{m}}} \]
This simplifies to:
\[ \lambda = \frac{h}{\sqrt{2m \cdot KE}} \]

### Step 4: Plug in the values
Now, we can substitute the known values into the equation:
- \( h = 6.6 \times 10^{-34} \, \text{kg m}^2/\text{s} \)
- \( m = 6.6 \times 10^{-27} \, \text{kg} \)
- \( KE = 7.425 \times 10^{-13} \, \text{J} \)

Substituting these values into the formula:
\[ \lambda = \frac{6.6 \times 10^{-34}}{\sqrt{2 \cdot (6.6 \times 10^{-27}) \cdot (7.425 \times 10^{-13})}} \]

### Step 5: Calculate the denominator
First, calculate the term inside the square root:
\[ 2 \cdot (6.6 \times 10^{-27}) \cdot (7.425 \times 10^{-13}) = 9.785 \times 10^{-39} \]

Now take the square root:
\[ \sqrt{9.785 \times 10^{-39}} \approx 3.128 \times 10^{-20} \]

### Step 6: Calculate the wavelength
Now, substitute this back into the equation for wavelength:
\[ \lambda = \frac{6.6 \times 10^{-34}}{3.128 \times 10^{-20}} \approx 2.11 \times 10^{-14} \, \text{m} \]

### Step 7: Final result
The wavelength of the particle is approximately:
\[ \lambda \approx 2.11 \times 10^{-14} \, \text{m} \]