The velocity of an electron having wavelenth of 0.15 nm will be

Options

A. `4.85 xx10^(14) cm S^(-1)`
B. `2.84 xx10^(5) cm s^(-1)`
C. `2.07 xx10^(6) cm s^(-1)`
D. `1.312 xx10^(7) cm s^(-1)`

Answer

To find the velocity of an electron with a wavelength of 0.15 nm, we can use the de Broglie wavelength formula, which relates the wavelength (λ) of a particle to its momentum (p):

\[ \lambda = \frac{h}{mv} \]

Where:
- \( \lambda \) is the wavelength,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( m \) is the mass of the electron (\( 9.1 \times 10^{-31} \, \text{kg} \)),
- \( v \) is the velocity of the electron.

### Step 1: Rearrange the formula to solve for velocity (v)
We can rearrange the formula to find the velocity:

\[ v = \frac{h}{m \lambda} \]

### Step 2: Convert the wavelength from nanometers to meters
Given that the wavelength \( \lambda = 0.15 \, \text{nm} \):

\[ \lambda = 0.15 \, \text{nm} = 0.15 \times 10^{-9} \, \text{m} = 1.5 \times 10^{-10} \, \text{m} \]

### Step 3: Substitute the values into the equation
Now we can substitute the values of \( h \), \( m \), and \( \lambda \) into the equation:

\[ v = \frac{6.626 \times 10^{-34} \, \text{Js}}{(9.1 \times 10^{-31} \, \text{kg}) \times (1.5 \times 10^{-10} \, \text{m})} \]

### Step 4: Calculate the denominator
First, calculate the denominator:

\[ 9.1 \times 10^{-31} \, \text{kg} \times 1.5 \times 10^{-10} \, \text{m} = 1.365 \times 10^{-40} \, \text{kg m} \]

### Step 5: Calculate the velocity
Now substitute this back into the equation for velocity:

\[ v = \frac{6.626 \times 10^{-34}}{1.365 \times 10^{-40}} \approx 4.85 \times 10^{6} \, \text{m/s} \]

### Step 6: Convert the velocity to cm/s
To convert the velocity from m/s to cm/s, we multiply by 100 (since 1 m = 100 cm):

\[ v = 4.85 \times 10^{6} \, \text{m/s} \times 100 = 4.85 \times 10^{8} \, \text{cm/s} \]

### Final Answer
Thus, the velocity of the electron is:

\[ \text{Velocity} = 4.85 \times 10^{8} \, \text{cm/s} \]