A particle X moving with a certain velocity has a de Broglie wavelength of `1 "Å"` . If particle Y has a mass of `25%` that of X and velocity `75%` that of X ,de Broglie wavelength of Y will be :

Options

A. 5.33 A
B. `3 A`
C. `6.88 A`
D. `48 A`

Answer

To solve the problem, we need to use the de Broglie wavelength formula, which is given by:

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

where:
- \(\lambda\) is the de Broglie wavelength,
- \(h\) is Planck's constant,
- \(m\) is the mass of the particle,
- \(v\) is the velocity of the particle.

### Step 1: Write the de Broglie wavelength for particle X

For particle X, the de Broglie wavelength is given as:

\[ \lambda_X = \frac{h}{m_X v_X} \]

Given that \(\lambda_X = 1 \, \text{Å}\) (1 angstrom), we can express this as:

\[ \lambda_X = 1 \, \text{Å} = \frac{h}{m_X v_X} \]

### Step 2: Write the de Broglie wavelength for particle Y

For particle Y, we know:
- The mass of Y is \(25\%\) of the mass of X, so \(m_Y = 0.25 m_X\).
- The velocity of Y is \(75\%\) of the velocity of X, so \(v_Y = 0.75 v_X\).

Now, substituting these values into the de Broglie wavelength formula for particle Y:

\[ \lambda_Y = \frac{h}{m_Y v_Y} = \frac{h}{(0.25 m_X)(0.75 v_X)} \]

### Step 3: Simplify the expression for \(\lambda_Y\)

Now we can simplify the expression:

\[ \lambda_Y = \frac{h}{0.25 m_X \cdot 0.75 v_X} = \frac{h}{0.1875 m_X v_X} \]

### Step 4: Relate \(\lambda_Y\) to \(\lambda_X\)

We know from the expression for \(\lambda_X\):

\[ \lambda_X = \frac{h}{m_X v_X} \]

Now we can express \(\lambda_Y\) in terms of \(\lambda_X\):

\[ \lambda_Y = \frac{h}{0.1875 m_X v_X} = \frac{1}{0.1875} \cdot \lambda_X \]

Calculating \( \frac{1}{0.1875} \):

\[ \frac{1}{0.1875} = \frac{1}{\frac{3}{16}} = \frac{16}{3} \]

Thus,

\[ \lambda_Y = \frac{16}{3} \lambda_X \]

### Step 5: Substitute the value of \(\lambda_X\)

Since \(\lambda_X = 1 \, \text{Å}\):

\[ \lambda_Y = \frac{16}{3} \cdot 1 \, \text{Å} = \frac{16}{3} \, \text{Å} \approx 5.33 \, \text{Å} \]

### Final Answer

The de Broglie wavelength of particle Y is approximately \(5.33 \, \text{Å}\).

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