Electron, proton and `He^(++)` are moving with same K.E.Then order of de-broglie wavelengths are:

Options

A. `lambda_e lt lambda_p lt lambda_(He^(++))`
B. `lambda_p gt lambda_e gt lambda_(He^(++))`
C. `lambda_e gt lambda_(He^(++)) gt lambda_p`
D. `lambda_e gt lambda_p gt lambda_(He^(++))`

Answer

To solve the problem, we need to find the de Broglie wavelengths of an electron, a proton, and a helium ion (He²⁺) that are all moving with the same kinetic energy (K.E.). The de Broglie wavelength (λ) is given by the formula:

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

where \(h\) is the Planck constant and \(p\) is the momentum of the particle. The momentum \(p\) can be expressed in terms of kinetic energy \(K\) and mass \(m\) as follows:

\[ p = \sqrt{2mK} \]

Substituting this into the de Broglie wavelength formula gives:

\[ \lambda = \frac{h}{\sqrt{2mK}} \]

Since the kinetic energy \(K\) is the same for all three particles (electron, proton, and He²⁺), we can see that the de Broglie wavelength is inversely proportional to the square root of the mass:

\[ \lambda \propto \frac{1}{\sqrt{m}} \]

### Step 1: Identify the masses of the particles
- Mass of electron \(m_e \approx 9.1 \times 10^{-31} \, \text{kg}\)
- Mass of proton \(m_p \approx 1.6 \times 10^{-27} \, \text{kg}\)
- Mass of He²⁺ (Helium ion) \(m_{He^{++}} = 4 \times m_p \approx 4 \times 1.6 \times 10^{-27} \, \text{kg} = 6.4 \times 10^{-27} \, \text{kg}\)

### Step 2: Write the expressions for the de Broglie wavelengths
Using the relationship derived earlier, we can express the wavelengths as:

\[ \lambda_e \propto \frac{1}{\sqrt{m_e}}, \quad \lambda_p \propto \frac{1}{\sqrt{m_p}}, \quad \lambda_{He^{++}} \propto \frac{1}{\sqrt{m_{He^{++}}}} \]

### Step 3: Calculate the ratios of the wavelengths
Now we can find the ratios of the de Broglie wavelengths:

\[ \frac{\lambda_e}{\lambda_p} = \frac{\sqrt{m_p}}{\sqrt{m_e}}, \quad \frac{\lambda_e}{\lambda_{He^{++}}} = \frac{\sqrt{m_{He^{++}}}}{\sqrt{m_e}}, \quad \frac{\lambda_p}{\lambda_{He^{++}}} = \frac{\sqrt{m_{He^{++}}}}{\sqrt{m_p}} \]

Substituting the masses into the ratios:

1. For electron to proton:

\[ \frac{\lambda_e}{\lambda_p} = \frac{\sqrt{1.6 \times 10^{-27}}}{\sqrt{9.1 \times 10^{-31}}} \]

2. For electron to He²⁺:

\[ \frac{\lambda_e}{\lambda_{He^{++}}} = \frac{\sqrt{6.4 \times 10^{-27}}}{\sqrt{9.1 \times 10^{-31}}} \]

3. For proton to He²⁺:

\[ \frac{\lambda_p}{\lambda_{He^{++}}} = \frac{\sqrt{6.4 \times 10^{-27}}}{\sqrt{1.6 \times 10^{-27}}} \]

### Step 4: Calculate the numerical values
Calculating these ratios gives us:

1. \(\lambda_e : \lambda_p : \lambda_{He^{++}} = \sqrt{\frac{1.6 \times 10^{-27}}{9.1 \times 10^{-31}}} : 1 : \sqrt{\frac{1.6 \times 10^{-27}}{6.4 \times 10^{-27}}}\)

After calculating these values, we find:

\[ \lambda_e : \lambda_p : \lambda_{He^{++}} \approx 84 : 2 : 1 \]

### Final Order
Thus, the order of de Broglie wavelengths is:

\[ \lambda_e > \lambda_p > \lambda_{He^{++}} \]

### Conclusion
The correct option for the order of de Broglie wavelengths is:

**Electron > Proton > He²⁺**