The minimum frequency of a `gamma`-ray that causes a deutron to disintegrate into a poton and a neutron is `(m_(d)=2.0141 am u, m_p=1.0078 am u, m_n=1.0087 am u.)`.

Options

A. `2.7 xx 10^(20)Hz`
B. `5.4 xx10^(20)Hz`
C. `10.8 xx10^(20)Hz`
D. `21.6xx10^(20)Hz`

Explanation

To solve the problem of finding the minimum frequency of a gamma-ray that causes a deuteron to disintegrate into a proton and a neutron, we will follow these steps:

### Step 1: Understand the Reaction
The reaction we are considering is:
\[ \text{Deuteron} \rightarrow \text{Proton} + \text{Neutron} \]

### Step 2: Calculate the Mass Defect
The mass defect (\(\Delta m\)) is calculated as the difference between the mass of the initial particle (deuteron) and the total mass of the products (proton and neutron).

Given:
- Mass of deuteron, \(m_d = 2.0141 \, \text{amu}\)
- Mass of proton, \(m_p = 1.0078 \, \text{amu}\)
- Mass of neutron, \(m_n = 1.0087 \, \text{amu}\)

The mass defect is given by:
\[ \Delta m = m_d - (m_p + m_n) \]
Substituting the values:
\[ \Delta m = 2.0141 - (1.0078 + 1.0087) = 2.0141 - 2.0165 = -0.0024 \, \text{amu} \]

### Step 3: Convert Mass Defect to Energy
The energy equivalent of the mass defect can be calculated using Einstein's equation:
\[ E = \Delta m c^2 \]
Where \(c\) is the speed of light (\(c = 3 \times 10^8 \, \text{m/s}\)) and we need to convert the mass defect from atomic mass units to kilograms. The conversion factor is:
\[ 1 \, \text{amu} = 1.66 \times 10^{-27} \, \text{kg} \]
Thus,
\[ \Delta m = -0.0024 \, \text{amu} = -0.0024 \times 1.66 \times 10^{-27} \, \text{kg} = -3.984 \times 10^{-30} \, \text{kg} \]

Now substituting into the energy equation:
\[ E = -3.984 \times 10^{-30} \, \text{kg} \times (3 \times 10^8 \, \text{m/s})^2 \]
Calculating this gives:
\[ E = -3.984 \times 10^{-30} \times 9 \times 10^{16} = -3.5856 \times 10^{-13} \, \text{J} \]

### Step 4: Relate Energy to Frequency
The energy of the gamma-ray can also be expressed in terms of its frequency (\(E = h \nu\)), where \(h\) is Planck's constant (\(h = 6.626 \times 10^{-34} \, \text{J s}\)). Setting the two expressions for energy equal gives:
\[ h \nu = E \]
Thus,
\[ \nu = \frac{E}{h} = \frac{-3.5856 \times 10^{-13}}{6.626 \times 10^{-34}} \]
Calculating this yields:
\[ \nu = 5.4 \times 10^{20} \, \text{Hz} \]

### Conclusion
The minimum frequency of a gamma-ray that causes a deuteron to disintegrate into a proton and a neutron is:
\[ \nu \approx 5.4 \times 10^{20} \, \text{Hz} \]