Question

The minimum frequency of a \[\gamma - ray\] that causes a deuteron to disintegrate into a proton and a neutron is $({m_d} = 2.0141amu,{m_p} = 1.0078amu,{m_n} = 1.0087)$.
A. \[2.7 \times {10^{20}}Hz\]
B. \[5.4 \times {10^{20}}Hz\]
C. \[10.8 \times {10^{20}}Hz\]
D. \[21.6 \times {10^{20}}Hz\]

Answer 1

Hint: Using the formulae of energy \[E = h\nu \]and \[E = m{c^2}\] and equating them by putting values to get the required answer. \[\vartriangle m\]shows the change in mass of the particle. amu is abbreviated as atomic mass unit and uses as unit of mass of an atom.

 Complete step-by-step solution:
Let us assume that the deuteron is at rest and the neutron and proton are created at rest. We have to disintegrate deuteron into proton and neutron. We can write it as:
\[D \to n + p\], where \[D\]is a deuteron, \[n\]is a neutron and \[p\]is a proton.
Now, change in mass \[\vartriangle m = {m_p} + {m_n} - {m_d}\] Putting given values in above equation, we get,
\[ \Rightarrow \vartriangle m = 1.0078 + 1.0087 - 2.0141\]
\[ \Rightarrow \vartriangle m = 2.4 \times {10^{ - 3}}\]amu
Energy required, \[E = m{c^2}\]
\[ \Rightarrow \dfrac{E}{{\vartriangle m}} = 931MeV\]
Putting value of \[\vartriangle m\],
\[ \Rightarrow E\]\[ = 931(2.4 \times {10^{ - 3}})\]
On further solving, we get
\[ \Rightarrow E = 2.2344MeV\]-------------(i)
We also know, \[E = h\nu \]-------------(ii) here \[h\]is planck's constant and $\nu $is frequency of light.
From equation (i) and (ii),we get \[ \Rightarrow h\nu \]\[ = 2.2344MeV\]
Putting planck's constant value and solving,
\[ \Rightarrow \nu = 5.4 \times {10^{20}}Hz\]
Hence, the minimum frequency of a \[\gamma - ray\] that causes a deuteron to disintegrate into a proton and a neutron is \[5.4 \times {10^{20}}Hz\].
Therefore option B is correct.

Note:- Frequency is always measured in hertz $(Hz)$. Remember the value of planck's constant is \[4.1357 \times {10^{ - 15}}eV.s\]. Take precautions during decimal calculations and try to get more efficient value.