Question

The minimum frequency of photon required to break a particle of mass $15.348$ amu into $4$ particles is __________ kHz.
[Mass of He nucleus $=4.002$ amu, $1$ amu $=1.66\times10^{-27}$ kg, $h=6.6\times10^{-34}$ J$\cdot$s and $c=3\times10^8$ m/s]

• A. $9\times10^{19}$
• B. $9\times10^{20}$
• C. $14.94\times10^{20}$
• D. $14.94\times10^{19}$

Hint:

For nuclear reactions, required photon energy is calculated using mass defect: $E=\Delta mc^2$.

Answer 1

The Correct Option is D

To find the minimum frequency of a photon required to break a particle of mass 15.348 amu into 4 particles, we need to apply the concept of energy conservation and Einstein's mass-energy equivalence principle, which is given by the equation:

\(E = mc^2\)

where \(E\) is the energy required to break the particle, \(m\) is the mass difference, and \(c\) is the speed of light.

Given the problem conditions:

• Mass of original particle \(= 15.348\) amu
• Mass of each of the 4 particles \(= 4.002\) amu (since they are helium nuclei)

The total mass of the products \(= 4 \times 4.002 = 16.008 \text{ amu}\)

The mass defect (loss) \(\Delta m = 16.008 \text{ amu} - 15.348 \text{ amu} = 0.660 \text{ amu}\)

Converting amu to kg:

\(0.660 \text{ amu} = 0.660 \times 1.66 \times 10^{-27} \text{ kg} = 1.0956 \times 10^{-27} \text{ kg}\)

The energy required is:

\(E = \Delta m c^2 = 1.0956 \times 10^{-27} \times (3 \times 10^8)^2 \text{ J}\)

Calculating this gives:

\(E = 1.0956 \times 10^{-27} \times 9 \times 10^{16} = 9.8604 \times 10^{-11} \text{ J}\)

To find the frequency of the photon, use the equation:

\(E = h \nu\)

where \(h = 6.6 \times 10^{-34} \text{ J}\cdot\text{s}\) is Planck's constant, and \(\nu\) is the frequency.

Rearranging for frequency:

\(\nu = \frac{E}{h} = \frac{9.8604 \times 10^{-11}}{6.6 \times 10^{-34}} \text{ Hz}\)

Calculating this gives:

\(\nu = 1.49397 \times 10^{23} \text{ Hz}\)

Convert Hz to kHz:

\(\nu = 1.49397 \times 10^{20} \text{ kHz}\)

Rounding off, we get:

\(\nu \approx 14.94 \times 10^{19} \text{ kHz}\)

Thus, the correct answer is 14.94 × 10¹⁹ kHz.