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In an alpha-decay, the kinetic energy of the alpha particle is 48 MeV and the Q-value of the reaction is 50 MeV. The mass number of the parent nucleus is 20n. Find n. (Assume the daughter nucleus is in the ground state.)

1. 5
2. 3
3. 7
4. 1

Correct answer: 5

Solution

In alpha decay, momentum is conserved: m_alpha * v_alpha = M_daughter * V_daughter. Kinetic energies: KE_alpha = (1/2)*m_alpha*v_alpha², KE_daughter = (1/2)*M_daughter*V_daughter². Since momenta are equal: KE_alpha/KE_daughter = M_daughter/m_alpha. Total Q = KE_alpha + KE_daughter = KE_alpha(1 + m_alpha/M_daughter). So KE_alpha = Q * M_daughter/(M_alpha + M_daughter). Plugging in: 48 = 50 * A_d/(A_d + 4), where A_d is mass number of daughter. Solving: 48(A_d + 4) = 50*A_d, 48*A_d + 192 = 50*A_d, A_d = 96. Parent mass number = 96 + 4 = 100 = 20n, so n = 5.

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