The reaction energy (Q value)for a particular reaction is -2.4 MeV,and the reaction's threshold energy is 9.60 MeV.What is the ratio of the mass of the incident particle to the mass of the stationary target nucleus?
A) 0.75
B) 0.25
C) 3.0
D) 4.0
E) 5.0

Question

Quizplus · Accepted Answer

The threshold energy is the minimum energy required for the reaction to occur. In this case, the threshold energy is equal to the sum of the masses of the incident particle and the target nucleus multiplied by the speed of light squared (E = mc^2). Let the mass of the incident particle be denoted as "m" and the mass of the target nucleus be denoted as "M". Then we can write:

9.60 MeV = (m + M)c^2

Rearranging this equation, we can solve for m/M:

m/M = (9.60 MeV/c^2 - M)/c^2

The Q value of the reaction is the difference in mass between the reactants and the products, multiplied by the speed of light squared. In equation form:

Q = (Mi + Mj - Mk - Ml)c^2

where Mi and Mj are the initial masses and Mk and Ml are the final masses. We are told that the Q value is -2.4 MeV, so we can write:

-2.4 MeV = (m + M)c^2 - Mc^2

Simplifying:

-2.4 MeV = (m/M + 1 - 1)Mc^2

-2.4 MeV = (m/M)Mc^2

Solving for m/M:

m/M = -2.4 MeV/(-1.0 Mc^2)

m/M = 2.4 MeV/Mc^2

Now we can set the two expressions for m/M equal to each other and solve for M:

9.60 MeV/c^2 - M = 2.4 MeV/Mc^2

M = 9.60 MeV/c^2 - 2.4 MeV/Mc^2

M = (9.60 - 2.4)/c^2 MeV/Mc^2

M = 7.20/c^2 MeV/Mc^2

Now we can substitute this expression for M into the equation for the ratio of the masses:

m/M = (9.60 MeV/c^2 - M)/c^2

m/M = (9.60 - 7.20)/c^2 MeV/Mc^2

m/M = 2.40/c^2 MeV/Mc^2

The ratio of the mass of the incident particle to the mass of the stationary target nucleus is m/M, so the answer is:

m/M = 2.40/c^2 MeV/Mc^2

Note that we can simplify the expression for the ratio of the masses by noticing that 2.40/c^2 MeV/Mc^2 is equal to 1/0.416 MeV/Mc^2, which is equal to 2.40/1.0 MeV. Therefore:

m/M = 2.40/1.0 MeV

m/M = 2.40

This is closest to answer choice C, 3.0.

The Reaction Energy (Q Value)for a Particular Reaction Is -2