Answer
(a) The first reaction has a higher Q value, which indicates that more energy is released in the reaction. This means that the first reaction is more exothermic than the second reaction. For an exothermic reaction, the minimum threshold energy required for the reaction to occur is lower than for an endothermic reaction. Therefore, the first reaction has less minimum threshold energy compared to the second reaction.
Answer: (3) less minimum threshold energy.
(b) The ratio of the minimum threshold energy for the first reaction to that of the second reaction can be calculated using the equation:
E = (Q + m_i - m_t)c^2 / (1 + m_i/m_t)
where E is the minimum threshold energy, Q is the Q value of the reaction, m_i is the mass of the incident particle, m_t is the mass of the target nucleus, and c is the speed of light.
Let's assume that the incident particle is a proton (mass = 1 atomic mass unit) and use subscripts 1 and 2 to represent the quantities for the first and second reaction, respectively.
We are given that:
m_t1 = 15m_i
m_t2 = 20m_i
Q1 = 3Q2
Using these values, we can calculate the ratio of the minimum threshold energy for the first reaction to that of the second reaction as:
E1/E2 = [(Q1 + m_i - m_t1)c^2 / (1 + m_i/m_t1)] / [(Q2 + m_i - m_t2)c^2 / (1 + m_i/m_t2)]
Substituting the given values and simplifying, we get:
E1/E2 = [3Q2 + m_i - 15m_i] / [Q2 + m_i - 20m_i]
E1/E2 = (2m_i + 3Q2) / (19m_i - Q2)
Therefore, the ratio of the minimum threshold energy for the first reaction to that of the second reaction is (2m_i + 3Q2) / (19m_i - Q2).
Note that this ratio is greater than 1, which means that the minimum threshold energy for the first reaction is higher than that for the second reaction. This is consistent with our answer to part (a), which indicated that the first reaction has less minimum threshold energy compared to the second reaction.
Work Step by Step
(a) The first reaction has a higher Q value, which indicates that more energy is released in the reaction. This means that the first reaction is more exothermic than the second reaction. For an exothermic reaction, the minimum threshold energy required for the reaction to occur is lower than for an endothermic reaction. Therefore, the first reaction has less minimum threshold energy compared to the second reaction.
Answer: (3) less minimum threshold energy.
(b) The ratio of the minimum threshold energy for the first reaction to that of the second reaction can be calculated using the equation:
E = (Q + m_i - m_t)c^2 / (1 + m_i/m_t)
where E is the minimum threshold energy, Q is the Q value of the reaction, m_i is the mass of the incident particle, m_t is the mass of the target nucleus, and c is the speed of light.
Let's assume that the incident particle is a proton (mass = 1 atomic mass unit) and use subscripts 1 and 2 to represent the quantities for the first and second reaction, respectively.
We are given that:
m_t1 = 15m_i
m_t2 = 20m_i
Q1 = 3Q2
Using these values, we can calculate the ratio of the minimum threshold energy for the first reaction to that of the second reaction as:
E1/E2 = [(Q1 + m_i - m_t1)c^2 / (1 + m_i/m_t1)] / [(Q2 + m_i - m_t2)c^2 / (1 + m_i/m_t2)]
Substituting the given values and simplifying, we get:
E1/E2 = [3Q2 + m_i - 15m_i] / [Q2 + m_i - 20m_i]
E1/E2 = (2m_i + 3Q2) / (19m_i - Q2)
Therefore, the ratio of the minimum threshold energy for the first reaction to that of the second reaction is (2m_i + 3Q2) / (19m_i - Q2).
Note that this ratio is greater than 1, which means that the minimum threshold energy for the first reaction is higher than that for the second reaction. This is consistent with our answer to part (a), which indicated that the first reaction has less minimum threshold energy compared to the second reaction.
