Quantum Physics

958 Chapter 29 Nuclear PhysicsThe total mass on the left side of the equation is the sum of the mass of2141H (2.014 102 u) and the mass of 7 N (14.003 074 u), which equals 16.017 176 u.Similarly, the mass on the right side of the equation is the sum of the mass of12 4(12.000 000 u) plus the mass of 2 He6 C(4.002 602 u), for a total of 16.002 602 u.Thus, the total mass before the reaction is greater than the total mass afterthe reaction. The mass difference in the reaction is equal to 16.017 176 u 16.002 602 u 0.014 574 u. This “lost” mass is converted to the kinetic energy ofthe nuclei present after the reaction. In energy units, 0.014 574 u is equivalent to13.576 MeV of kinetic energy carried away by the carbon and helium nuclei.The energy required to balance the equation is called the Q value of the reaction.In Equation 29.22, the Q value is 13.576 MeV. Nuclear reactions in which there is arelease of energy—that is, positive Q values—are said to be exothermic reactions.The energy balance sheet isn’t complete, however: We must also consider thekinetic energy of the incident particle before the collision. As an example, assumethat the deuteron in Equation 29.22 has a kinetic energy of 5 MeV. Adding this toour Q value, we find that the carbon and helium nuclei have a total kinetic energyof 18.576 MeV following the reaction.Now consider the reaction42 He 14 7 N : 17 8O 1 1H[29.23]Before the reaction, the total mass is the sum of the masses of the alpha particle andthe nitrogen nucleus: 4.002 602 u 14.003 074 u 18.005 676 u. After the reaction,the total mass is the sum of the masses of the oxygen nucleus and the proton:16.999 133 u 1.007 825 u 18.006 958 u. In this case, the total mass after the reactionis greater than the total mass before the reaction. The mass deficit is 0.001 282 u,equivalent to an energy deficit of 1.194 MeV. This deficit is expressed by the negativeQ value of the reaction, 1.194 MeV. Reactions with negative Q values are calledendothermic reactions. Such reactions won’t take place unless the incoming particlehas at least enough kinetic energy to overcome the energy deficit.At first it might appear that the reaction in Equation 29.23 can take place if theincoming alpha particle has a kinetic energy of 1.194 MeV. In practice, however,the alpha particle must have more energy than this. If it has an energy of only1.194 MeV, energy is conserved but careful analysis shows that momentum isn’t.This can be understood by recognizing that the incoming alpha particle has somemomentum before the reaction. However, if its kinetic energy is only 1.194 MeV,the products (oxygen and a proton) would be created with zero kinetic energy andthus zero momentum. It can be shown that in order to conserve both energy andmomentum, the incoming particle must have a minimum kinetic energy given byKE min 1 m M Q [29.24]where m is the mass of the incident particle, M is the mass of the target, and theabsolute value of the Q value is used. For the reaction given by Equation 29.23, wefind thatKE min 1 4.002 602 1.194 MeV 1.535 MeV14.003 074This minimum value of the kinetic energy of the incoming particle is called thethreshold energy. The nuclear reaction shown in Equation 29.23 won’t occur if theincoming alpha particle has a kinetic energy of less than 1.535 MeV, but can occurif its kinetic energy is equal to or greater than 1.535 MeV.Quick Quiz 29.5If the Q value of an endothermic reaction is 2.17 MeV, then the minimum kineticenergy needed in the reactant nuclei if the reaction is to occur must be (a) equalto 2.17 MeV, (b) greater than 2.17 MeV, (c) less than 2.17 MeV, or (d) exactly halfof 2.17 MeV.