As a consequence of the conservation of linear momentum in...

As a consequence of the conservation of linear momentum in the laboratory coordinate axis system, the compound system C that is formed in the reaction a $+\mathrm{X} \rightarrow$ $\mathrm{C} \rightarrow \mathrm{b}+\mathrm{Y}$ will move with a given velocity $v_{\mathrm{C}}$. Show that the kinetic energy needed for the particle a to induce an endothermic reaction ( $Q<0$ ) has a threshold value which can be expressed as $E_{\mathrm{thr}}=(-Q)\left(1+m_2 / m_{\mathrm{x}}\right)$. Discuss this result.

Key Concepts

Conservation of Linear Momentum

This principle states that the total momentum of a closed system remains constant if no external forces act on it. In the context of nuclear or particle reactions, it determines how the momentum is distributed among the reaction products, especially when a compound system is formed, leading to specific velocities of the different particles following the reaction.

Conservation of Energy

The law of conservation of energy requires that the total energy, including kinetic and potential (or binding) energy, remains constant in an isolated system. In nuclear reactions, it is applied by relating the kinetic energies of the reactants and products to the reaction Q-value, which represents the net energy change due to differences in binding energy.

Reaction Q-Value

The Q-value of a reaction is the net energy released or absorbed during the process. A negative Q-value indicates an endothermic reaction where energy must be provided for the reaction to occur, impacting the threshold kinetic energy required for the initiating particle to induce the reaction.

Threshold Energy in Endothermic Reactions

For endothermic reactions (where Q is negative), a minimum or threshold kinetic energy is necessary to overcome the energy deficit represented by -Q. This threshold energy ensures that sufficient energy is available for the reaction to proceed, taking into account the distribution of energy among the products as dictated by both conservation of momentum and energy.

Mass Ratios and Reaction Kinematics

The masses of the involved particles are crucial in determining the energy and momentum distribution during the reaction. In threshold energy calculations, the ratio of masses (such as m2/mx in the given expression) influences the kinetic energy required by the incident particle, highlighting the role of mass in factoring the necessary energy allocation between translational motion and reaction feasibility.

Transcript

00:01 For this question, we are trying to find the expression for the minimum kinetic energy that is required.

00:10 For the case when we have a mass m1 that is firing towards a stationary mass m2 to produce the final products m3.

00:23 M3 consists of all the masses of the final products.

00:26 And the minimum amounts of kinetic energy required for m1 for this entire reaction to occur would be the case when the final products all have minimum amounts of kinetic energy and this will imply that they must all be moving together at the same velocity so that the total kinetic energy is a minimum at the end so at this condition, we want to find what is the kinetic energy of our initial m1.

01:01 And to start off, we're going to use conservation of energy.

01:07 So conservation of energy tells us that the total initial energy would just be the kinetic energy of m1, plus the rest energy of m1, kinetic energy, plus rest energy of m1, plus rest energy of m2 because m2 is at rest so that's all the initial kinetic energy this is equal to the final energy of the total masses of our products and this is from the rest energy relationship sorry this is from the energy momentum relationship right where e square is equals to p square c square plus m square c4 so if you take the square roots we get the energy right so from here what is our p square c square right we don't know what is this but what we can do is to relate this p square c square to the initial momentum of the system right so the p square over here is the final momentum of the system by energy conservation this must also be the initial momentum of the system and the initial momentum only comes from mass m1 this is the only contributor of the initial momentum.

02:30 And this momentum expression can actually be derived from the energy momentum relationship of our mass m1, right, where it's total mass, sorry, total energy of m1.

02:46 This equals to p -square c -square plus m -square -c -4.

02:50 So this is energy momentum relationship again for m1.

02:58 And this p is the same p as this p, right, by conservation of momentum.

03:06 So what we can do is to replace this p -square -c -square with e1 -square minus m -1 -c -4.

03:17 Now after doing this substitution, we need to then substitute.

03:22 What is e1 right so e1 by conservation of energy is just the kinetic energy plus the rest energy very simple we can then try to substitute this in and then simplifying this equation right this plus this is equals to this and we can simplify that by squaring both sides so we're gonna square both sides to get rid of the square root.

03:56 So for the left -hand side, squaring the left -hand side first, this square.

04:02 This is the expression that we will get.

04:06 Right to be expanded out.

04:08 And then squaring the right -hand side, we just get rid of the square root.

04:12 So nothing much except for substituting in what is e1 square.

04:18 So e1 square would be this expression here squared...

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