Problem 77 - University Physics with Modern Physics

Question

(a) Prove that when a particle with mass \(m\) and kinetic energy \(K\) collides with a stationary particle with mass \(M,\) the total kinetic energy \(K_{\mathrm{cm}}\) in the center-of-mass coordinate system (the energy available to cause reactions) is $$K_{\mathrm{cm}}=\frac{M}{M+m} K$$ Assume that the kinetic energies of the particles and nuclei are much lower than their rest energies. (b) If \(K_{\text { th }}\) is the minimum, or threshold, kinetic energy to cause an endoergic reaction to occur in the situation of part (a), show that $$K_{\mathrm{th}}=-\frac{M+m}{M} Q$$

Step-by-Step Solution

Answer

(a) \( K_{\mathrm{cm}} = \frac{M}{M+m} K \); (b) \( K_{\text{th}} = -\frac{M+m}{M} Q \).

Step 1: Understand Kinetic Energy in Center-of-Mass Frame

The total kinetic energy in the center-of-mass (CM) frame, \( K_{\mathrm{cm}} \), is defined as the kinetic energy of the system of particles as observed from the center-of-mass reference frame. When a particle with mass \( m \) and kinetic energy \( K \) collides with a stationary particle with mass \( M \), the velocities in the center-of-mass frame need to be adjusted accordingly.

Step 2: Calculate the Velocity of the Center of Mass

The velocity of the center of mass \( v_{\text{cm}} \) is given by the formula:\[v_{\text{cm}} = \frac{mv}{M+m}\]where \( v \) is the initial velocity of the particle of mass \( m \). This velocity accounts for the movement caused by the colliding particles in the CM frame.

Step 3: Calculate Kinetic Energy in Lab Frame and Transform to CM Frame

The total kinetic energy in the lab frame equals the initial kinetic energy, \( K = \frac{1}{2} mv^2 \). In the CM frame, the kinetic energy of the particles is changed, and it is given by:\[K_{\mathrm{cm}} = \frac{1}{2} (m + M) v_{\text{cm}}^2 + \frac{1}{2} \frac{Mm^2}{(M+m)^2} v^2\]Substituting \( v_{\text{cm}} \) from Step 2, the expression simplifies to \( K_{\mathrm{cm}} = \frac{M}{M+m} K \).

Step 4: Understand the Threshold Kinetic Energy Condition

\( K_{\text{th}} \) is the minimum energy required in the incident frame to initiate an endoergic reaction. This energy must compensate for the intrinsic energy requirement of the reaction, defined as the reaction's Q-value. The threshold condition relates the CM energy and the reaction's Q-value.

Step 5: Derive the Threshold Energy Equation

Using the energy conservation principle and endoergic condition, the negative Q-value, \( Q \), signifies energy absorption. Therefore, the threshold energy \( K_{\text{th}} \) is derived from setting the available energy in the CM frame equal to \(-Q\), leading to:\[K_{\mathrm{th}} = -\frac{(M+m)}{M}Q\]

Key Concepts

Kinetic Energy Transformation

Kinetic energy transformation is an essential concept when examining collisions between particles. Initially, a particle with mass \(m\) and kinetic energy \(K\) moves toward a stationary particle with mass \(M\).

• The kinetic energy in the lab frame is straightforward: \( K = \frac{1}{2}mv^2 \).
• Once we shift our perspective to the center-of-mass (CM) frame, the total kinetic energy in that frame changes.

The transformation of kinetic energy arises because the relative motion between the particles is different in the CM frame, which considers the movement of both particles. The total kinetic energy available to produce reactions, denoted as \(K_{\mathrm{cm}}\), becomes:\[ K_{\mathrm{cm}} = \frac{M}{M+m} K \] This formula tells us that only a portion of the initial kinetic energy in the lab frame is available for reactions when viewed from the CM frame.

Center-of-Mass Coordinate System

The center-of-mass (CM) coordinate system is a crucial reference frame in understanding particle impacts. This frame allows us to track the movement of composite systems of particles more intuitively. By definition:

• The CM coordinate system moves with the average velocity of the mass in the system.
• For a particle with mass \(m\) impacting a stationary particle \(M\), the velocity of the CM is:

\[ v_{\text{cm}} = \frac{mv}{M+m} \] This velocity is essential because it shifts our observation point to move with the system's center of gravity, simplifying the analysis of energy distribution. In this ‘neutral’ frame, we see how kinetic energy distributes between the particles during a collision, especially important for precision in collision and reaction calculations.

Threshold Kinetic Energy

Threshold kinetic energy, often denoted as \(K_{\text{th}}\), is the crucial amount of energy required to instigate a specific reaction. In collisions involving particles with low kinetic energies, achieving the necessary activation energy becomes vital.

• \(K_{\text{th}}\) is the minimum energy that must be present for an endoergic reaction process to happen.
• Such reactions need this energy to overcome inherent energetic barriers depicted by the reaction’s Q-value, which often signifies energy absorption.

The equation for \(K_{\text{th}}\) in a scenario with a negative Q-value (indicative of endoergicity) can be derived as:\[ K_{\mathrm{th}} = -\frac{M+m}{M}Q \] Understanding this relationship is key to knowing when a collision will have sufficient energy to be successful in initiating reactions.

Endoergic Reactions

Endoergic reactions are chemical or nuclear processes that require an input of external energy to proceed. Unlike exoergic reactions, which release energy, an endoergic reaction's energy profile includes:

• The necessity for more energy in the reactant state than is released in the product state.
• A positive change in energy, which means the reaction absorbs energy (represented as a negative Q-value).

When particles collide and form a reaction, the available kinetic energy in the system must meet this energy deficit for the reaction to occur. Therefore, understanding the concept of threshold kinetic energy is vital when analyzing such reactions. Endoergic reaction studies are important for fields like nuclear chemistry and physics, where energy considerations are essential for applications like energy production and materials synthesis.