(ii) Three Distinguishable Particles Are Initially Sealed In The Right Side Of A Two-compartment Container.
Introduction to the System
Understanding the behavior of particles confined within a container is fundamental in statistical mechanics and thermodynamics. The scenario where three distinguishable particles are initially sealed in the right side of a two-compartment container provides a rich context for exploring concepts such as probability distributions, particle dynamics, and equilibrium processes. This setup serves as an ideal model for analyzing how particles distribute themselves over time when allowed to move freely between compartments, subject to physical constraints and thermal agitation.Physical Description of the System
Structure of the Two-Compartment Container
The container is divided into two sections—left and right—by an impermeable partition that initially prevents particles from moving between them. The compartments are assumed to be identical in volume, denoted as \( V \), and the entire system is isolated, meaning no particles or energy are exchanged with the surroundings initially. The initial condition specifies that all three distinguishable particles are confined within the right compartment.Characteristics of the Particles
The particles are considered distinguishable, meaning each particle has unique properties that can be identified individually. For example, they could be different types of molecules or labeled particles in an experimental setting. The distinguishability affects the counting of microstates and the calculation of probabilities during the analysis.Initial Conditions and State Variables
Initial Particle Distribution
At the outset:- All three particles are in the right compartment.
- The left compartment is empty.
- The particles are free to move within their compartment but cannot cross the partition in this initial state.
Initial Energy and Temperature
Typically, such problems assume a thermal environment at temperature \( T \), allowing particles to possess kinetic energy distributed according to the Maxwell-Boltzmann distribution. The initial energy states are constrained by the compartment's volume and the particles' thermal energies.Dynamics of Particle Movement
Removing the Partition and Allowing Movement
The key physical process involves removing or bypassing the partition, enabling particles to diffuse between the two compartments. The movement is driven by thermal agitation, and over time, the particles tend to distribute themselves randomly across the entire volume.Transition Probabilities and Random Walks
The particles' movement can be modeled as a stochastic process, where each particle has a probability per unit time of crossing from one compartment to the other. This is often described by a Markov process with transition rates depending on factors like temperature and the energy barrier (if any) at the partition.Statistical Analysis of the System
Microstates and Macrostates
- Microstate: A specific arrangement of particles with defined positions and energy states.
- Macrostate: Defined by the number of particles in each compartment, regardless of their individual identities.
Possible Distributions of Particles
Initially, all particles are in the right compartment:- Macrostate: (Right: 3 particles, Left: 0 particles)
- (3, 0): All in the right
- (2, 1): Two in the right, one in the left
- (1, 2): One in the right, two in the left
- (0, 3): All in the left
Probability Calculations and Equilibrium
Initial Probability Distribution
At the start, the probability that all particles are in the right compartment is 1, with other configurations having zero probability.Evolution to Equilibrium
Over time, the system tends toward an equilibrium where the particles are distributed according to the statistical weight of each macrostate. The equilibrium probability \( P_{n} \) that \( n \) particles are in the right compartment can be derived assuming equal likelihood of particles being in either compartment if the process is symmetric and ergodic.For distinguishable particles, the probability that exactly \( n \) particles are in the right compartment is:
\[
P(n) = \frac{\binom{3}{n}}{2^3} = \frac{\binom{3}{n}}{8}
\]
where \( \binom{3}{n} \) is the binomial coefficient.
Thus:
- \( P(3) = \frac{1}{8} \)
- \( P(2) = \frac{3}{8} \)
- \( P(1) = \frac{3}{8} \)
- \( P(0) = \frac{1}{8} \)
This distribution indicates that, at equilibrium, the particles are most likely to be found in configurations with two or one particle in the right compartment.
