Statistical Ensembles: Identifying Correct Statements
This section details the correct statements regarding statistical ensembles, fundamental concepts in thermodynamics and statistical mechanics.
Statement A Analysis
Statement A is correct. An ensemble is defined as a hypothetical collection of a vast number of identical systems (replicas). These systems share the same macroscopic properties but differ in their microscopic configurations (microstates).
- An ensemble represents imaginary replications of the system under study.
- It allows for statistical averaging over possible microstates consistent with macroscopic constraints.
Statement C Analysis
Statement C is correct. This statement relates to the ergodic hypothesis. It postulates that, for a system in equilibrium, the average value of a macroscopic property calculated over all possible microstates in the ensemble (ensemble average) is equal to the average value of that property calculated over a long time interval for a single system (time average).
- Ensemble average $\langle X \rangle_{ens}$ = Time average $\overline{X}_{time}$
- This equality is crucial for connecting theoretical calculations with experimental observations.
Statements B and D Analysis (Incorrect)
Statement B is incorrect because while systems in an ensemble are macroscopically identical (sharing properties like temperature, pressure, volume), they exist in different microstates. Each microstate represents a specific configuration of particles.
Statement D is incorrect. A canonical ensemble is specifically defined by systems with a fixed number of particles (N), fixed volume (V), and fixed temperature (T). Therefore, all systems within a canonical ensemble must have the same composition (i.e., the same number of particles, N).
- Canonical Ensemble: Fixed \{N, V, T\}.
- Composition is determined by N, which is constant across ensemble members.
Thus, statements A and C accurately describe properties related to statistical ensembles.
Important Questions from Statistical Thermodynamics
- The molar partition Function of the system is :
- An approximate partition function $Q(N,V,T)$ of a gas is given below.
$Q(N,V,T) = \frac{1}{N!} (\frac{2\pi mk_B T}{h^2})^{3N/2} (V-Nb)^N$
The equation of state(s) for this gas is/are
[Note: $b$ is a parameter independent of volume.] - The moment of inertia of a homonuclear diatomic molecule is $7.5 \times 10^{-45} \text{ kg m}^2$. Its rotational partition function at 500 K is ________.
- Consider $N$ particles at temperature $T$, pressure $P$, volume $V$ and chemical potential $\mu$ having energy $E$. The parameters that are kept constant for a canonical ensemble are
- For ortho-hydrogen, the nuclear wavefunction and the rotational quantum number, respectively, are
