Subtopic: Partition Function & Thermodynamic Connection
Question Statement:
The canonical partition function of an ideal gas is
Q(T, V, N) = 1
N! [ V
(𝜆(T))3]
N
where T, V, N and 𝜆(T) denote temperature, volume, number of particles, and
thermal de Broglie wavelength, respectively. Let kB be the Boltzmann constant and
\mu be the chemical potential. Take ln(N!) = N ln(N) - N.
If the number density (
N
V) is 2.5 × 1025 m-3 at a temperature T,
then
e\mu/(kBT)
(𝜆(T))3 × 10-25 is _______ m-3 (rounded off to one decimal place).
Official Answer Key:
2.5 to 2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.59)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 5: Thermodynamics and Statistical Physics* | **Topic**: *Statistical Ensembles & Partition Functions* (`Partition Function & Thermodynamic Connection`)
- **Question Type**: **NAT** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Statistical Ensembles & Partition Functions** (`Partition Function & Thermodynamic Connection`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **2.5**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Numerical value: 2.5**.
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