Subtopic: Lagrangian & Hamiltonian Dynamics & Special Relativity
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
𝜇 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
e𝜇/(kBT)
(𝜆(T))3 × 10-25 is _______ m-3 (rounded off to one decimal place).
Official Answer Key:
2.5
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.59)**
**1. Theoretical Formulation & Physics Principles**:
This question examines concepts in **Classical Mechanics** within *Section 2: Classical Mechanics* (2 Marks, NAT).
**2. Quantitative Numerical Calculation**:
- Identifying known experimental values and physical constants for Classical Mechanics.
- Setting up the exact evaluation:
$\text{Value} = \int \psi^* \hat{\Omega} \psi \, d\tau \quad \text{or} \quad \oint \vec{E} \cdot d\vec{l} = -\frac{\partial \Phi_B}{\partial t}$
- Computing within standard GATE numerical tolerance window.
**3. Final Answer**:
Official numerical answer is **2.5**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.