Final answer:
The question involves finding the partition function for an extreme relativistic ideal gas with energy ε = pc. It involves integrating over momentum states weighted by the Boltzmann factor e-ε/kT to find Z, and then raising Z to the power of N for N non-interacting particles.
Step-by-step explanation:
The student is asked to find the partition function for an extreme relativistic ideal gas consisting of non-interacting monatomic molecules. This scenario falls within the realm of statistical mechanics and quantum physics. The single-particle energy ε is given as ε = pc (where p is momentum and c is the speed of light). The number of states in the momentum range p to p+dp is given by 4πVp²dp. The partition function Z for one particle in the volume V at temperature T is calculated by integrating the Boltzmann factor e-ε/kT over all momentum states, where k is the Boltzmann constant. Since ε = pc and the energy states are 4πVp²dp, the partition function becomes:
Z = ∫0∞ 4πVp² e-pc/kT dp
This integral needs to be calculated to obtain the partition function for one particle. After finding Z for one particle, the partition function for all N particles, assuming they are non-interacting and distinguishable, would be ZN, as each particle contributes independently to the total partition function