The (canonical) partition function of the blackbody radiation may be written as

\[
Q(V, T)=\prod_{\omega} Q_{1}(\omega, T)
\]

so that

\[
\ln Q(V, T)=\sum_{\omega} \ln Q_{1}(\omega, T) \approx \int_{0}^{\infty} \ln Q_{1}(\omega, T) g(\omega) d \omega
\]

here, \(Q_{1}(\omega, T)\) is the single-oscillator partition function given by equation (3.8.14) and \(g(\omega)\) is the density of states given by equation (7.3.2). Using this information, evaluate the Helmholtz free energy of the system and derive other thermodynamic properties such as the pressure \(P\) and the (thermal) energy density \(U / V\). Compare your results with the ones derived in Section 7.3 from the \(q\)-potential of the system.