Consider an ideal gas contained in a volume V at temperature T. If all particles are identical the Grand canonical partition function can be calculated using $$Z_g(V,T,z) := \sum_{N=0}^\infty z^N Z_c(N,V,T)$$ where $z$ is the fugacity, and $$Z_c(N,V,T):= \frac{1}{N!h^{3N}} \int_\Gamma e^{- \beta H(X)} dX$$ is the canonical partition function.
For identical particles I can compute the Grand canonical partition function. Now I am asked about two different kind of particles with masses $m_2= 2m_1$ and energies of $$h_1(p) = \frac{p^2}{2m_1}$$ and $$h_2(p) = \frac{p^2}{2m_2} + \Delta$$ where $\Delta > 0$ is a constant. The fugacitys are given as $z_{1,2}=e^{\beta \mu_{1,2}}$, where $\mu_{1,2}$ are the corresponding chemical potentials. The task is to find the partition function.
I am sorry to admit, but I have absolutly no clue where to start. Since no potential is present the Hamiltionian should be $$H(X) = \sum_{i=1}^{N_1} \frac{p_i^2}{2m_1} + \sum_{i=1}^{N_2} \frac{p_j^2}{2m_2} + N_2 \Delta$$ where $N_1+N_2=N$ are the numbers of particles of the differen kind. First I don't understand where that $\Delta$ comes from? And I have no idea how to continue. Simply inserting $H$ in $Z_c$ does not give any usefull results.