<p>The thermodynamics of a Bose system with mutual interactions, in both the dilute and dense limits, is developed by means of cluster expansion and Mayer’s generating function formalism. This method allows for the derivation of the equation of state, accounting for deviations from ideal Bose gas behaviour due to inter particle interactions. The critical (saturation) density marking the onset of Bose Einstein condensation is determined by identifying the singular point of the series representing the grand partition function. By invoking Mayer’s convergence criterion, the phase boundary and other key thermodynamic parameters for a Bose gas with hard core repulsion are calculated. This formalism is applied to specific Bose Einstein condensate systems that have been realized in experiments, and it reveals a measurable increase in the saturation density compared to that of the ideal gas model, indicating a positive interaction-induced correction. Additionally, the pressure of the ground state obtained from this statistical approach aligns well with results independently derived using quantum field theoretic quantization techniques, affirming the consistency and reliability of the method.</p>