Work done by inflated balloon

1. The problem statement, all variables and given/known data



An initially deflated and flat balloon is connected by a valve to a storage tank containing helium gas at 1 MPa at ambient temperature of 20°C. The valve is opened and the balloon is inflated at constant pressure of 100 kPa (atmospheric pressure) until it becomes spherical at D₁ = 1 m. If the balloon is larger than this, the balloon material is stretched giving a pressure inside as:



P = P₀ + C (1-D₁/D) D₁/D



The balloon is slowly inflated to a final diameter of 4 m, at which point the pressure inside is 400 kPa. The temperature remains constant at 20°C. Determine the work done during the overall process.



2. Relevant equations



W = ∫PdV

PV = nRT



3. The attempt at a solution



I solved for the volumes of the ballon at D₁ = 1 m and at D₁ = 4 m to use for the definite integral. Then I the formula for P in terms of V using the ideal gas law:



P = nRT/V



At this point, I realize that I don't know the number of moles in the balloon and that I haven't used the expression given in the problem for P. I know how to evaluate the calculus, but I'm terrible at setting up the integrals. Anyone have a hint for me? Thanks!