The initial volume of the balloon is given by V1 = (4/3)πr1³ = (4/3)π(3cm)³ = 113.1 cm³. The final volume of the balloon is given by V2 = (4/3)πr2³ = (4/3)π(2cm)³ = 33.5 cm³.
According to the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. We can use this equation to find the number of moles of air in the balloon before and after the expansion.
n1 = (P1V1)/(RT1) = (740 mm)(0.1131 L)/(0.08206 L atm/mol K)(293 K) = 0.00438 mol
n2 = (P2V2)/(RT2) = (740 mm)(0.0335 L)/(0.08206 L atm/mol K)(293 K) = 0.00163 mol
The difference in the number of moles is the amount of air that needs to be added to the balloon to expand it to the final volume at the same temperature and pressure.
Δn = n2 - n1 = 0.00163 mol - 0.00438 mol = -0.00275 mol
Since the volume of the balloon is changing at constant temperature and pressure, we can assume that the amount of air added is at the same temperature and pressure. At NTP (0°C and 1 atm), the volume of one mole of gas is 22.4 L. Therefore, the volume of air required for the expansion is:
V = nRT/P = (-0.00275 mol)(0.08206 L atm/mol K)(293 K)/(1 atm) = -0.06 L
The negative sign indicates that the volume of air removed from the balloon is greater than the volume of air added. This is because the balloon is contracting as it is being filled with less air. Therefore, the volume of air required for the expansion is 0.06 L.