Here it is
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In the  a  part, you calculated mole fraction of water and nitrogen gas by taking the ratio of the partial pressure with P total
Now the increase in weight is 0.15 gram it means this is the amount of water absorbed by drying agent
Convert these grams into moles and that will be the moles' water
Calculate moles of nitrogen gas because you already know all the mole fractions
Now calculate the total number of moles by adding moles of  water and moles of nitrogen gas
Use the ideal gas equation
V = nRT/P
P n R T all are given
Get the volume