How many grams of diphosphorus trioxide, #P_2O3#, are required to produce 10.2 moles of phosphorous acid, #H_3PO_3# in the reaction #P_2O_3 + 3H_2O -> 2H_3PO_3#?

Balanced Equation
#"P"_2"O"_3 + "3H"_2"O"##rarr##"2H"_3"PO"_3"#

First use the balanced equation to determine the mole ratio between #"H"_3"PO"_3"# and #"P"_2"O"_3"#. This ratio will be used to determine the moles #"P"_2"O"_3"# required to produce #10.2# moles #"H"_3"PO"_3"#.

Mole Ratio Between #"P"_2"O"_3"# and #"H"_3"PO"_3"# from the balanced equation.

#(1"mol P"_2"O"_3)/(2"mol H"_3"PO"_3)# and #(2"mol H"_3"PO"_3)/(1"mol P"_2"O"_3)#

Multiply the moles #"H"_3"PO"_3"# by the molar mass that cancels #"H"_3"PO"_3"# and leaves #"P"_2"O"_3"#

#10.2color(red)cancel(color(black)("mol H"_3"PO"_3))xx(1"mol P"_2"O"_3)/(2color(red)cancel(color(black)("mol H"_3"PO"_3)))="5.10 mol P"_2"O"_3#

Now that the moles #"P"_2"O"_3"# required to produce #"10.2 mol H"_3"PO"_3"# are known, multiply the number of moles by its molar mass, #"109.945 g/mol"#. https://www.ncbi.nlm.nih.gov/pccompound?term=P2O3 This will give the mass in grams needed for #"P"_2"O"_3"# to produce #"10.2 mol H"_3"PO"_3"#.

#5.10color(red)cancel(color(black)("mol P"_2"O"_3))xx(109.945"g P"_2"O"_3)/(1color(red)cancel(color(black)("mol P"_2"O"_3)))="561 g P"_2"O"_3# rounded to three significant figures