Aqueous solution of $ 0.5 m H_2So_4 is more concentrated then 0.5 m H_2So_4 $ solution; then what will be the possible density of that solution ?
If x molar solution of any substance is more concentrated than x molul solution so its molarity value is less than the molality value
$ \therefore { molarity \over molarity } \lt 1 $
$$ Now { molarity \over molarity } = { 1000 \times w \over M \times V } \times { M \times W_0 \over 1000 \times W} $$
$$ \therefore { Molarity \over Molarity } = { w_0 gm \over V ml } \therefore { w_0 \over V} \lt {gm \over ml} $$
$$ Now { W_o \over V } = { W_0 w - w \over V} = { w_0 + W \over V} - { w \over v } = density of soln ( d { gm \over ml } ) - { w \over v } $$
$$ \therefore { W_0 gm \over V ml } = d { gm \over ml } + { molarity \times M \over 1000} $$
( M = mol.mass of solute )
$$ \therefore d { gm \over ml} + { molarity \times M \over 1000} \lt 1 {gm \over ml } \therefore d { gm \over ml } \lt 1 { gm \over ml } + { molarity \times M \over 1000 } $$
$$ Here for 0.5 M H_2 SO_4 aqueous solution d \lt ( 1 + { 0.5 \times 98 \over 1000 } ) { gm \over ml } \therefore d \lt 1.049 { gm \over ml } $$