What mass of metallic sodium must be dissolved in 200 ml of 10% sodium hydroxide solution

What mass of metallic sodium must be dissolved in 200 ml of 10% sodium hydroxide solution (p = 1.1 g / ml) so that the concentration of the substance in the solution increases to 30%

Given:
V1 solution (NaOH) = 200 g
ω1 (NaOH) = 10%
ρ1 solution (NaOH) = 1.1 g / ml
ω3 (NaOH) = 30%

Find:
m (Na) -?

Solution:
1) 2Na + 2H2O => 2NaOH + H2 ↑;
2) m1 solution (NaOH) = ρ1 solution (NaOH) * V1 solution (NaOH) = 1.1 * 200 = 220 g;
3) m1 (NaOH) = ω1 (NaOH) * m1 solution (NaOH) / 100% = 10% * 220/100% = 22 g;
4) Let m (Na) = (x) r;
5) n (Na) = m (Na) / M (Na) = (x / 23) mol;
6) n2 (NaOH) = n (Na) = (x / 23) mol;
7) m2 (NaOH) = n2 (NaOH) * M (NaOH) = (40x / 23) g;
8) n (H2) = n (Na) / 2 = ((x / 23) / 2) mol;
9) m (H2) = n (H2) * M (H2) = ((x / 23) / 2) * 2 = (x / 23) r;
10) m3 (NaOH) = m1 (NaOH) + m2 (NaOH) = (22 + (40x / 23)) g;
11) m3 solution (NaOH) = m1 solution (NaOH) + m (Na) – m (H2) = (220 + x – (x / 23)) g;
12) ω3 (NaOH) = m3 (NaOH) * 100% / m3 solution (NaOH);
30% = (22 + (40x / 23)) * 100% / (220 + x – (x / 23));
0.3 = (22 + (40x / 23)) / (220 + x – (x / 23));
22 + (40x / 23) = 66 + 0.3x – (0.3x / 23);
506 + 40x = 1518 + 6.9 – 0.3x;
33.4x = 1012;
x = 30.3
m (Na) = x = 30.3 g.

Answer: The mass of Na is 30.3 g.