How many grams of metallic sodium should be dissolved in 1000 ml of water to obtain a 20% alkali solution?

Given:
V (H2O) = 1000 ml
ω (NaOH) = 20%
ρ (H2O) = 1 g / ml

Find:
m (Na) -?

Solution:
1) 2Na + 2H2O => 2NaOH + H2 ↑;
2) Let m (Na) = (x) r;
3) n (Na) = m (Na) / M (Na) = (x / 23) mol;
4) n (NaOH) = n (Na) = (x / 23) mol;
5) m (NaOH) = n (NaOH) * M (NaOH) = (x / 23) * 40 = (40x / 23) g;
6) n (H2) = n (Na) = (x / 23) / 2 = (x / 46) mol;
7) m (H2) = n (H2) * M (H2) = (x / 46) * 2 = (x / 23) r;
8) m (H2O) = ρ (H2O) * V (H2O) = 1 * 1000 = 1000 g;
9) m solution (NaOH) = m (Na) + m (H2O) – m (H2) = (x + 1000 – (x / 23)) g;
10) m solution (NaOH) = m (NaOH) * 100% / ω (NaOH) = (40x / 23) * 100% / 20% = (200x / 23) g;
11) x + 1000 – (x / 23) = 200x / 23;
23x + 23000 – x = 200x;
178x = 23000;
x = 129.2;
m (Na) = x = 129.2 g.

Answer: The mass of Na is 129.2 g.