# 4. In a plant, sulfuric acid is purchased with a concentration of 95%.    The specific gravity of the   95% sulfuric acid is 1.8358.     A solution of 40 % sulfuric acid is required in the digester.  The required volume of 40% acid solution is 1500 liters and its specific gravity is 1.3070.  .   The procedure requires the complete addition of water before the acid.  What is the required volume of water and acid to make the 40% solution. Hello!

The mass conservation law will help us in this problem. Denote `rho_1` the known density of 95% sulfuric acid, `rho_2` the known density of 40% sulfuric acid (the given numbers are in  `g/(cm^3)` ), `rho_w=1g/(cm^3)`  the (known) density of pure water. Let the unknown volumes of 95% acid and water be `V_a` and `V_w,` and the known volume of resulting solution `V_2.`

Then the mass before the mixing is  `V_w rho_w + V_a rho_1`  and the mass after the mixing is  `V_2 rho_2,` and they must be equal:

`V_w rho_w + V_a rho_1 =V_2 rho_2.`

The mass of pure sulfuric acid also remains the same, thus

`0.95 V_a rho_1 = 0.40 V_2 rho_2.`

From this equation we can find the required volume of 95% acid:  `V_a = 0.40/0.95 *V_2 *rho_2/rho_1.`

Substitute it into the first equation and obtain

`V_w rho_w +0.40/0.95 V_2 rho_2 =V_2 rho_2, or V_w = V_2rho_2/rho_w (1 - 0.40/0.95).`