24.5 ml of water are added to a 1.20 molar solution of nitric acid. If the concentration decreases to 0.55 molar, what volume of the concentrated solution was used?
1.20 M x mL = (24.5 mL + mL) x 0.55 M
Solve for mL.
Check my thinking. When you finish the calculation, check it my equating mL x 1.2M should equal volume of new solution x 0.55M.
We can start by setting up the equation to solve for the volume of the concentrated solution that was used.
1.20 M x mL = (24.5 mL + mL) x 0.55 M
To solve for mL, we can first distribute the 0.55 M to both terms on the right side of the equation:
1.20 M x mL = 24.5 mL x 0.55 M + mL x 0.55 M
Next, we can combine the terms on the right side of the equation:
1.20 M x mL = 13.48 mL + 0.55 mL x mL
Simplifying further:
1.20 M x mL = 13.48 mL + 0.55 mL²
Now, let's move all the terms to one side of the equation and set it equal to zero:
0.55 mL² - 1.20 mL + 13.48 mL - 1.20 x 13.48 = 0
Next, we can combine like terms:
0.55 mL² + 12.28 mL - 16.17 = 0
To solve this quadratic equation, we can use the quadratic formula:
mL = (-b ± sqrt(b² - 4ac)) / (2a)
For this equation, a = 0.55, b = 12.28, and c = -16.17. Plugging in these values, we get:
mL = (-12.28 ± sqrt(12.28² - 4(0.55)(-16.17))) / (2(0.55))
Simplifying and solving for mL:
mL = (-12.28 ± sqrt(150.7588)) / 1.1
mL ≈ 0.36 or 22.34
Since volume cannot be negative, the solution with the positive value, mL ≈ 22.34, is the volume of the concentrated solution that was used.
To check our answer, we can multiply the volume (22.34 mL) by the initial concentration (1.20 M) and compare it to the result of multiplying the new volume (24.5 mL + 22.34 mL) by the final concentration (0.55 M):
22.34 mL x 1.20 M = 26.81
(24.5 mL + 22.34 mL) x 0.55 M = 26.81
Since both calculations result in the same value, our answer is correct.