The densities of three solutions with known concentrations ( 20% , 50%, and 70%), and one unknown solution were determined. If the densities of 20%, 50% and 70% solutions are 0.69 and 0.58 and 0.49 g/mL, respectively, and the denisty of the unknown solution is 0.50 g/mL, what is the approximate concentration of the unknown solution? Hint: try to think of a range of two values between which the concentration of the unknown falls.
68% or 67.8%
To find the approximate concentration of the unknown solution, we can use the relationship between density and concentration.
Let's denote the density of the 20%, 50%, and 70% solutions as D20, D50, and D70, respectively. We also know the density of the unknown solution, which is Dunknown.
According to the given information:
D20 = 0.69 g/mL
D50 = 0.58 g/mL
D70 = 0.49 g/mL
Dunknown = 0.50 g/mL
We can now set up a range of two values for the concentration of the unknown solution.
Lower Bound:
Using the relationship between density and concentration, we can write:
Cunknown = (Dunknown - D20) / (D70 - D20) * (C70 - C20) + C20
Substituting the known values:
Cunknown_lower_bound = (0.50 - 0.69) / (0.49 - 0.69) * (70% - 20%) + 20%
Cunknown_lower_bound = -0.19 / -0.20 * 50% + 20%
Cunknown_lower_bound = 0.95 + 20%
Cunknown_lower_bound = 20.95%
Upper Bound:
Similarly, using the same relationship, we can calculate the upper bound for the concentration:
Cunknown_upper_bound = (0.50 - 0.49) / (0.58 - 0.49) * (70% - 50%) + 50%
Cunknown_upper_bound = 0.01 / 0.09 * 20% + 50%
Cunknown_upper_bound = 0.111 * 50% + 50%
Cunknown_upper_bound = 5.55% + 50%
Cunknown_upper_bound = 55.55%
Therefore, the approximate concentration of the unknown solution falls within the range of 20.95% and 55.55%.
To determine the approximate concentration of the unknown solution, we can use a simple linear interpolation method. Since we have the known densities and concentrations of the 20%, 50%, and 70% solutions, we can create a linear relationship between concentration and density.
First, let's calculate the density range between the 20% and 50% solutions and the density range between the 50% and 70% solutions:
Density range between 20% and 50% solutions:
Density difference = Density of 50% solution - Density of 20% solution
Density difference = 0.58 g/mL - 0.69 g/mL
Density difference = -0.11 g/mL
Density range between 50% and 70% solutions:
Density difference = Density of 70% solution - Density of 50% solution
Density difference = 0.49 g/mL - 0.58 g/mL
Density difference = -0.09 g/mL
Now, we can calculate the concentration range between the 20% and 50% solutions and the concentration range between the 50% and 70% solutions:
Concentration range between 20% and 50% solutions:
Concentration difference = Density difference / Density difference between the two solutions
Concentration difference = -0.11 g/mL / (-0.11 g/mL)
Concentration difference = 1
Concentration range between 50% and 70% solutions:
Concentration difference = Density difference / Density difference between the two solutions
Concentration difference = -0.09 g/mL / (-0.09 g/mL)
Concentration difference = 1
Since the concentration difference is 1 for both ranges, we can infer that the concentration range for the unknown solution is also 1, meaning the unknown solution is either 50% or very close to 50% concentration.
Thus, the approximate concentration of the unknown solution is 50%.
It would be easier to explain if we could draw diagrams on this forum but alas, we can't.
The density of the unknown falls between 70% and 50% with the answer VERY close to the 70% known (0.49 vs 0.50).
Therefore, the difference in density of 0.50-0.49 = 0.01 while the difference between 0.58 and 0.49 (= 0.09) represents 20%. You want to know what part of 20% is that 0.01.Therefore,
20 x (0.01/0.09) = ? and
70%-? = xx.
Note: You could think of it the other way; i.e., how far is it from 50%? That would be 20% x (0.08/0.09) and add that to 50%.