Question

Solution A has a concentration of \(0.05 \mathrm{g} / \mathrm{mL}\) and solution \(\mathrm{B}\) has a concentration of 0.01 g/mL. You start with \(200 \mathrm{mL}\) of solution \(\mathrm{A},\) and pour in \(x\) millilitres of solution \(B\). a) Write an equation for the concentration, \(C(x),\) of the solution after \(x\) millilitres have been added. b) You need to make a solution with a concentration of 0.023 g/mL. How can you use your function equation to determine how many millilitres need to be added?

Short answer

C(x) = \frac{10 + 0.01x}{200 + x}; approximately 415 mL of Solution B is needed.

Step-by-step solution

- Determine the total amount of solute in Solution A

Calculate the amount of solute in the initial 200 mL of Solution A. Use the concentration formula: \text{concentration} = \frac{\text{mass of solute}}{\text{volume of solution}}. For Solution A, concentration is 0.05 g/mL, and volume is 200 mL. Therefore, the mass of solute is \[0.05 \text{ g/mL} \times 200 \text{ mL} = 10 \text{ g}\]

- Determine the amount of solute in Solution B

Calculate the amount of solute in x mL of Solution B, which has a concentration of 0.01 g/mL. Therefore, the amount of solute in Solution B is \[0.01 \text{ g/mL} \times x \text{ mL} = 0.01x \text{ g}\]

- Write the total volume and mass of the combined solution

When x mL of Solution B is added to 200 mL of Solution A, the total volume is \[200 \text{ mL} + x \text{ mL} = (200 + x) \text{ mL}\] The combined mass of the solute from both solutions is \[10 \text{ g} + 0.01x \text{ g}\]

- Write the equation for the concentration C(x)

The concentration of the combined solution, \(C(x)\), is given by the total mass of the solute divided by the total volume of the solution. Therefore, the equation is \[C(x) = \frac{10\text{ g} + 0.01x\text{ g}}{200 \text{ mL} + x \text{ mL}}\]

- Solve for x when the desired concentration is 0.023 g/mL

Set the concentration equation equal to 0.023 g/mL and solve for x: \[0.023 = \frac{10 + 0.01x}{200 + x}\] Cross-multiply to solve: \[0.023(200 + x) = 10 + 0.01x\] Expand and simplify: \[4.6 + 0.023x = 10 + 0.01x \0.013x = 5.4 \ x = \frac{5.4}{0.013} \ x \approx 415.38 \text{ mL}\]

Key concepts

Solute Mass Calculation

Understanding how to calculate the mass of a solute in a solution is crucial for various chemistry problems, including concentration calculations. Initially, we need to determine how much solute is present in each solution. For Solution A, with a concentration of 0.05 g/mL and a volume of 200 mL, the solute mass is calculated as follows: \[0.05 \text{ g/mL} \times 200 \text{ mL} = 10 \text{ g}\] Similarly, for Solution B, with a concentration of 0.01 g/mL and a chosen volume of x mL, the solute mass would be: \[0.01 \text{ g/mL} \times x \text{ mL} = 0.01x \text{ g} \] These straightforward calculations help in determining the total solute before mixing the solutions.

Volume Mixing

When you mix different solutions, the volumes add up. This is important for finding the combined volume once the solutions are mixed. Here, we start with 200 mL of Solution A and add x mL of Solution B. The total volume of the mixed solution would be: \[ 200 \text{ mL} + x \text{ mL} = (200 + x) \text{ mL} \] This calculation is vital because the concentration of a solution is dependent on both the total mass of the solute and the total volume of the solution.

Concentration Equations

We can derive an equation to find the concentration of the mixed solution by combining our previous calculations. The concentration of a solution is given by the mass of the solute divided by the volume of the solution. After pouring x mL of Solution B into Solution A, the concentration equation becomes: \[ C(x) = \frac{10 \text{ g} + 0.01x \text{ g}}{200 \text{ mL} + x \text{ mL}} \] This formula helps us understand how the concentration changes as the volume of Solution B increases or decreases.

Linear Equations

To determine the required volume of Solution B to achieve a certain concentration, we use the concentration equation and set it equal to the desired concentration. In this problem, we need a final concentration of 0.023 g/mL. The equation then becomes: \[ 0.023 = \frac{10 + 0.01x}{200 + x} \] Solving this step-by-step, we get: Cross-multiply: \[0.023(200 + x) = 10 + 0.01x \] Expand and simplify: \[4.6 + 0.023x = 10 + 0.01x \] Isolate x: \[0.013x = 5.4 \] Solve for x: \[x = \frac{5.4}{0.013} \] Therefore, \[ x \text{is approximately } 415.38 \text{ mL}\] This solution denotes how many milliliters of Solution B are needed to reach the desired concentration.