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Sagot :
Let's solve this step-by-step. Given:
1. Stock solution concentration: [tex]\(50 \, \text{mg/mL}\)[/tex]
2. Final solution concentration: [tex]\(1\% = 0.01 \, \text{(fraction)}\)[/tex]
3. Final volume required: [tex]\(35 \, \text{mL}\)[/tex]
### Step 1: Determine the final concentration in terms of stock solution concentration
A 1% solution means [tex]\(1\)[/tex] gram (or [tex]\(1000\)[/tex] mg) of solute per [tex]\(100\)[/tex] mL of solution.
So, the final concentration in terms of actual mass is:
[tex]\[ 1\% = \frac{1000 \, \text{mg}}{100 \, \text{mL}} = 10 \, \text{mg/mL} \][/tex]
Since the volume is [tex]\(35\)[/tex] mL, we need a total of:
[tex]\[ 35 \, \text{mL} \times 10 \, \text{mg/mL} = 350 \, \text{mg} \][/tex]
### Step 2: Calculate the volume of stock solution needed
Using the formula [tex]\( C_1 V_1 = C_2 V_2 \)[/tex], where:
- [tex]\(C_1\)[/tex] is the stock concentration ([tex]\(50 \, \text{mg/mL}\)[/tex])
- [tex]\(V_1\)[/tex] is the volume of the stock solution we need to determine
- [tex]\(C_2\)[/tex] is the final concentration ([tex]\(10 \, \text{mg/mL}\)[/tex])
- [tex]\(V_2\)[/tex] is the final volume ([tex]\(35 \, \text{mL}\)[/tex])
Hence,
[tex]\[ 50 \, \text{mg/mL} \times V_1 = 10 \, \text{mg/mL} \times 35 \, \text{mL} \][/tex]
Solving for [tex]\(V_1\)[/tex]:
[tex]\[ V_1 = \frac{10 \, \text{mg/mL} \times 35 \, \text{mL}}{50 \, \text{mg/mL}} \][/tex]
[tex]\[ V_1 = \frac{350 \, \text{mg}}{50 \, \text{mg/mL}} \][/tex]
[tex]\[ V_1 = 7 \, \text{mL} \][/tex]
### Step 3: Calculate the volume of the diluent needed
The total final volume is [tex]\(35 \, \text{mL}\)[/tex], and we have determined that [tex]\(7 \, \text{mL}\)[/tex] of it will be the stock solution. Therefore, the volume of the diluent needed is:
[tex]\[ 35 \, \text{mL} - 7 \, \text{mL} = 28 \, \text{mL} \][/tex]
### Conclusion
The correct way to prepare 35 mL of a 1% solution from a 50 mg/mL stock solution is to add:
- [tex]\(7 \, \text{mL}\)[/tex] of the stock solution to [tex]\(28 \, \text{mL}\)[/tex] of diluent.
Therefore, the correct option is:
Add 7 mL of stock solution to 28 mL of diluent.
Given the provided options:
- Add 7 mL of stock solution to 28 mL of diluent.
- Add 17 mL of stock solution to 18 mL of diluent.
- Add 17 mL of stock solution to 35 mL of diluent.
- Add 7 mL of stock solution to 35 mL of diluent.
Since none of these match, if the calculated solution is indeed correct but isn't an available option, there might be an error or miscommunication in the options presented.
1. Stock solution concentration: [tex]\(50 \, \text{mg/mL}\)[/tex]
2. Final solution concentration: [tex]\(1\% = 0.01 \, \text{(fraction)}\)[/tex]
3. Final volume required: [tex]\(35 \, \text{mL}\)[/tex]
### Step 1: Determine the final concentration in terms of stock solution concentration
A 1% solution means [tex]\(1\)[/tex] gram (or [tex]\(1000\)[/tex] mg) of solute per [tex]\(100\)[/tex] mL of solution.
So, the final concentration in terms of actual mass is:
[tex]\[ 1\% = \frac{1000 \, \text{mg}}{100 \, \text{mL}} = 10 \, \text{mg/mL} \][/tex]
Since the volume is [tex]\(35\)[/tex] mL, we need a total of:
[tex]\[ 35 \, \text{mL} \times 10 \, \text{mg/mL} = 350 \, \text{mg} \][/tex]
### Step 2: Calculate the volume of stock solution needed
Using the formula [tex]\( C_1 V_1 = C_2 V_2 \)[/tex], where:
- [tex]\(C_1\)[/tex] is the stock concentration ([tex]\(50 \, \text{mg/mL}\)[/tex])
- [tex]\(V_1\)[/tex] is the volume of the stock solution we need to determine
- [tex]\(C_2\)[/tex] is the final concentration ([tex]\(10 \, \text{mg/mL}\)[/tex])
- [tex]\(V_2\)[/tex] is the final volume ([tex]\(35 \, \text{mL}\)[/tex])
Hence,
[tex]\[ 50 \, \text{mg/mL} \times V_1 = 10 \, \text{mg/mL} \times 35 \, \text{mL} \][/tex]
Solving for [tex]\(V_1\)[/tex]:
[tex]\[ V_1 = \frac{10 \, \text{mg/mL} \times 35 \, \text{mL}}{50 \, \text{mg/mL}} \][/tex]
[tex]\[ V_1 = \frac{350 \, \text{mg}}{50 \, \text{mg/mL}} \][/tex]
[tex]\[ V_1 = 7 \, \text{mL} \][/tex]
### Step 3: Calculate the volume of the diluent needed
The total final volume is [tex]\(35 \, \text{mL}\)[/tex], and we have determined that [tex]\(7 \, \text{mL}\)[/tex] of it will be the stock solution. Therefore, the volume of the diluent needed is:
[tex]\[ 35 \, \text{mL} - 7 \, \text{mL} = 28 \, \text{mL} \][/tex]
### Conclusion
The correct way to prepare 35 mL of a 1% solution from a 50 mg/mL stock solution is to add:
- [tex]\(7 \, \text{mL}\)[/tex] of the stock solution to [tex]\(28 \, \text{mL}\)[/tex] of diluent.
Therefore, the correct option is:
Add 7 mL of stock solution to 28 mL of diluent.
Given the provided options:
- Add 7 mL of stock solution to 28 mL of diluent.
- Add 17 mL of stock solution to 18 mL of diluent.
- Add 17 mL of stock solution to 35 mL of diluent.
- Add 7 mL of stock solution to 35 mL of diluent.
Since none of these match, if the calculated solution is indeed correct but isn't an available option, there might be an error or miscommunication in the options presented.
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