Three solutions X,Y,Z of HCl are mixed to produce 100mL of 0.1M solution . The molarities of X,Y and Z are 0.07M, 0.12M and 0.15M respectively. What respective volumes of X,Y and Z should be mixed?

Options

A. `50mL, 25mL,25mL`
B. `20mL, 60mL,20mL`
C. `40mL,30mL,30mL`
D. `55mL,20mL,25mL`

Answer

To solve the problem of mixing three solutions of HCl (X, Y, and Z) to achieve a final volume of 100 mL with a molarity of 0.1 M, we can follow these steps:

### Step 1: Define Variables
Let:
- \( V_X \) = volume of solution X (in mL)
- \( V_Y \) = volume of solution Y (in mL)
- \( V_Z \) = volume of solution Z (in mL)

### Step 2: Set Up the Volume Equation
From the problem, we know that the total volume of the mixed solution should be 100 mL:
\[ V_X + V_Y + V_Z = 100 \, \text{mL} \tag{1} \]

### Step 3: Set Up the Molarity Equation
The total number of moles of HCl contributed by each solution can be expressed as:
- Moles from X = \( 0.07 \times V_X \)
- Moles from Y = \( 0.12 \times V_Y \)
- Moles from Z = \( 0.15 \times V_Z \)

The total moles in the final solution should equal the molarity times the volume:
\[ 0.07 V_X + 0.12 V_Y + 0.15 V_Z = 0.1 \times 100 \tag{2} \]

This simplifies to:
\[ 0.07 V_X + 0.12 V_Y + 0.15 V_Z = 10 \tag{3} \]

### Step 4: Solve the Equations
Now we have a system of equations:
1. \( V_X + V_Y + V_Z = 100 \)
2. \( 0.07 V_X + 0.12 V_Y + 0.15 V_Z = 10 \)

From equation (1), we can express \( V_Z \) in terms of \( V_X \) and \( V_Y \):
\[ V_Z = 100 - V_X - V_Y \tag{4} \]

Substituting equation (4) into equation (3):
\[ 0.07 V_X + 0.12 V_Y + 0.15(100 - V_X - V_Y) = 10 \]

Expanding this gives:
\[ 0.07 V_X + 0.12 V_Y + 15 - 0.15 V_X - 0.15 V_Y = 10 \]

Combining like terms:
\[ -0.08 V_X - 0.03 V_Y + 15 = 10 \]

Rearranging:
\[ -0.08 V_X - 0.03 V_Y = -5 \]
\[ 0.08 V_X + 0.03 V_Y = 5 \tag{5} \]

### Step 5: Solve for One Variable
Now we can express \( V_Y \) in terms of \( V_X \):
\[ 0.03 V_Y = 5 - 0.08 V_X \]
\[ V_Y = \frac{5 - 0.08 V_X}{0.03} \tag{6} \]

### Step 6: Substitute Back
Substituting equation (6) back into equation (1):
\[ V_X + \frac{5 - 0.08 V_X}{0.03} + V_Z = 100 \]

This will give us a linear equation in terms of \( V_X \) and \( V_Z \). However, for simplicity, we can test possible values for \( V_X \), \( V_Y \), and \( V_Z \) based on the molarities given.

### Step 7: Testing Values
We can test the options provided in the question:
1. **Option A**: \( V_X = 50 \, \text{mL}, V_Y = 25 \, \text{mL}, V_Z = 25 \, \text{mL} \)
2. **Option B**: \( V_X = 20 \, \text{mL}, V_Y = 60 \, \text{mL}, V_Z = 20 \, \text{mL} \)
3. **Option C**: \( V_X = 40 \, \text{mL}, V_Y = 30 \, \text{mL}, V_Z = 30 \, \text{mL} \)
4. **Option D**: \( V_X = 55 \, \text{mL}, V_Y = 20 \, \text{mL}, V_Z = 25 \, \text{mL} \)

Calculating the moles for each option and checking if they equal 10 moles will lead us to the correct answer.

### Final Answer
After testing the options, we find that:
- **Option D**: \( V_X = 55 \, \text{mL}, V_Y = 20 \, \text{mL}, V_Z = 25 \, \text{mL} \) is the correct combination.