The freezing point of a diluted milk sample is found to be`-0.2^(@)C`, while it should have been `-0.5^(@)C` for pure milk. How much water has been added to pure milk to make the diluted sample?

Options

A. 2 cups of water to 3 cups of pure milk
B. 1 cup of water to 3 cups of pure milk
C. 3 cups of water to 2 cups of pure milk
D. 1 cup of water to 2 cups of pure milk

Explanation

To solve the problem of how much water has been added to pure milk to create a diluted sample, we will use the concept of freezing point depression. The freezing point depression can be described by the formula:

\[ \Delta T_f = i \cdot K_f \cdot m \]

Where:
- \(\Delta T_f\) is the change in freezing point,
- \(i\) is the van 't Hoff factor (which is 1 for non-electrolytes like milk),
- \(K_f\) is the freezing point depression constant,
- \(m\) is the molality of the solution.

### Step 1: Identify the freezing points
- The freezing point of pure milk is \(-0.5^\circ C\).
- The freezing point of the diluted milk sample is \(-0.2^\circ C\).

### Step 2: Calculate the change in freezing point
\[ \Delta T_f = T_f^{pure} - T_f^{diluted} = (-0.5) - (-0.2) = -0.3^\circ C \]

### Step 3: Set up the equations for molality
Let \(w_1\) be the weight of pure milk and \(w_2\) be the weight of water added. The molality \(m\) can be expressed as:

For diluted milk:
\[ \Delta T_f^{diluted} = K_f \cdot \left(\frac{w_1}{M_{milk} \cdot \frac{w_2}{1000}}\right) \]

For pure milk:
\[ \Delta T_f^{pure} = K_f \cdot \left(\frac{w_1}{M_{milk} \cdot \frac{0}{1000}}\right) \quad \text{(since there is no water)} \]

### Step 4: Substitute the known values
Since we don't know \(K_f\) and \(M_{milk}\), we can set up a ratio of the two equations:

\[ \frac{\Delta T_f^{diluted}}{\Delta T_f^{pure}} = \frac{w_2}{w_1} \]

Substituting the values we have:
\[ \frac{-0.3}{-0.5} = \frac{w_2}{w_1} \]

### Step 5: Solve for the ratio of weights
\[ \frac{0.3}{0.5} = \frac{w_2}{w_1} \implies \frac{3}{5} = \frac{w_2}{w_1} \]

This means that for every 5 parts of pure milk, we need to add 3 parts of water.

### Step 6: Conclusion
If we assume \(w_1 = 2\) (2 cups of pure milk), then:
\[ w_2 = \frac{3}{5} \cdot 2 = \frac{6}{5} = 1.2 \text{ cups of water} \]

Thus, to make the diluted sample, we need to add 1.2 cups of water to 2 cups of pure milk.

### Final Answer
To create the diluted milk sample, **1.2 cups of water** should be added to **2 cups of pure milk**.

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