which of the following has been arranged in order of decreasing freezing point?

Options

A. `0.05MKNO_(3)gt0.04MCaCl_(2)gt0.140M` sugar `gt0.075M CuSO_(4)`
B. `0.04M BaCl_(2)gt0.140M` sucrose `gt0.075M CuSO_(4)gt0.05M KNO_(3)`
C. `0.075M CuSO_(4)gt0.140M` sucrose `gt0.04M BaCl_(2)gt0.05M KNO_(3)`
D. `0.075M CuSO_(4)gt0.05M NaNO_(3)gt0.140M` sucrose `gt0.04M BaCl_(2)`

Answer

To determine which of the following compounds has been arranged in order of decreasing freezing point, we need to analyze the depression in freezing point (ΔTf) for each compound. The depression in freezing point is given by the formula:

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

Where:
- \( K_f \) is the cryoscopic constant (a property of the solvent),
- \( i \) is the van't Hoff factor (the number of particles the solute dissociates into),
- \( m \) is the molality of the solution.

### Step 1: Identify the Compounds and Their Concentrations
We have the following compounds with their respective concentrations:
1. KNO3 (0.05 M)
2. CaCl2 (0.04 M)
3. Sugar (0.140 M)
4. CuSO4 (0.075 M)

### Step 2: Calculate the Van't Hoff Factor (i) for Each Compound
- **KNO3**: Dissociates into K⁺ and NO3⁻, so \( i = 2 \).
- **CaCl2**: Dissociates into Ca²⁺ and 2 Cl⁻, so \( i = 3 \).
- **Sugar**: Does not dissociate, so \( i = 1 \).
- **CuSO4**: Dissociates into Cu²⁺ and SO4²⁻, so \( i = 2 \).

### Step 3: Calculate ΔTf for Each Compound
Now, we can calculate ΔTf for each compound using the formula:

1. **KNO3**:
\[ \Delta T_f = K_f \cdot 2 \cdot 0.05 = 0.1 K_f \]

2. **CaCl2**:
\[ \Delta T_f = K_f \cdot 3 \cdot 0.04 = 0.12 K_f \]

3. **Sugar**:
\[ \Delta T_f = K_f \cdot 1 \cdot 0.140 = 0.140 K_f \]

4. **CuSO4**:
\[ \Delta T_f = K_f \cdot 2 \cdot 0.075 = 0.15 K_f \]

### Step 4: Compare ΔTf Values
Now we compare the ΔTf values:
- KNO3: \( 0.1 K_f \)
- CaCl2: \( 0.12 K_f \)
- Sugar: \( 0.14 K_f \)
- CuSO4: \( 0.15 K_f \)

### Step 5: Determine the Order of Freezing Points
The freezing point decreases as ΔTf increases. Thus, the order of decreasing freezing point (highest to lowest) is:
1. KNO3 (lowest ΔTf = 0.1 K_f)
2. CaCl2 (ΔTf = 0.12 K_f)
3. Sugar (ΔTf = 0.14 K_f)
4. CuSO4 (highest ΔTf = 0.15 K_f)

### Final Answer
The order of decreasing freezing point is:
- KNO3 > CaCl2 > Sugar > CuSO4