Which will have largest `DeltaT_(b)`?

Options

A. `180`g glucose in `1`kg water
B. `342`g sucrose in `1,000`kg water
C. `18`g glucose in `100`kg water
D. `65`g urea in `1`kg water

Explanation

To determine which option will have the largest \(\Delta T_b\) (elevation in boiling point), we can follow these steps:

### Step 1: Understand the Formula
The elevation in boiling point is given by the formula:
\[ \Delta T_b = K_b \cdot m \]
where \(K_b\) is the ebullioscopic constant (which is constant for a given solvent) and \(m\) is the molality of the solution.

### Step 2: Recognize that the Solvent is the Same
In this question, all options use water as the solvent, and the weight of the solvent is the same (1 kg). Therefore, the molality \(m\) will depend solely on the number of moles of solute.

### Step 3: Calculate Moles of Solute for Each Option
We will calculate the number of moles of solute for each option using the formula:
\[ \text{Moles} = \frac{\text{Weight of solute (g)}}{\text{Molecular weight of solute (g/mol)}} \]

1. **Option 1: Glucose (180 g)**
- Molecular weight of glucose = 180 g/mol
- Moles = \(\frac{180 \text{ g}}{180 \text{ g/mol}} = 1 \text{ mole}\)

2. **Option 2: Solute (342 g)**
- Molecular weight of solute = 342 g/mol
- Moles = \(\frac{342 \text{ g}}{342 \text{ g/mol}} = 1 \text{ mole}\)

3. **Option 3: Glucose (18 g)**
- Molecular weight of glucose = 180 g/mol
- Moles = \(\frac{18 \text{ g}}{180 \text{ g/mol}} = 0.1 \text{ moles}\)

4. **Option 4: Urea (65 g)**
- Molecular weight of urea = 60.06 g/mol
- Moles = \(\frac{65 \text{ g}}{60.06 \text{ g/mol}} \approx 1.0833 \text{ moles}\)

### Step 4: Compare the Moles
Now we compare the number of moles calculated for each option:
- Option 1: 1 mole
- Option 2: 1 mole
- Option 3: 0.1 moles
- Option 4: 1.0833 moles

### Step 5: Determine the Largest \(\Delta T_b\)
Since \(\Delta T_b\) is directly proportional to the number of moles of solute, the option with the largest number of moles will have the largest \(\Delta T_b\).

From our calculations, Option 4 (65 g of urea) has the largest number of moles (approximately 1.0833 moles).

### Conclusion
Thus, the correct answer is:
**Option D: 65 grams of urea in 1 kg of water will have the largest \(\Delta T_b\).**

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