1.032 kg of aqueous, one molal methanol is added with 500 mL water . Calculate the molality of dilute solution.

Answer

To calculate the molality of the dilute solution, we can follow these steps:

### Step 1: Understand the given data
We have:
- 1.032 kg of a one molal methanol solution.
- 500 mL of water is added to this solution.

### Step 2: Calculate the mass of methanol in the solution
Since the solution is one molal, it means there is 1 mole of methanol in 1 kg of solvent (water).

The molar mass of methanol (CH₃OH) is calculated as follows:
- Carbon (C): 12 g/mol
- Hydrogen (H): 4 g/mol (3 from CH₃ and 1 from OH)
- Oxygen (O): 16 g/mol

Total molar mass of methanol = 12 + 4 + 16 = 32 g/mol.

Since we have 1 mole of methanol, the mass of methanol in the solution is 32 g or 0.032 kg.

### Step 3: Calculate the mass of the solvent (water) in the original solution
The total mass of the aqueous solution is 1.032 kg. Since this includes both the mass of methanol and the mass of water, we can find the mass of the solvent (water) as follows:

Mass of solvent (water) = Total mass of solution - Mass of methanol
= 1.032 kg - 0.032 kg
= 1.000 kg.

### Step 4: Add the mass of additional water
We are adding 500 mL of water to the solution. The density of water is approximately 1 kg/L, so 500 mL of water has a mass of:

Mass of additional water = 0.500 kg.

Now, we can find the total mass of the solvent:

Total mass of solvent = Mass of original solvent + Mass of additional water
= 1.000 kg + 0.500 kg
= 1.500 kg.

### Step 5: Calculate the molality of the dilute solution
Molality (m) is defined as the number of moles of solute per kilogram of solvent.

We have:
- Number of moles of methanol = 1 mole (from the original solution).
- Mass of solvent = 1.500 kg.

Now, we can calculate the molality:

Molality (m) = Number of moles of solute / Mass of solvent (kg)
= 1 mole / 1.500 kg
≈ 0.6667 mol/kg.

### Final Answer:
The molality of the dilute solution is approximately **0.667 mol/kg**.

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