What volume of `NH_(3)` gas at STP would be needed to prepare 100 ml of 2.5 molal (2.5 m) ammonium hydroxide solution?

Options

A. 5.6 lit
B. 0.056 lit
C. 11.2 lit
D. 0.56 lit

Explanation

To solve the problem of determining the volume of `NH3` gas at STP needed to prepare 100 ml of a 2.5 molal (2.5 m) ammonium hydroxide solution, we can follow these steps:

### Step 1: Understand the relationship between molality and moles
Molality (m) is defined as the number of moles of solute per kilogram of solvent. In this case, we need to find the number of moles of ammonium hydroxide (NH4OH) in the solution.

### Step 2: Calculate the mass of the solvent
Given that we have 100 ml of water, we can convert this volume into mass using the density of water, which is approximately 1 g/ml.

\[ \text{Mass of water} = \text{Density} \times \text{Volume} = 1 \, \text{g/ml} \times 100 \, \text{ml} = 100 \, \text{g} = 0.1 \, \text{kg} \]

### Step 3: Calculate the number of moles of ammonium hydroxide
Using the molality formula:

\[ \text{Molality} = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}} \]

We can rearrange this to find the moles of NH4OH:

\[ \text{Moles of NH4OH} = \text{Molality} \times \text{mass of solvent (kg)} = 2.5 \, \text{mol/kg} \times 0.1 \, \text{kg} = 0.25 \, \text{moles} \]

### Step 4: Relate moles of ammonium hydroxide to moles of ammonia
The reaction between ammonia (NH3) and water to form ammonium hydroxide (NH4OH) is a 1:1 ratio. Therefore, the moles of ammonia needed will be equal to the moles of ammonium hydroxide produced.

\[ \text{Moles of NH3 needed} = 0.25 \, \text{moles} \]

### Step 5: Calculate the volume of ammonia gas at STP
At standard temperature and pressure (STP), 1 mole of any gas occupies 22.4 liters. Therefore, we can calculate the volume of 0.25 moles of ammonia gas:

\[ \text{Volume of NH3} = \text{moles} \times \text{volume per mole} = 0.25 \, \text{moles} \times 22.4 \, \text{L/mole} = 5.6 \, \text{L} \]

### Final Answer
The volume of `NH3` gas at STP needed to prepare 100 ml of a 2.5 molal ammonium hydroxide solution is **5.6 liters**.

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