If `10^(-4) dm^(3)` of water is introduced into a `1.0 dm^(3)` flask to `300K` how many moles of water are in the vapour phase when equilibrium is established ? (Given vapour pressure of `H_(2)O` at `300K` is `3170Pa R =8.314 JK^(-1) mol^(-1))` .

Options

A. `5.56 xx 10^(-3) mol e`
B. `1.53 xx 10^(-3) mol e`
C. `4.46 xx 10^(-3) mol e`
D. `1.27 xx 10^(-3) mol e`

Explanation

To solve the problem step by step, we will use the ideal gas law and the given data.

### Step 1: Understand the given data
- Volume of water introduced: \( V_{water} = 10^{-4} \, \text{dm}^3 \)
- Volume of the flask: \( V_{flask} = 1.0 \, \text{dm}^3 \)
- Temperature: \( T = 300 \, K \)
- Vapor pressure of water at \( 300 \, K \): \( P_{vapor} = 3170 \, \text{Pa} \)
- Ideal gas constant: \( R = 8.314 \, \text{J K}^{-1} \text{mol}^{-1} \)

### Step 2: Convert the volume of the flask to cubic meters
Since \( 1 \, \text{dm}^3 = 0.001 \, \text{m}^3 \):
\[ V_{flask} = 1.0 \, \text{dm}^3 = 0.001 \, \text{m}^3 \]

### Step 3: Use the ideal gas law to find the number of moles of water vapor
The ideal gas law is given by:
\[ PV = nRT \]
Where:
- \( P \) = pressure (vapor pressure of water)
- \( V \) = volume of the flask
- \( n \) = number of moles of water vapor
- \( R \) = ideal gas constant
- \( T \) = temperature in Kelvin

Rearranging the equation to find \( n \):
\[ n = \frac{PV}{RT} \]

### Step 4: Substitute the values into the equation
Substituting the known values:
\[ n = \frac{3170 \, \text{Pa} \times 0.001 \, \text{m}^3}{8.314 \, \text{J K}^{-1} \text{mol}^{-1} \times 300 \, K} \]

### Step 5: Calculate the number of moles
Calculating the numerator:
\[ 3170 \, \text{Pa} \times 0.001 \, \text{m}^3 = 3.17 \, \text{Pa m}^3 \]

Calculating the denominator:
\[ 8.314 \, \text{J K}^{-1} \text{mol}^{-1} \times 300 \, K = 2494.2 \, \text{J mol}^{-1} \]

Now, substituting these values into the equation for \( n \):
\[ n = \frac{3.17}{2494.2} \approx 0.00127 \, \text{mol} \]

### Step 6: Final result
Thus, the number of moles of water in the vapor phase when equilibrium is established is approximately:
\[ n \approx 1.27 \times 10^{-3} \, \text{mol} \]