A vessel has `N_(2)` gas and water vapours at a total pressure of `1 atm`. The partial pressure of water vapours is `0.3atm`. The contents of this vessel are transferred to another vessel having one-third of the capacity of original volume, completely at the same temperature the total pressure of this system in the new vessel is

Options

A. 3 atm
B. 1 atm
C. 3.33 atm
D. 2.4 atm

Answer

To solve the problem step by step, we will follow the principles of Dalton's Law of Partial Pressures and Boyle's Law.

### Step 1: Identify Given Information
We know:
- Total pressure in the original vessel, \( P_{\text{total}} = 1 \, \text{atm} \)
- Partial pressure of water vapors, \( P_{\text{H}_2O} = 0.3 \, \text{atm} \)

### Step 2: Calculate the Partial Pressure of Nitrogen Gas
Using Dalton's Law of Partial Pressures:
\[ P_{\text{total}} = P_{\text{N}_2} + P_{\text{H}_2O} \]
Substituting the known values:
\[ 1 \, \text{atm} = P_{\text{N}_2} + 0.3 \, \text{atm} \]
Rearranging to find \( P_{\text{N}_2} \):
\[ P_{\text{N}_2} = 1 \, \text{atm} - 0.3 \, \text{atm} = 0.7 \, \text{atm} \]

### Step 3: Apply Boyle's Law
The contents are transferred to a new vessel with one-third the capacity of the original vessel. Boyle's Law states:
\[ P_1 V_1 = P_2 V_2 \]
Where:
- \( P_1 \) is the initial pressure of nitrogen, \( P_{\text{N}_2} = 0.7 \, \text{atm} \)
- \( V_1 \) is the original volume
- \( P_2 \) is the new pressure of nitrogen
- \( V_2 = \frac{1}{3} V_1 \)

Substituting into Boyle's Law:
\[ 0.7 \, \text{atm} \cdot V_1 = P_2 \cdot \left(\frac{1}{3} V_1\right) \]
Cancelling \( V_1 \) from both sides:
\[ 0.7 = \frac{P_2}{3} \]
Multiplying both sides by 3 to solve for \( P_2 \):
\[ P_2 = 3 \cdot 0.7 = 2.1 \, \text{atm} \]

### Step 4: Calculate Total Pressure in the New Vessel
The partial pressure of water vapors remains unchanged at \( 0.3 \, \text{atm} \). Therefore, the total pressure in the new vessel is:
\[ P_{\text{total, new}} = P_2 + P_{\text{H}_2O} = 2.1 \, \text{atm} + 0.3 \, \text{atm} = 2.4 \, \text{atm} \]

### Final Answer
The total pressure of the system in the new vessel is \( 2.4 \, \text{atm} \).

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