1 mole of each of `X_(1),X_(2),X_(3)` with van der Waal's constants a (in atm `L^(3) mol^(-2)`) 1.0, 3.8, 2.1 respectively is kept separately in three different vessels of equal volume at identical temperature. Their pressures are observed to `P_(1),P_(2),` and `P_(3)` respectively. On the basis of this data alone, select the correct option (neglect the effect of 'b') :

Options

A. `P_(1)ltP_(2)ltP_(3)`
B. `P_(2)ltP_(1)ltP_(3)`
C. `P_(2)ltP_(3)ltP_(1)`
D. `P_(1)=P_(2)=P_(3)`

Explanation

To solve the problem, we need to analyze the relationship between the Van der Waals constant \( a \) and the pressure exerted by the gases \( X_1, X_2, \) and \( X_3 \). Here’s a step-by-step solution:

### Step 1: Understand the Van der Waals Equation
The Van der Waals equation for real gases is given by:
\[ \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \]
Since we are neglecting the effect of \( b \), we can simplify this to:
\[ P + \frac{a n^2}{V^2} = \frac{nRT}{V} \]
Where:
- \( P \) = pressure of the gas
- \( a \) = Van der Waals constant (specific to each gas)
- \( n \) = number of moles (1 mole for each gas)
- \( V \) = volume of the gas
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 2: Rearranging the Equation
Rearranging the equation gives us:
\[ P = \frac{nRT}{V} - \frac{a n^2}{V^2} \]
Since \( n = 1 \) for each gas, the equation simplifies to:
\[ P = \frac{RT}{V} - \frac{a}{V^2} \]

### Step 3: Analyze the Effect of \( a \)
From the equation, we can see that as the value of \( a \) increases, the term \( \frac{a}{V^2} \) becomes larger. This means that the pressure \( P \) will decrease because we are subtracting a larger value from the ideal pressure \( \frac{RT}{V} \).

### Step 4: Compare the Values of \( a \)
Given the values of \( a \):
- For \( X_1 \), \( a_1 = 1.0 \)
- For \( X_2 \), \( a_2 = 3.8 \)
- For \( X_3 \), \( a_3 = 2.1 \)

### Step 5: Determine the Order of Pressures
Since \( a_2 \) is the largest, \( X_2 \) will have the lowest pressure. Next, \( a_3 \) is larger than \( a_1 \), so \( X_3 \) will have a lower pressure than \( X_1 \). Thus, the order of pressures will be:
\[ P_2 < P_3 < P_1 \]
This means:
- \( P_2 \) (for \( X_2 \)) is the lowest,
- \( P_3 \) (for \( X_3 \)) is in the middle,
- \( P_1 \) (for \( X_1 \)) is the highest.

### Final Answer
The correct order of pressures is:
\[ P_2 < P_3 < P_1 \]