Calculate the pressure exerted by 4 mole `CO_2` ​of gas at 273 K, if the van der Waal's constat "a"=3.592`dm^6atm^(-2)` . Assume that the volume occupied by `CO_2` molecules is negligible.

Answer

To calculate the pressure exerted by 4 moles of CO₂ gas at 273 K using the van der Waals equation, we will follow these steps:

### Step 1: Write down the van der Waals equation
The van der Waals equation is given by:

\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]

Since we are assuming the volume occupied by CO₂ molecules is negligible, we can set \(b = 0\). Thus, the equation simplifies to:

\[ P + \frac{a n^2}{V^2} = \frac{nRT}{V} \]

### Step 2: Rearrange the equation to solve for pressure (P)
Rearranging the equation gives us:

\[ P = \frac{nRT}{V} - \frac{a n^2}{V^2} \]

### Step 3: Identify and substitute the known values
We have:
- Number of moles, \(n = 4\)
- Temperature, \(T = 273 \, K\)
- van der Waals constant, \(a = 3.592 \, \text{dm}^6 \text{atm}^{-2}\)
- Gas constant, \(R = 0.0821 \, \text{atm L K}^{-1} \text{mol}^{-1}\)
- At standard temperature and pressure (STP), the volume occupied by 1 mole of gas is approximately \(22.4 \, L\). Therefore, for 4 moles, the volume \(V = 4 \times 22.4 \, L = 89.6 \, L\).

### Step 4: Substitute the values into the equation
Now substituting the values into the rearranged equation:

\[ P = \frac{4 \times 0.0821 \times 273}{89.6} - \frac{3.592 \times 4^2}{89.6^2} \]

### Step 5: Calculate the first term
Calculating the first term:

\[ \frac{4 \times 0.0821 \times 273}{89.6} = \frac{89.7364}{89.6} \approx 1.000 \]

### Step 6: Calculate the second term
Calculating the second term:

\[ \frac{3.592 \times 16}{89.6^2} = \frac{57.472}{8025.76} \approx 0.00715 \]

### Step 7: Combine the results
Now, substituting these results back into the equation for pressure:

\[ P = 1.000 - 0.00715 \approx 0.99285 \, \text{atm} \]

### Step 8: Final pressure calculation
Thus, the pressure exerted by 4 moles of CO₂ gas at 273 K is approximately:

\[ P \approx 0.99285 \, \text{atm} \]

### Summary
The final pressure exerted by 4 moles of CO₂ at 273 K is approximately **0.993 atm**.

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