Henry's law constant for `CO_(2)` in water is `1.67xx10^(8) Pa` at `298 K`. Calculate the quantity of `CO_(2)` in `500mL` of soda water when packed under `2.5atm CO_(2)` pressure at `298 K`.

Explanation

To solve the problem, we need to determine the quantity of CO₂ dissolved in 500 mL of soda water under a pressure of 2.5 atm at 298 K, given that the Henry's law constant for CO₂ in water is \(1.67 \times 10^8 \, \text{Pa}\).

### Step-by-Step Solution:

1. **Convert the pressure from atm to Pa:**
\[ P_{\text{CO}_2} = 2.5 \, \text{atm} \times 1.01 \times 10^5 \, \text{Pa/atm} = 2.525 \times 10^5 \, \text{Pa} \]

2. **Apply Henry's Law:**
According to Henry's Law:
\[ P_{\text{CO}_2} = K_H \cdot X_{\text{CO}_2} \]
where \(P_{\text{CO}_2}\) is the partial pressure of CO₂, \(K_H\) is the Henry's law constant, and \(X_{\text{CO}_2}\) is the mole fraction of CO₂.

Rearrange to solve for \(X_{\text{CO}_2}\):
\[ X_{\text{CO}_2} = \frac{P_{\text{CO}_2}}{K_H} = \frac{2.525 \times 10^5 \, \text{Pa}}{1.67 \times 10^8 \, \text{Pa}} = 0.00151 \]

3. **Calculate the number of moles of water:**
Assume the density of water is \(1 \, \text{g/mL}\), so the mass of 500 mL of water is:
\[ \text{Mass of water} = 500 \, \text{g} \]
The molar mass of water (\(\text{H}_2\text{O}\)) is 18 g/mol. Therefore, the number of moles of water is:
\[ n_{\text{H}_2\text{O}} = \frac{500 \, \text{g}}{18 \, \text{g/mol}} \approx 27.78 \, \text{mol} \]

4. **Relate mole fraction to moles of CO₂ and water:**
The mole fraction of CO₂ is given by:
\[ X_{\text{CO}_2} = \frac{n_{\text{CO}_2}}{n_{\text{CO}_2} + n_{\text{H}_2\text{O}}} \]
Since \(n_{\text{CO}_2}\) is very small compared to \(n_{\text{H}_2\text{O}}\), we approximate:
\[ X_{\text{CO}_2} \approx \frac{n_{\text{CO}_2}}{n_{\text{H}_2\text{O}}} \]
Thus:
\[ n_{\text{CO}_2} = X_{\text{CO}_2} \times n_{\text{H}_2\text{O}} = 0.00151 \times 27.78 \approx 0.042 \, \text{mol} \]

5. **Calculate the mass of dissolved CO₂:**
The molar mass of CO₂ is 44 g/mol. Therefore, the mass of dissolved CO₂ is:
\[ \text{Mass of CO}_2 = n_{\text{CO}_2} \times \text{Molar mass of CO}_2 = 0.042 \, \text{mol} \times 44 \, \text{g/mol} = 1.848 \, \text{g} \]

### Final Answer:
The quantity of CO₂ dissolved in 500 mL of soda water is \(1.848 \, \text{g}\).