\(CO_2\) gas is bubbled through water during a soft drink manufacturing process at 298 K. If \(CO_2\) exerts a partial pressure of 0.835 bar then \(x \text{ m mol}\) of \(CO_2\) would dissolve in 0.9 L of water. The value of \(x\) is ________. (Nearest integer) (Henry's law constant for \(CO_2\) at 298 K is \(1.67 \times 10^3\) bar)
Show Hint
For very dilute solutions (like gas in liquid), always use the approximation \(n_{gas} + n_{solvent} \approx n_{solvent}\) to avoid solving complex quadratic equations.
Step 1: Understanding the Concept:
Henry's Law describes the solubility of a gas in a liquid.
It states that the partial pressure of a gas is proportional to its mole fraction in the solution. Step 2: Key Formula or Approach:
1. Henry's Law: \(p = K_H \cdot \chi\), where \(\chi\) is the mole fraction of the gas.
2. \(\chi \approx \frac{n_{gas}}{n_{solvent}}\) for dilute solutions. Step 3: Detailed Explanation:
1. Calculate Mole Fraction (\(\chi\)):
\[ \chi_{CO_2} = \frac{p}{K_H} = \frac{0.835 \text{ bar}}{1.67 \times 10^3 \text{ bar}} = 0.5 \times 10^{-3} = 5 \times 10^{-4} \]
2. Calculate moles of water in 0.9 L:
Volume = 900 mL. Mass = 900 g (taking density = 1 g/mL).
\[ n_{H_2O} = \frac{900 \text{ g}}{18 \text{ g/mol}} = 50 \text{ mol} \]
3. Calculate moles of \(CO_2\) dissolved:
\[ \chi_{CO_2} = \frac{n_{CO_2}}{n_{CO_2} + n_{H_2O}} \approx \frac{n_{CO_2}}{n_{H_2O}} \]
\[ n_{CO_2} = \chi_{CO_2} \times n_{H_2O} = 5 \times 10^{-4} \times 50 = 250 \times 10^{-4} \text{ mol} \]
\[ n_{CO_2} = 0.025 \text{ mol} = 25 \text{ mmol} \] Step 4: Final Answer:
The value of \(x\) is 25.
