Question

$CO_2$ 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 mmol 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 × $10^3$ bar)

Answer 1

25

Answer 2

We are given the partial pressure of $CO_2$ gas ($p_{CO_2}$), the volume of water, and Henry's law constant ($K_H$) for $CO_2$ in water at 298 K. We need to find the amount of $CO_2$ dissolved in water in millimoles.

According to Henry's law, the partial pressure of a gas above a solution is proportional to its mole fraction in the solution: $p_{CO_2} = K_H \cdot x_{CO_2}$

Given $p_{CO_2} = 0.835$ bar and $K_H = 1.67 \times 10^3$ bar, we can calculate the mole fraction of $CO_2$ in the solution: $x_{CO_2} = \frac{p_{CO_2}}{K_H} = \frac{0.835 \text{ bar}}{1.67 \times 10^3 \text{ bar}} = \frac{0.835}{1670} = 0.0005$

The mole fraction of $CO_2$ in the solution is defined as: $x_{CO_2} = \frac{\text{moles of } CO_2}{\text{moles of } CO_2 + \text{moles of water}} = \frac{n_{CO_2}}{n_{CO_2} + n_{water}}$

We need to find the number of moles of water in 0.9 L. Assuming the density of water is 1 g/mL or 1 kg/L, the mass of 0.9 L of water is 0.9 kg = 900 g. The molar mass of water ($H_2O$) is approximately 18 g/mol. The number of moles of water is: $n_{water} = \frac{\text{mass of water}}{\text{molar mass of water}} = \frac{900 \text{ g}}{18 \text{ g/mol}} = 50 \text{ mol}$

Since the solubility of $CO_2$ in water is relatively low, the number of moles of dissolved $CO_2$ is much smaller than the number of moles of water ($n_{CO_2} \ll n_{water}$). Thus, we can approximate the mole fraction as: $x_{CO_2} \approx \frac{n_{CO_2}}{n_{water}}$

Now, we can calculate the number of moles of $CO_2$: $n_{CO_2} = x_{CO_2} \times n_{water} = 0.0005 \times 50 \text{ mol} = 0.025 \text{ mol}$

We are asked to find the amount of $CO_2$ in millimoles. 1 mol = 1000 mmol $n_{CO_2} \text{ in mmol} = 0.025 \text{ mol} \times 1000 \text{ mmol/mol} = 25 \text{ mmol}$

So, x mmol of $CO_2$ would dissolve in 0.9 L of water, where x is the number of millimoles. The value of x is 25.

The question asks for the nearest integer value of x. Since the calculated value is 25, the nearest integer is 25.