Question

Henry's law constant for CO$_2$ in water is 1.67 × 10$^8$ Pa at 298 K. Calculate the quantity of CO$_2$ in 500 mL of soda water when packed under 2.5 atm CO$_2$ pressure at 298 K.

Answer 1

0.042 moles

Answer 2

Solution:

Henry's law states that the partial pressure of a gas in the vapor phase ($P$) is proportional to the mole fraction of the gas in the solution ($x$). The relationship is given by $P = k_H x$, where $k_H$ is Henry's law constant.

Given:

• Henry's law constant, $k_H = 1.67 \times 10^8$ Pa at 298 K.
• Partial pressure of CO$_2$, $P_{CO_2} = 2.5$ atm.
• Volume of soda water (assume volume of water) = 500 mL.
• Temperature = 298 K.

First, convert the pressure from atmospheres (atm) to Pascals (Pa). 1 atm = 101325 Pa $P_{CO_2} = 2.5 \text{ atm} \times 101325 \frac{\text{Pa}}{\text{atm}} = 253312.5 \text{ Pa}$

Using Henry's law, $P_{CO_2} = k_H x_{CO_2}$, where $x_{CO_2}$ is the mole fraction of CO$_2$ in the solution. $x_{CO_2} = \frac{P_{CO_2}}{k_H} = \frac{253312.5 \text{ Pa}}{1.67 \times 10^8 \text{ Pa}}$ $x_{CO_2} \approx 1.51684 \times 10^{-3}$

Next, calculate the number of moles of water in 500 mL. Assume the density of water is 1 g/mL. Mass of water = Volume × Density = 500 mL × 1 g/mL = 500 g. The molar mass of water (H$_2$O) is approximately 18 g/mol. Number of moles of water, $n_{H_2O} = \frac{\text{Mass of water}}{\text{Molar mass of water}} = \frac{500 \text{ g}}{18 \text{ g/mol}} \approx 27.7778 \text{ moles}$.

The mole fraction of CO$_2$ in the solution is defined as $x_{CO_2} = \frac{n_{CO_2}}{n_{CO_2} + n_{H_2O}}$. Since the solubility of CO$_2$ in water is low, $n_{CO_2} \ll n_{H_2O}$. Therefore, we can approximate the mole fraction as $x_{CO_2} \approx \frac{n_{CO_2}}{n_{H_2O}}$.

Now, calculate the number of moles of CO$_2$, $n_{CO_2}$. $n_{CO_2} = x_{CO_2} \times n_{H_2O}$ $n_{CO_2} \approx (1.51684 \times 10^{-3}) \times 27.7778 \text{ moles}$ $n_{CO_2} \approx 0.04213$ moles.

Rounding to two significant figures (based on the pressure value 2.5 atm), the quantity of CO$_2$ is approximately 0.042 moles.