Question

Consider a carbonated drink in a bottle at \(37^{\circ} \mathrm{C}\) and \(130 \mathrm{kPa}\). Assuming the gas space above the liquid consists of a saturated mixture of \(\mathrm{CO}_{2}\) and water vapor and treating the drink as water, determine \((a)\) the mole fraction of the water vapor in the \(\mathrm{CO}_{2}\) gas and \((b)\) the mass of dissolved \(\mathrm{CO}_{2}\) in a 200-ml drink.

Short answer

Answer: The mole fraction of water vapor in the CO₂ gas is 0.0503, and the mass of dissolved CO₂ in a 200-ml drink is 2.30 g.

Step-by-step solution

Find the partial pressures of CO₂ gas and water vapor.

Assuming the gas space above the liquid consists of a saturated mixture of CO₂ and water vapor, we can find the partial pressures of each component using Dalton's law of partial pressures, which states that the total pressure is equal to the sum of the partial pressures of each component: \(P_{total} = P_{CO_2} + P_{H_2O}\) The total pressure in the bottle is given as 130 kPa. We can find the partial pressure of water vapor, \(P_{H_2O}\), by using the saturation pressure, \(p_g\), at the given temperature: \(p_g (37^{\circ}C) = 6.54 \mathrm{kPa}\) (from steam tables) This means that the partial pressure of the CO₂ gas can be determined as follows: \(P_{CO_2} = P_{total} - P_{H_2O} = 130\,\mathrm{kPa} - 6.54\, \mathrm{kPa} = 123.46\,\mathrm{kPa}\)

Calculate the mole fraction of water vapor in the gas space.

Now that we have the partial pressures of both CO₂ and water vapor, we can calculate the mole fraction of water vapor, which is given by: \(x_{H_2O} = \frac{P_{H_2O}}{P_{total}}\) Substituting the values: \(x_{H_2O} = \frac{6.54 \, \mathrm{kPa}}{130\, \mathrm{kPa}} = 0.0503\) So, the mole fraction of water vapor in the CO₂ gas is 0.0503.

Calculate the mass of dissolved CO₂ in the drink using Henry's law.

To determine the mass of dissolved CO₂, we will use Henry's law, which states that the concentration of a gas in a liquid is proportional to the partial pressure of the gas above the liquid: \(C = k_H P_{CO_2}\) Here, \(C\) is the concentration of dissolved CO₂ in the drink, \(k_H\) is Henry's law constant, and \(P_{CO_2}\) is the partial pressure of CO₂ determined in Step 1. Given the temperature of 37°C and considering the drink as water, we find the Henry's law constant for CO₂ from reference tables: \(k_H (37^{\circ}C) = 2.1 \times 10^{-2} \, \mathrm{mol/kg \cdot kPa}\) Now we can calculate the concentration of dissolved CO₂: \(C = (2.1 \times 10^{-2}\,\mathrm{mol/kg \cdot kPa})(123.46\,\mathrm{kPa}) = 2.5897\, \mathrm{mol/kg}\) To find the mass of dissolved CO₂ in a 200-ml drink, we need to multiply the concentration by the mass of the drink (assuming a density of 1 kg/L for water at room temperature): \(\mathrm{Mass\,of\,CO_2} = (2.5897\, \mathrm{mol/kg}) * (200\, \mathrm{ml}) * \frac{1\, \mathrm{kg}}{1000\, \mathrm{ml}} * \frac{44.01\, \mathrm{g}}{1\, \mathrm{mol}}\) \(\mathrm{Mass\,of\,CO_2} = 2.30\, \mathrm{g}\) So, the mass of dissolved CO₂ in a 200-ml drink is 2.30 g.

Key concepts

Dalton's Law of Partial Pressures

When dealing with gases, Dalton's Law of Partial Pressures helps us to understand the behavior of gas mixtures. It states that the total pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of each individual gas in the mixture. This principle is essential, especially when we study carbonated drinks, because it allows us to calculate the partial pressures of gases like carbon dioxide and water vapor in the space above the liquid.

Dalton's Law can be expressed mathematically as:

• \( P_{\text{total}} = P_{\text{CO}_2} + P_{\text{H}_2O} \)

In the context of carbonated drinks, applying this law helps us isolate the pressure contributions from the water vapor and the CO₂, which is critical in determining other properties such as mole fraction and solubility.

Henry's Law

Henry's Law is integral when studying how gases dissolve in liquids. This law states that the concentration of a gas dissolved in a liquid is proportional to the partial pressure of that gas above the liquid. For carbonated beverages, this means the solubility of CO₂ gas in the drink is dependent on its partial pressure.

The relationship as per Henry's Law can be written as:

• \( C = k_H \times P_{\text{CO}_2} \)

where \( C \) is the concentration of dissolved gas, \( k_H \) is the Henry's law constant, and \( P_{\text{CO}_2} \) is the partial pressure of CO₂. This is crucial for determining how much CO₂ is actually dissolved in the drink, affecting both the taste and carbonation level.

Mole Fraction

The mole fraction signifies the ratio of moles of one component to the total moles of the mixture. This is a useful concept when assessing the composition of gas mixtures in a carbonated beverage environment. Specifically, it tells us how much of a particular gas (such as water vapor) is present compared to the entire gas mixture.

To find the mole fraction of water vapor (\( x_{\text{H}_2O} \)), we use the formula:

• \( x_{\text{H}_2O} = \frac{P_{\text{H}_2O}}{P_{\text{total}}} \)

This ratio is critical in understanding the proportion and behavior of water vapor in gaseous mixtures, particularly above a carbonated drink, to maintain the drink’s quality.

Partial Pressure

Partial pressure is an important concept in understanding the behavior of gas mixtures. It refers to the pressure that a single gas in a mixture would exert if it occupied the entire volume alone. This concept is essential in scenarios such as in carbonated drinks, where multiple gases coexist above the liquid.

When calculating partial pressures, it is done in the context of the whole mixture. Each gas contributes to the total pressure based on its proportion in the mixture. The ability to calculate partial pressures, such as \( P_{\text{CO}_2} \) or \( P_{\text{H}_2O} \), allows us to further determine the mole fractions and solubility levels, creating a symphony of relationships in the gases trapped in carbonated drinks.

Saturation Pressure

Saturation pressure is the pressure at which a vapor is in equilibrium with its liquid form at a given temperature. It tells us the maximum amount of vapor that can coexist with the liquid without further evaporation or condensation. For water vapor in the context of carbonated drinks, knowing the saturation pressure is essential to determine how much water vapor can be present above the liquid.

When a carbonated drink reaches a certain temperature, the saturation pressure will dictate the maximum possible pressure of the vapor present there. This is critical because if the actual vapor pressure exceeds this value, condensation will occur. Conversely, if it's less, more liquid can still evaporate until the equilibrium is achieved.