Problem 84 - Chemistry and Chemical Reactivity

Question

If a carbonated beverage is bottled under 1.5 bar \(\mathrm{CO}_{2}\) pressure, what will be the concentration of dissolved \(\mathrm{CO}_{2}\) in that beverage? \((k_{\mathrm{H}} \text { for } \mathrm{CO}_{2}\) is \(0.034 \mathrm{mol} / \mathrm{kg}\) bar. After the pressure is released, what fraction of the dissolved gas will escape before equilibrium with the \(\mathrm{CO}_{2}\) in the atmosphere is reached?

Step-by-Step Solution

Answer

Approximately 0.051 mol/kg \(\mathrm{CO}_2\) is dissolved; almost all \(\mathrm{CO}_2\) escapes upon pressure release.

Step 1: Understand the Henry's Law Equation

Henry's Law states that the concentration of a gas dissolved in a liquid is directly proportional to the partial pressure of the gas above the liquid. The formula is given by: \[ C = k_H \cdot P \] Where \( C \) is the concentration of the gas, \( k_H \) is the Henry's Law constant, and \( P \) is the partial pressure of the gas.

Step 2: Apply Henry's Law to Calculate Initial \(\mathrm{CO}_2\) Concentration

Given that the pressure \( P \) is 1.5 bar and \( k_H \) for \(\mathrm{CO}_2\) is 0.034 mol/kg bar, we substitute into the Henry's Law equation: \[ C = 0.034\, \text{mol/kg bar} \times 1.5\, \text{bar} = 0.051\, \text{mol/kg} \] This is the concentration of dissolved \( \mathrm{CO}_2 \) when the beverage is bottled.

Step 3: Determine the Equilibrium \(\mathrm{CO}_2\) Concentration After Pressure Release

Once the pressure is released, the beverage will equilibrate with atmospheric \(\mathrm{CO}_2\), with a very low partial pressure of about 0.0004 bar. Using the same formula with \( P = 0.0004\, \text{bar} \), we calculate: \[ C_{\text{new}} = 0.034\, \text{mol/kg bar} \times 0.0004\, \text{bar} = 0.0000136\, \text{mol/kg} \] This is the concentration of \( \mathrm{CO}_2 \) once the beverage reaches equilibrium with the atmosphere.

Step 4: Calculate the Fraction of Escaped \( \mathrm{CO}_2\)

The amount of \(\mathrm{CO}_2\) that escapes is the difference between the initial and equilibrium concentrations. Thus, the escaped \(\mathrm{CO}_2\) is:\[ C_{\text{escaped}} = 0.051\, \text{mol/kg} - 0.0000136\, \text{mol/kg} = 0.0509864\, \text{mol/kg} \] The fraction of escaped \( \mathrm{CO}_2 \) is the ratio of escaped \( \mathrm{CO}_2 \) to initial pressure dissolved \( \mathrm{CO}_2 \): \[ \text{Fraction} = \frac{0.0509864}{0.051} \approx 0.9997 \] Almost all the gas escapes.

Key Concepts

Carbonated Beverages

Carbonated beverages are a popular type of drink that contains dissolved carbon dioxide (CO₂) under high pressure. This added CO₂ gives these drinks their fizz and can make them quite refreshing. For example, soda and sparkling water are common examples of carbonated beverages.

When a carbonated beverage is bottled, the CO₂ is dissolved in the liquid at a specific pressure. This pressure keeps the gas in solution. As the bottle is sealed, the CO₂ remains dissolved until the bottle is opened. Once you remove the cap, the pressure is released, and the CO₂ starts escaping, which causes the fizzing sound.

The amount of CO₂ that escapes plays a big role in the taste and texture of the beverage. Over time and with exposure to air, the drink can become "flat," as the CO₂ dissolves into the atmosphere.

Partial Pressure

Partial pressure is a key concept in understanding how gases behave in mixtures, like in a carbonated beverage. It refers to the pressure that a specific gas in a mixture would exert if it alone occupied the entire volume.

In the case of carbonated beverages, the partial pressure of CO₂ is crucial. When the drink is sealed, the CO₂ is at a higher partial pressure. This is why it's able to dissolve in the liquid according to Henry's Law. The equation for this law is: \[ C = k_H \cdot P \] where \( C \) is the concentration of the dissolved gas, \( k_H \) is the Henry's Law constant, and \( P \) is the partial pressure.

When you open the bottle, the partial pressure of CO₂ drops significantly, causing the gas to escape as it finds equilibrium with the atmospheric CO₂, which is at much lower pressure.

Gas Solubility

Gas solubility refers to the ability of a gas to dissolve in a liquid. This property depends on factors like temperature, pressure, and the nature of the liquid and the gas. In simpler terms, it explains how much and how easily a gas can mix with a liquid.

In the context of carbonated beverages, Henry's Law explains the relationship between pressure and solubility. If the pressure of a gas above a liquid is increased, more gas will dissolve in the liquid. That's why carbonated beverages lose their fizz once opened—the pressure decreases, and CO₂ is less soluble, causing it to escape into the air.

The solubility is also affected by temperature; gases are generally more soluble in cold liquids. This is why cold carbonated beverages retain their carbonation better than those that are warm.

Equilibrium

Equilibrium in the context of gas solubility is the state where the rate at which the gas is dissolving in the liquid equals the rate at which it is escaping. This means the concentration of the gas in the liquid remains constant over time.

When a carbonated beverage is initially sealed, it is in equilibrium with a high CO₂ pressure. When opened, the equilibrium is disrupted because the pressure drops.

Eventually, the beverage will reach a new equilibrium with the atmospheric CO₂ at a much lower concentration than when bottled. This new equilibrium is what causes the drink to eventually go "flat," as the CO₂ concentration decreases.

• High pressure maintained in sealed bottles keeps high CO₂ concentration (fizz).
• Opening the bottle releases pressure, breaking equilibrium.
• New equilibrium is with atmospheric CO₂, usually leading to a flat drink.