Problem 82 - 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? \(\left(k_{\mathrm{H}} \text { for } \mathrm{CO}_{2} \text { is } 0.034 \mathrm{mol} / \mathrm{kg} \text { bar. }\right)\) 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

The concentration of dissolved CO2 is 0.051 mol/kg; about 33.3% escapes.

Step 1: Calculate Initial Concentration of CO2

Use Henry's Law to find the concentration of CO2 in the beverage. Henry’s Law is given by: \[ C = k_H \times P \] where \( C \) is the concentration, \( k_H \) is the Henry's constant, and \( P \) is the pressure. Here, \( k_H = 0.034 \mathrm{mol}/\mathrm{kg}\text{ bar} \) and \( P = 1.5 \text{ bar} \). Therefore, \( C = 0.034 \times 1.5 = 0.051 \mathrm{mol}/\mathrm{kg} \).

Step 2: Determine CO2 Concentration at Atmospheric Pressure

Assume atmospheric pressure of CO2 is 1 bar. Using Henry's Law again, with \( P = 1 \text{ bar} \), we find the concentration at equilibrium with the atmosphere: \[ C_{atm} = k_H \times P_{atm} = 0.034 \times 1 = 0.034 \mathrm{mol}/\mathrm{kg} \].

Step 3: Calculate Fraction of CO2 Released

The fraction of CO2 released is the difference between the initial and final concentration over the initial concentration. Thus, it is given by: \[ \text{Fraction} = \frac{C_{initial} - C_{atm}}{C_{initial}} = \frac{0.051 - 0.034}{0.051} \approx 0.333 \].

Key Concepts

Carbonation Process

When we talk about the carbonation process, we're referring to the method of dissolving carbon dioxide (CO₂) into a liquid, usually under high pressure. This is what gives carbonated beverages their signature fizz. In the bottling process, beverages are exposed to CO₂ at a pressure above atmospheric levels, often measured in bars.

This high pressure is crucial because it forces more CO₂ into the liquid than would naturally dissolve at atmospheric pressure. Once the beverage is sealed, this extra CO₂ remains dissolved until the bottle is opened. Understanding how much CO₂ stays in the liquid involves Henry's Law, a concept central to both the carbonation process and determining gas solubility.

Gas Solubility Calculations

Gas solubility calculations involve determining how much of a given gas will dissolve in a liquid at specific conditions of pressure and temperature. This is where Henry's Law comes in handy. Henry’s Law states:

• \( C = k_H \times P \)

Where \( C \) is the concentration of the gas, \( k_H \) is Henry's constant for the gas, and \( P \) is the pressure of the gas above the liquid.

Using this law, we can calculate the concentration of dissolved CO₂ in our beverage. With a Henry’s constant \( k_H \) of 0.034 mol/kg bar and a pressure \( P \) of 1.5 bar, the concentration \( C \) is computed as 0.051 mol/kg. This process helps in determining how effervescent a carbonated drink will be when it is initially opened.

Atmospheric Pressure Effects

Atmospheric pressure effects play a significant role in the behavior of gases in liquids. When the pressure above a liquid is changed—such as when a beverage bottle is opened and exposed to atmospheric pressure—equilibrium gets disrupted.

Carbon dioxide starts escaping from the liquid to re-establish equilibrium with the atmospheric pressure. This release leads to the iconic fizz or bubbling sound.

To quantify how much CO₂ is lost, we use Henry’s Law to calculate the concentration of CO₂ at atmospheric pressure, which is typically 1 bar. Here, \( C_{atm} \) is 0.034 mol/kg. Comparing this to the concentration before opening (0.051 mol/kg), we see that approximately 33.3% of the CO₂ escapes. The understanding of this phenomenon is key to optimizing carbonation in beverages.