Question

A student carried out the following experiment to measure the pressure of carbon dioxide in the space above the carbonated soft drink in a bottle. First, she weighed the bottle \((853.5 \mathrm{~g})\). Next, she carefully removed the cap to let the \(\mathrm{CO}_{2}\) gas escape. She then reweighed the bottle with the cap \((851.3 \mathrm{~g})\). Finally, she measured the volume of the soft drink (452.4 mL). Given that the Henry's law constant for \(\mathrm{CO}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is \(3.4 \times 10^{-2} \mathrm{~mol} / \mathrm{L} \cdot \mathrm{atm},\) calculate the pressure of \(\mathrm{CO}_{2}\) over the soft drink in the bottle before it was opened. Explain why this pressure is only an estimate of the true value.

Short answer

The pressure of \( \mathrm{CO}_2 \) is approximately \( 3.757 \times 10^{-3} \) atm, an estimate due to non-ideal conditions.

Step-by-step solution

Calculate the Mass of Escaped CO2

Subtract the final mass of the bottle with cap from the initial mass to find the mass of the CO2 that escaped. \( ext{Mass of } \mathrm{CO}_2 = 853.5 \text{ g} - 851.3 \text{ g} = 2.2 \text{ g} \)

Convert the Mass of CO2 to Moles

Use the molar mass of \( \mathrm{CO}_2 \) \((44.01 \text{ g/mol})\) to calculate the moles of \( \mathrm{CO}_2 \). \[ \text{Moles of } \mathrm{CO}_2 = \frac{2.2 \text{ g}}{44.01 \text{ g/mol}} \approx 0.05 \text{ mol} \]

Calculate the Concentration of CO2 in Solution

Find the concentration of \( \mathrm{CO}_2 \) using the volume of the soft drink. \[ \text{Concentration} = \frac{0.05 \text{ mol}}{0.4524 \text{ L}} \approx 0.1105 \text{ mol/L} \]

Apply Henry's Law to Find the Pressure

Use Henry's Law \( P = k_H \times C \) where \( k_H = 3.4 \times 10^{-2} \text{ mol/L} \cdot \text{atm} \) and \( C \) is the concentration. \[ P = 3.4 \times 10^{-2} \cdot 0.1105 \approx 3.757 \times 10^{-3} \text{ atm} \]

Assess the Estimate of Pressure

The calculated pressure is an estimate because the Henry's Law constant assumes ideal conditions, which don't account for other gases or temperature variations besides the given \( 25^{\circ} \text{C} \). Additionally, some CO2 might have remained in solution.

Key concepts

carbon dioxide

Carbon dioxide ( CO_2 ) is a colorless gas that is naturally present in the Earth's atmosphere. It plays a significant role in various chemical and biological processes. One common everyday encounter with carbon dioxide is in carbonated beverages, where it is dissolved under pressure. When the cap of a bottle is removed, the pressure inside decreases, allowing the CO_2 to escape and form bubbles.

Understanding the behavior of carbon dioxide in solutions is essential in chemistry, as it affects calculations related to its release from liquids. For students, it's important to remember that the interaction between CO2 and water can be quantified using principles like Henry's Law.

pressure calculation

Pressure calculation is crucial when dealing with gases in any chemical scenario. In our experiment, the primary goal is to determine the pressure of carbon dioxide in a soft drink before it is opened. This is done by applying Henry's Law, which relates the pressure of a gas above a liquid to its concentration within the liquid.

To apply Henry's Law, we use the formula \( P = k_H \times C \), where \( P \) denotes the pressure, \( k_H \) is the Henry's Law constant, and \( C \) represents the concentration of the gas in the liquid. The challenge here is ensuring that all units are consistent and accurately reflect the conditions described in the problem, such as temperature. This helps us calculate an estimated pressure, keeping in mind that real-world scenarios might introduce additional variables.

moles of gas

The concept of moles is a fundamental part of chemistry, providing a bridge between macroscopic and microscopic properties. One mole represents \( 6.022 \times 10^{23} \) entities, usually atoms or molecules. For gases, this concept is particularly useful in linking their mass to their chemical behavior.

In the context of the exercise, once we've determined the mass of carbon dioxide that has escaped (2.2 ext{ g}), we convert this mass to moles using the molar mass of carbon dioxide (44.01 ext{ g/mol}). The calculation is simple: divide the mass by the molar mass, yielding about \( 0.05 \text{ mol} \). Understanding this conversion is crucial for further calculations like determining concentration or applying laws such as Henry's Law.

mass of gas

Determining the mass of a gas involves careful measurement, especially in experiments like the one described. When the cap is removed, the mass of carbon dioxide that escapes can be determined by subtracting the new mass of the bottle from the initial one.

The mass of any gas can be linked back to its pressure and concentration calculations. Here, the process begins with measuring the bottle's mass before and after releasing the gas, providing a direct way to calculate the mass of the carbon dioxide. With that figured out ( 2.2 ext{ g} ), this value becomes essential in converting to moles, which then leads to concentration calculations.

concentration calculation

Concentration calculation allows us to understand how much of a gas is dissolved in a liquid. This is often expressed in \( \text{mol/L} \), indicating moles of solute per liter of solution.

To calculate the concentration of carbon dioxide in the soft drink, we use the number of moles of CO_2 and the volume of the drink in liters (0.4524 ext{ L}). By dividing the moles (0.05 \text{ mol}) by the volume, we obtain a concentration of approximately \( 0.1105 \text{ mol/L} \). This concentration is crucial when applying Henry's Law to estimate the pressure of carbon dioxide above the solution. Understanding such calculations enables students to approach similar problems with confidence.