Question

Soft Drinks The partial pressure of \(\mathrm{CO}_{2}\) inside a bottle of soft drink is 4.0 \(\mathrm{atm}\) at \(25^{\circ} \mathrm{C}\) . The solubility of \(\mathrm{CO}_{2}\) is 0.12 \(\mathrm{mol} / \mathrm{L}\) . When the bottle is opened, the partial pressure drops to \(3.0 \times 10^{-4} \mathrm{atm} .\) What is the solubility of \(\mathrm{CO}_{2}\) in the open drink? Express your answer in grams per liter.

Short answer

The solubility of CO₂ in the open drink is approximately \(3.96 \times 10^{-4} \: g/L\).

Step-by-step solution

Understand Henry's Law

Henry's Law states that the solubility of a gas in a liquid is proportional to the partial pressure of the gas above the liquid. Mathematically, it can be represented as: \(C = kP\) where C is the concentration of the gas in the liquid (moles/L), k is Henry's Law constant, and P is the partial pressure of the gas.

Solve for Henry's Law constant (k)

Using Henry's Law equation, we can solve for the constant k by substituting the given values of the initial partial pressure (\(P_{i}\)) and solubility (\(C_{i}\)): \(C_{i} = kP_{i}\) \(k = \frac{C_{i}}{P_{i}}\) Substituting the given values: \(k = \frac{0.12\: mol/L}{4.0\: atm} = 0.03\: (mol/L) / (atm)\)

Calculate the solubility after opening the bottle

Now that we have the value of Henry's Law constant (k), we can find the solubility of CO₂ after opening the bottle (\(C_{f}\)), using the new partial pressure (\(P_{f}\)). Using the formula, we can substitute the values we know: \(C_{f} = kP_{f}\) \(C_{f} = 0.03\: (\dfrac{mol}{L.atm})\: \times\: 3.0 \times 10^{-4} \: atm\) \(C_{f} = 9.0 \times 10^{-6}\: mol/L\)

Convert solubility from moles per liter to grams per liter

Finally, we need to express the solubility in grams per liter (g/L). We need to multiply the molar solubility by the molar mass of CO₂: Molar mass of CO₂ = 12.01 g/mol (C) + 2 × 16.00 g/mol (O) = 44.01 g/mol \(grams\: per\: liter = molarity \times molar\: mass\) \(C_{f} = 9.0 \times 10^{-6}\: mol/L \times 44.01\: g/mol = 3.96 \times 10^{-4}\: g/L\) So, the solubility of CO₂ in the open drink is approximately \(3.96 \times 10^{-4} \: g/L\).

Key concepts

Partial Pressure

Imagine you have a mixture of gases, like the air we breathe. Each type of gas in the mixture exerts a force on the walls of its container. This force is what we call pressure. Now, 'partial pressure' is the term we use to describe the pressure that just one type of gas in the mixture would exert if it were alone in the container, taking up the same volume at the same temperature. Think of it like this: if you were in a crowd, and you only focused on the conversation of one person, that individual's voice is like the 'partial pressure' amidst the overall noise level.

When we look at beverages like soft drinks, they're bottled with carbon dioxide gas, which creates that fizzy sensation we all love. The carbon dioxide in the drink is under high pressure, and when you open the bottle, this pressure above the liquid decreases rapidly. According to Henry's Law, that change in partial pressure also changes the amount of gas dissolved in the drink – which in our case, affects the fizziness of the soda.

Solubility of Gases

So what does 'solubility of gases' really mean? Simply put, it's a measure of how much gas can dissolve in a certain amount of liquid. You know when you make a cup of tea and add sugar? The sugar 'dissolves' in the tea. Gases dissolve in liquids in a similar way. But unlike sugar in tea, the solubility of gases depends on both the temperature of the liquid and the partial pressure of the gas above the liquid. This is exactly what Henry's Law describes – the relationship between gas solubility and partial pressure.

To understand it better, think of a closed soda bottle. The solubility of carbon dioxide is high because the pressure inside the bottle is kept high on purpose. When you open the bottle, that pressure above the liquid decreases, and some of the dissolved gas escapes, resulting into fewer bubbles in your drink. It's like releasing the air out of a balloon – the carbon dioxide escapes from the liquid just as air whooshes out of the balloon.

Concentration Calculations

Now, we're getting down to the nitty-gritty: 'concentration calculations.' Concentration here refers to how much solute is present in a given amount of solution. In our soft drink example, we're talking about how much carbon dioxide is dissolved in the soda. To calculate concentration, we often use moles per liter (mol/L), which is also known as molarity. Molarity gives us an exact idea of the number of moles of a certain substance within one liter of solution.

When dealing with gases like CO₂ in a drink, the concentration tells us about the gas's solubility under certain conditions. If we use Henry's Law, as the exercise outlines, we can see how the solubility changes with pressure. But sometimes we want the concentration in more practical units, like how many grams are dissolved in each liter of liquid, especially when cooking or following a recipe. By using the molecular weight of CO₂, we can convert our concentration from moles to grams per liter, making it a lot easier to visualize how much gas is present in our bottle of soda.