Problem 52 - Chemistry The Molecular Nature of Matter

Question

If the solubility of a gas in water is \(0.010 \mathrm{~g} \mathrm{~L}^{-1}\) at \(25^{\circ} \mathrm{C}\) with the partial pressure of the gas 1. 0 atm, predict the solubility of the gas at the same temperature but at double the pressure.

Step-by-Step Solution

Answer

The new solubility of the gas at double the pressure (2.0 atm) is 0.020 g/L.

Step 1: Understanding Henry's Law

Henry's Law states that the solubility of a gas in a liquid at a given temperature is directly proportional to the pressure of the gas above the liquid. Mathematically, this can be expressed as S = kP, where S is the solubility, k is Henry's law constant, and P is the pressure.

Step 2: Using given solubility to find Henry's law constant (k)

We can find the value of k using the given solubility 0.010 g/L at 1.0 atm. Plugging in the values into the equation S = kP, we get 0.010 g/L = k * 1.0 atm. Solving this gives us k = 0.010 g/L / 1.0 atm.

Step 3: Calculating the new solubility

Now, with the pressure doubled (2.0 atm), we use the same value for k to find the new solubility. Using S = kP, the new solubility S becomes S = 0.010 g/L / atm * 2.0 atm which equals 0.020 g/L.

Key Concepts

Henry's Law

When it comes to understanding how gases dissolve in liquids, Henry's Law is the go-to principle. It's a straightforward concept: the amount of gas that dissolves in a liquid is proportionate to the pressure of that gas in the air above the liquid. Think of it like a party where the number of guests inside depends on how many are waiting outside; the higher the pressure or the bigger the crowd, the more will want to come in.

However, not all parties are the same. Similarly, the 'invitation rule' (Henry's Law constant, '\( k \)') varies with different gases and liquids, which is why seltzer gets its fizz from carbon dioxide under pressure, while oxygen bubbles out of your fish tank. This rule helps us predict how gases will behave when bottled up, or even in our bloodstreams when we go scuba diving!

Gas Solubility and Pressure

Imagine squeezing more air into a bicycle tire. The more you pump, the tighter the air is packed, and just like that, the solubility of gases in a liquid cranks up as the pressure rises. If you're hiking up a mountain, the air is thinner, meaning less oxygen is available to cram into your water bottle. On the flip side, deep-sea creatures are soaking in oxygen because there's so much more crammed into the water at those high pressures.

What's fascinating is this relationship isn't linear for all gases. For some, as you pump up the pressure, they don't just dissolve a bit more; they can vastly more soluble, depending on the unique traits of the gas and the liquid. This concept isn't just academic; it has practical implications, from carbonated drinks to how divers must avoid 'the bends.'

Calculating Solubility

‘Calculating solubility’ is a bit like following a recipe. Just as you need the right ingredients in the right amounts, you need specific information to calculate solubility. With gases, it comes down to the pressure of the gas and the nature of the liquid, defined by Henry's Law constant, '\( k \)'—a unique value for each gas-liquid pair. Knowing this constant allows you to stir the pot: change the pressure, and you can whip up a prediction of how much more or less gas will dissolve.

Just like doubling the sugar in a cake makes it sweeter, doubling the pressure, according to Henry's Law, doubles the gas's solubility in a liquid. To predict these changes, you apply the ‘recipe’ '\( S = kP \)‘, where '\( S \)‘ is solubility and '\( P \)‘ is pressure. It's as easy as pie, as long as you know your constants and pressures!

Chemical Equilibrium

Chemical equilibrium arrives at a party when guests are entering and leaving at the same pace. Applied to solubility, it's the state achieved when the rate at which the gas molecules dissolve in the liquid equals the rate at which they escape back into the gas phase. It's like a dance floor with a constant number of dancers, even though some may step off for a break while others join in.

When you first crack open a soda, you can think of equilibrium as the lively fizzing. With time, that dynamic changes—equal going in as coming out—and the fizz fades. But equilibrium isn't just for sodas; it's a pivotal concept in chemistry that keeps reactions in balance, from industrial processes to cellular respiration. It's all about that perfect balance, the 'Goldilocks' state, where everything is just right.