Question

At sea level and \(25^{\circ} \mathrm{C}\), the solubility of oxygen gas in water is \(1.25\) \(\times 10^{-3} \mathrm{M}\). In a city in the United States that lies high above sea level, the atmospheric pressure is \(0.800 \mathrm{~atm}\). What is the solubility of oxygen in water in this city? A. \(1.05 \times 10^{-3} \mathrm{M}\) B. \(1.56 \times 10^{-3} \mathrm{M}\) C. \(1.00 \times 10^{-3} \mathrm{M}\) D. \(1.25 \times 10^{-3} \mathrm{M}\)

Short answer

C. \(1.00 \times 10^{-3} \mathrm{M}\)

Step-by-step solution

Understand Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas over the liquid. Mathematically, it can be expressed as \[ S_1 = k_H \times P_1 \] and \[ S_2 = k_H \times P_2 \] where \( S \) is the solubility, \( P \) is the pressure, and \( k_H \) is Henry's constant.

Set up the ratio using Henry's Law

We can set up a ratio since Henry's constant \( k_H \) remains the same for a given gas at a constant temperature. Thus:\[ \frac{S_1}{S_2} = \frac{P_1}{P_2} \]

Assign known values

Given that \(S_1 = 1.25 \times 10^{-3} \mathrm{M} \), \( P_1 = 1.00 \mathrm{~atm} \), and \( P_2 = 0.800 \mathrm{~atm} \), we need to find \( S_2 \).

Rearrange the ratio to solve for \( S_2 \)

Rearrange the equation to solve for \( S_2 \):\[ S_2 = S_1 \times \frac{P_2}{P_1} \]

Substitute and solve for \( S_2 \)

Substituting in the known values:\[ S_2 = (1.25 \times 10^{-3} \mathrm{M}) \times \frac{0.800 \mathrm{~atm}}{1.00 \mathrm{~atm}} \]\[ S_2 = 1.00 \times 10^{-3} \mathrm{M} \]

Choose the correct answer

The calculated solubility of oxygen is \(1.00 \times 10^{-3} \mathrm{M}\). Thus, the correct answer is option C.

Key concepts

Solubility of Gases

The solubility of gases in liquids is a fascinating concept tied to everyday life. Imagine how fish breathe underwater. It's mainly due to gases like oxygen dissolving in water. Solubility refers to the amount of gas that can be dissolved in a liquid under specific conditions.
Several factors impact the solubility of gases:

• Temperature: Higher temperatures often decrease gas solubility in liquids.
• Pressure: Increasing pressure typically increases gas solubility.

Henry's Law is vital here. It quantifies the relationship between the pressure of a gas and its solubility in a liquid.

Partial Pressure

Partial pressure helps us understand how different gases behave in a mixture. When breathing, for example, the air we inhale is a mix of gases, each contributing to the total pressure.
Partial pressure is defined as the pressure exerted by a single gas in a mixture. Imagine different gases sharing space in a room, each one pushing against the walls.
According to Henry's Law, the solubility of a gas in a liquid is directly proportional to this partial pressure. So, if the atmospheric pressure decreases, the solubility of gases like oxygen in water also decreases.
It is mathematically expressed as:
\[ S = k_H \times P \]
where \( S \) is solubility, \( k_H \) is Henry's constant, and \( P \) is the partial pressure.

Solubility Calculations

Let's break down solubility calculations using Henry's Law in a simple way.
The exercise essentially involves determining how the solubility of oxygen in water changes with altitude, where pressures are lower.

• Start by noting the provided solubility at sea level: \(1.25 \times 10^{-3} \) M at \(1.00 \) atm.
• Establish that partial pressure at higher altitudes is \(0.800 \) atm.
• Use the relationship:\[ \frac{S_1}{S_2} = \frac{P_1}{P_2} \]

Rearrange to find \( S_2 \):\[ S_2 = S_1 \times \frac{P_2}{P_1} \]
Substitute the known values:
\( S_2 = (1.25 \times 10^{-3} \text{ M}) \times \frac{0.800}{1.00} \)
The new solubility, \( S_2 \), is \( 1.00 \times 10^{-3} \text{ M} \).
This is the final answer, showing how gas solubility decreases with reduced partial pressure.