Problem 38 - Chemistry The Central Science

Question

The partial pressure of \(\mathrm{O}_{2}\) in air at sea level is 0.21 atm. Using the data in Table \(13.1,\) together with Henry's law, calculate the molar concentration of \(\mathrm{O}_{2}\) in the surface water of a mountain lake saturated with air at \(20^{\circ} \mathrm{C}\) and an atmospheric pressure of 650 torr.

Step-by-Step Solution

Answer

The molar concentration of \(\mathrm{O}_{2}\) in the surface water of the mountain lake is approximately \(2.34 \times 10^{-4}\, \frac{\text{mol}}{\text{L}}\).

Step 1: Convert the pressure to atm

We need to convert the atmospheric pressure given in torr to atmospheres (atm), as we are going to use atm in our calculations. The conversion factor is 1 atm = 760 torr. So, we have \[650\,\text{torr} \times \frac{1\,\text{atm}}{760\,\text{torr}} \approx 0.855\,\text{atm}\]

Step 2: Calculate the partial pressure of O₂ in mountain lake air

We know that the partial pressure of \(\mathrm{O}_{2}\) in air at sea level is 21% (0.21 atm). Now, we must calculate the partial pressure of \(\mathrm{O}_{2}\) at the mountain lake air using the atmospheric pressure value we found in Step 1: \[0.21\,\text{atm}\times 0.855\,\text{atm}\approx 0.1796\,\text{atm}\]

Step 3: Obtain the Henry's law constant for O₂

From Table 13.1, the Henry's law constant for \(\mathrm{O}_{2}\) in water at \(20^{\circ} \mathrm{C}\) is: \[k_{\mathrm{O}_{2}} = 1.3 \times 10^{-3}\, \frac{\text{mol}}{\text{L}\cdot\text{atm}}\]

Step 4: Use Henry's law to calculate the molar concentration of O₂

Now, we can use Henry's law to find the molar concentration of \(\mathrm{O}_{2}\) in the surface water of the mountain lake: \[C_{\mathrm{O}_{2}} = k_{\mathrm{O}_{2}} \times P_{\mathrm{O}_{2}}\] \[C_{\mathrm{O}_{2}} = (1.3 \times 10^{-3}\, \frac{\text{mol}}{\text{L}\cdot\text{atm}}) \times 0.1796\, \text{atm}\] \[C_{\mathrm{O}_{2}} \approx 2.34 \times 10^{-4}\, \frac{\text{mol}}{\text{L}}\] The molar concentration of \(\mathrm{O}_{2}\) in the surface water of the mountain lake is approximately \(2.34 \times 10^{-4} \frac{\text{mol}}{\text{L}}\).

Key Concepts

Partial Pressure in Gases

Partial pressure refers to the pressure exerted by a single gas within a mix of gases. Think of it as the contribution each gas makes to the total pressure. In a mixture of gases, each component gas will exert its own partial pressure as if it were the only gas present.
For example, oxygen in the atmosphere contributes about 21% of the total atmospheric pressure at sea level. This is why its partial pressure is typically 0.21 atm under these conditions.
The concept is important when studying gas interactions and reactions, as it allows us to focus on one gas of interest without interference from others.

Understanding Molar Concentration

Molar concentration, often called molarity, is a way to express the concentration of a solute in a solution. It tells us how many moles of a substance are present per liter of solution. This is expressed as mol/L.
It's crucial in chemistry because reactions involve particles reacting in solution. By knowing the molar concentration, we can determine how much of a substance is available to participate in a reaction, which is essential for stoichiometric calculations.
In our context, calculating the molar concentration of \({O}_{2}\) in water helps in understanding how much oxygen is dissolved and available, for example, for aquatic life.

Atmospheric Pressure Conversion

Converting atmospheric pressure between units like torr and atm is sometimes necessary because different regions and scientific studies may use different units.
The conversion factor between torr and atm is directly linked to their definitions: 1 atm is approximately 760 torr. To convert torr to atm, divide the pressure in torr by 760.
This step is important in calculations involving gases because it standardizes the pressure, making it easier to apply laws like Henry's Law consistently.

The Role of Gas Solubility

Gas solubility is the ability of a gas to dissolve in a liquid. This is influenced by temperature, pressure, and the nature of the gas and liquid.
Henry's Law gives us a way to quantify gas solubility, stating that the amount of dissolved gas is proportional to its partial pressure above the liquid. This relationship is described by \(C = k \times P\), where \(C\) is concentration, \(k\) is Henry's constant, and \(P\) is partial pressure.
Understanding solubility helps us predict and explain real-world phenomena, such as why soda fizzes when opened or how aquatic life can obtain oxygen from water.

Sea Level Air Pressure Explained

Sea level air pressure is the reference atmospheric pressure at Earth's surface, approximately 1 atm or 760 torr. It's used as a standard reference point because it's where most weather measurements and standard conditions are defined.
This standard pressure helps compare atmospheric readings from different altitudes, allowing us to understand how pressure changes with elevation.
For example, at higher altitudes like a mountain lake, atmospheric pressure decreases. This affects gas solubility in water, influencing ecological systems and human activities.