The Henry's law constant for the solubility of oxygen in water is \(3.30 \times 10^{-4} \mathrm{M} / \mathrm{atm}\) at \(12^{\circ} \mathrm{C}\) and \(2.85 \times 10^{-4} \mathrm{M} / \mathrm{atm}\) at \(22^{\circ} \mathrm{C}\). Air is \(21 \mathrm{~mol} \%\) oxygen. (a) How many grams of axygen can be dissolved in one liter of a trout stream at \(12^{\circ} \mathrm{C}\left(54^{\circ} \mathrm{F}\right)\) at an air pressure of \(1.00 \mathrm{~atm} ?\) (b) How many grams of oxygen can be dissolved per liter in the same trout stream at \(22^{\circ} \mathrm{C}\left(72^{\circ} \mathrm{F}\right)\) at the same pressure as in (a)? (c) A nuclear power plant is responsible for the stream's increase in temperature. What percentage of dissolved oxygen is lost by this increase in the stream's temperature?

Understanding the solubility of oxygen in water is pivotal for a wide range of scientific applications, from aquatic biology to chemical engineering. The solubility refers to the amount of oxygen that can be dissolved in water at a given temperature and pressure. Henry's law offers a precise way to quantify this solubility. It states that at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid.

In the provided exercise, we are particularly interested in how much oxygen dissolves in water from the air under specific conditions. This is directly relevant to environmental studies, such as assessing the health of a trout stream. The solubility is calculated using the Henry's law constant, which represents the solubility of oxygen at a specific temperature, expressed in molarity per atmosphere (\text{M/atm}). For example, at 12°C the constant is given as \(3.30 \times 10^{-4} \text{M/atm}\). By applying the law, we can estimate the amount of oxygen in moles that can be dissolved per liter of water and subsequently convert that number to grams.

Partial pressure plays a crucial role in understanding gas solubility in liquids, such as the solubility of oxygen in water. The term refers to the pressure exerted by a single type of gas in a mixture of gases. For example, in the Earth's atmosphere, which is a mixture of several gases, oxygen makes up approximately 21% of the volume, and therefore also contributes to 21% of the total atmospheric pressure.

To apply Henry's law, we must calculate the partial pressure of oxygen, abbreviated as \(P_{O_2}\). This is done by multiplying the percentage of oxygen in the air by the total atmospheric pressure, which is usually around 1 atm at sea level. Therefore, \(P_{O_2}\) can be estimated as \(21\% \times 1 \text{atm}\). Knowing the partial pressure allows us to use Henry's law to find out how much oxygen will dissolve in water at a specified temperature, effectively connecting the gas phase to the aqueous phase and enabling accurate predictions and calculations of gas solubility.

Temperature has a significant impact on the solubility of gases in liquids, with solubility typically decreasing as temperature increases. This inverse relationship is important in natural settings, such as streams and lakes, where the temperature can dictate the amount of oxygen available to aquatic organisms.

In our specific exercise, we examine the change in water temperature from 12°C to 22°C. As the temperature rises, the kinetic energy of the water molecules increases, resulting in more collisions between water and gas molecules. These collisions can cause the gas molecules to escape from the water, leading to decreased solubility. Different gases have different solubility dependencies on temperature, and oxygen is one such gas where solubility decreases noticeably with temperature increase.

The exercise requires us to calculate the percentage of dissolved oxygen lost due to a temperature increase, driven by a hypothetical increase from a nearby nuclear power plant. This calculation is essential for environmental studies and managing aquatic ecosystems, as even small changes in dissolved oxygen levels can have dramatic effects on the organisms living in the water.