The Henry's law constant of oxygen in water at \(25^{\circ} \mathrm{C}\) is \(1.3 \times 10^{-3} \mathrm{~mol} / \mathrm{L} \cdot\) atm. Calculate the molarity of oxygen in water under 1 atmosphere of air. Comment on the prospect for our survival without hemoglobin molecules. (Recall from previous problems that the total volume of blood in an adult human is about \(5 \mathrm{~L}\).)

Henry's Law is a fundamental principle that defines the relationship between the solubility of a gas in a liquid and the pressure of the gas above the liquid. The equation mentioned in Henry's Law is simply qualified as \( C = kP \) where \( C \) is the concentration of the gas dissolved in the liquid (expressed as molarity), \( k \) represents the Henry's Law constant for the particular gas-liquid combination, and \( P \) is the partial pressure of the gas over the liquid.

Henry's Law constant is specific to the gas in question and also depends on the temperature. A higher constant indicates a greater solubility of the gas in the liquid at a given pressure. It's crucial for various scientific and industrial processes — such as when calculating how much carbon dioxide can dissolve in a soda bottle under certain pressure conditions. In the aforementioned exercise, the Henry's Law constant for oxygen in water at 25°C is \( 1.3 \times 10^{-3} \mathrm{~mol/L \cdot atm} \), which can be used to predict how much oxygen will dissolve in water at a certain pressure.

Molarity refers to the concentration of a solute in a solution and is defined as the number of moles of the solute per liter of solution. In the context of our exercise, when calculating the molarity of oxygen in water, we're determining how many moles of oxygen would be present in each liter of water under a specific set of conditions, such as temperature and pressure.

According to Henry's Law, given that the pressure of the atmosphere is 1 atm and knowing the Henry's law constant for oxygen, we can determine the molarity of oxygen in water by simply multiplying these two values together. From our step by step solution, we get the molarity of oxygen in water to be \( 1.3 \times 10^{-3} \text{mol/L} \), under atmospheric conditions. This measurement is critical for assessing the amount of oxygen available in aquatic environments, which has implications for the survival of aquatic life and the operation of ecosystems.

Gas solubility in liquids is the property that defines how well a gas dissolves in a liquid. This solubility can be greatly affected by factors such as temperature, pressure, and the nature of the liquid and gas involved. Generally, gases tend to be less soluble in liquids as the temperature increases, because higher temperatures provide more energy for the gas molecules to escape into the air.

In practical terms, this means that cold water can hold more dissolved oxygen than warm water, which is why aquatic organisms are often more stressed in warmer temperatures due to lower oxygen levels. Additionally, the solubility of gases in liquids is directly proportional to the pressure of the gas above the liquid, as encapsulated in Henry's Law. This principle explains why pressurized containers, like soda cans, can hold more carbonated gas than if the liquid were simply left open to the air. From a biological perspective, the solubility of oxygen in our blood is essential for our respiratory system and the transport of oxygen to various tissues, highlighting the profound implications gas solubility has on everyday life and various industrial and environmental processes.