Question

The small bubbles that form on the bottom of a water pot that is being heated (before boiling) are due to dissolved air coming out of solution. Use Henry's law and the solubilities given to calculate the total volume of nitrogen and oxygen gas that should bubble out of 1.5 \(\mathrm{L}\) of water upon warming from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) . Assume that the water is initially saturated with nitrogen and oxygen gas at \(25^{\circ} \mathrm{C}\) and a total pressure of 1.0 atm. Assume that the gas bubbles out at a tem- perature of \(50^{\circ} \mathrm{C}\) . The solubility of oxygen gas at \(50^{\circ} \mathrm{C}\) is 27.8 \(\mathrm{mg} / \mathrm{L}\) at an oxygen pressure of 1.00 atm. The solubility of nitrogen gas at \(50^{\circ} \mathrm{C}\) is 14.6 \(\mathrm{mg} / \mathrm{L}\) at a nitrogen pressure of 1.00 atm. Assume that the air above the water contains an oxygen partial pressure of 0.21 \(\mathrm{atm}\) and a nitrogen partial pressure of 0.78 \(\mathrm{atm.}\)

Short answer

To calculate the total volume of nitrogen and oxygen gas that bubbles out, first calculate the initial volume of gas using Henry's law at 25 degrees Celsius based on the partial pressures. Then calculate the final volume of gas at 50 degrees Celsius using the given solubilities and convert to liters using the ideal gas law. Finally, subtract the final volumes from the initial volumes to determine the amount of gas that has bubbled out.

Step-by-step solution

Calculating the initial amount of oxygen and nitrogen dissolved in water

Initially at 25 degrees Celsius, the solubility of gases in water is the product of the solubility at this temperature (which is not provided but can be assumed to be saturated) and the partial pressures. Assuming saturation, and knowing air consists of approximately 21% oxygen and 78% nitrogen by volume, we can calculate the solubility using Henry's law, which states that the concentration of a gas in a liquid at constant temperature is directly proportional to the partial pressure of the gas in the equilibrium with the liquid. Thus, the initial amounts of O2 and N2 are the product of their respective partial pressures and the total volume of water.

Calculating the final amount of oxygen and nitrogen at 50 degrees Celsius

At 50 degrees Celsius, the solubility of gases in water is different based on Henry's law. Given the solubility of O2 and N2 at this temperature and 1 atm pressure, we can adjust these values to reflect the actual partial pressures found in the air above the water by multiplying the solubilities by the respective partial pressure ratios. The volume of each gas at 50 degrees Celsius can be calculated by multiplying the adjusted solubility value (in mg/L) by the water volume (1.5 L) and converting to liters of gas using the ideal gas law. For the calculations, we need to convert mg to g and then use molar masses of O2 (32.00 g/mol) and N2 (28.02 g/mol) to find the moles of each gas. Then using the ideal gas law (\( PV = nRT \)), we can calculate the volume the gas will occupy.

Determining the volume of gas that bubbles out

The gas that bubbles out will be the difference between the volume of gas dissolved at the initial temperature and the volume of gas at the final temperature. We can calculate this by subtracting the final volume (calculated in Step 2) from the initial volume (assuming initial saturation, as calculated in Step 1). The final answer will be the sum of the volumes of nitrogen and oxygen that bubble out.

Key concepts

Gas Solubility in Liquids

Understanding gas solubility in liquids is crucial for many scientific and industrial processes. Solubility is the capacity of a gas to dissolve in a liquid. Henry's law is commonly used to express this concept; it states that at a constant temperature, the amount of gas that dissolves in a liquid is directly proportional to the partial pressure of the gas above the liquid.

For instance, when you heat a pot of water, the water's ability to hold dissolved gases like nitrogen and oxygen decreases. This is because increased temperature often leads to a decrease in solubility, which is why you observe bubbles forming before the water boils.

In a practical sense, when solubility calculations are made, one must consider both the gas's solubility in the liquid at a specific temperature and the partial pressure of the gas. An essential point to remember is that each gas has its unique solubility in a particular liquid, which can change with temperature and pressure.

Partial Pressure

Relation to Gas Solubility

Partial pressure is a fundamental concept when discussing the solubility of gases in liquids. It refers to the pressure exerted by a single type of gas in a mixture of gases. The air we breathe, for example, is a mixture primarily composed of nitrogen and oxygen, each exerting its own partial pressure.

Through Henry's law, we can understand how the partial pressure of gas like oxygen or nitrogen above a liquid affects its dissolved concentration within the liquid. As the partial pressure of the gas increases, so does its solubility in the liquid, provided the temperature remains constant. This plays a significant role in calculations and observations, as seen in the bubbles forming in a pot of heating water. Here, the partial pressures of nitrogen and oxygen are critical to determining how much gas will escape as the water heats up.

Implication in Everyday Phenomena

Consider carbonated beverages, which are bottled under high CO₂ partial pressure to increase the gas's solubility, ensuring the drink remains fizzy until you open it and decrease the pressure, causing the gas to escape. Similarly, in our exercise, the gas escapes from the water with temperature increase due to the change in solubility and partial pressures.

Ideal Gas Law

Applying the Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates the pressure (P), volume (V), amount (in moles, n), and temperature (T) of an ideal gas, through the formula: \( PV = nRT \), where R is the ideal gas constant. This law is incredibly useful for calculations involving gases, especially when predicting the behavior of a gas under different conditions.

In the case of our exercise, we use the Ideal Gas Law to convert the mass of gases that have dissolved in the water at a certain temperature to a volume at another temperature. Here, it helps in computing the final volume of gas that escapes as the water temperature increases from 25°C to 50°C. Knowing the molar masses of oxygen and nitrogen, we can find the number of moles and apply the Ideal Gas Law to find the new volume the gases will occupy.

This law is assumed to hold accurately for most practical situations despite gases not being perfectly ideal. Deviations can occur at high pressures or low temperatures, but for the conditions like those in our exercise, the Ideal Gas Law provides a solid approximation for solving the problem at hand.