AP Chemistry: Gas Collection Over Water

Accurately measuring the properties of a gas is a cornerstone of chemical analysis, yet common lab techniques like water displacement introduce a hidden variable: water vapor. Understanding how to correct for this vapor pressure is not just a testable skill for the AP Chemistry exam; it's a fundamental practice in engineering fields dealing with gas systems and in pre-med contexts like analyzing respiratory gases.

The Setup: Why Water Displacement Complicates Measurements

When you collect a gas by water displacement, you typically bubble the gas through water into an inverted, water-filled container like a eudiometer or graduated cylinder. The gas rises and displaces the water, allowing you to measure its volume. However, the collected gas is not pure; it becomes saturated with water vapor. This means the total pressure you measure (often atmospheric pressure) is actually the sum of the pressure from the gas you produced and the pressure from the water vapor that evaporated into the space. If you use this total pressure directly in the ideal gas law to find moles of your target gas, your answer will be erroneously high because you're accounting for extra particles that aren't part of your chemical reaction.

Foundational Principle: Dalton's Law of Partial Pressures

To correct for the water vapor, you apply Dalton's Law of Partial Pressures. This law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of the individual gases. A partial pressure is the pressure a gas would exert if it alone occupied the entire volume. For a gas collected over water, the law is expressed as: $$P_{\text{total}}=P_{\text{dry gas}}+P_{\text{water vapor}}$$ Here, $P_{\text{total}}$ is usually the atmospheric pressure in the lab, $P_{\text{water vapor}}$ is the pressure due to water molecules in the gas phase, and $P_{\text{dry gas}}$ is the pressure attributable solely to the gas you're studying (e.g., hydrogen, oxygen, or carbon dioxide). Your primary goal is to rearrange this equation to solve for the dry gas pressure: $P_{\text{dry gas}}=P_{\text{total}}-P_{\text{water vapor}}$. Think of it like a shared bill: the total cost (atmospheric pressure) is the sum of your share (dry gas) and your friend's share (water vapor); to find what you owe, you subtract their known portion.

The Key Variable: Water Vapor Pressure and How to Find It

Water vapor pressure is not a constant; it depends almost exclusively on temperature. As temperature increases, more water molecules evaporate, increasing the vapor pressure inside the collection container. You cannot calculate this value from first principles in a typical lab problem; instead, you must look it up in a provided vapor pressure table. These tables list the pressure of water vapor (in units like mmHg, torr, or kPa) in equilibrium with liquid water at various temperatures. For example, at 22.0°C, the vapor pressure of water is approximately 19.8 mmHg. It is critical that you use the vapor pressure corresponding to the exact temperature of the water during the experiment, as even a few degrees difference can change the value significantly. On the AP exam, such a table will be provided if needed.

The Step-by-Step Correction Method

With the principles in hand, follow this systematic approach to analyze a gas collected over water.

1. Identify Known Quantities: Determine the total pressure ($P_{\text{total}}$), the volume of the collected gas ($V$), the temperature ($T$ in Kelvin), and the temperature of the water bath (for the vapor pressure lookup).
2. Find $P_{\text{water vapor}}$: Using the given temperature of the water, consult the vapor pressure table to find the partial pressure of water vapor at that temperature.
3. Calculate $P_{\text{dry gas}}$: Apply Dalton's Law: $P_{\text{dry gas}}=P_{\text{total}}-P_{\text{water vapor}}$. Ensure all pressure units are consistent (e.g., all in torr or all in atm).
4. Solve for the Amount of Gas: Use the ideal gas law, $PV=nRT$, but only with the dry gas values. Insert $P_{\text{dry gas}}$, $V$, $T$ (in K), and the appropriate gas constant $R$ to solve for $n$, the number of moles of the dry, target gas.

This corrected mole amount is then used in subsequent stoichiometry calculations to determine yields, purities, or molar masses.

Worked Example and AP-Style Application

Let's walk through a typical problem. In an experiment, 455 mL of hydrogen gas is collected over water at 25.0°C. The atmospheric pressure in the lab is 755 mmHg. The vapor pressure of water at 25.0°C is 23.8 mmHg. How many moles of hydrogen gas were produced?

Step 1: Knowns.

• $P_{\text{total}}=755\text{ mmHg}$
• $V=455\text{ mL}=0.455\text{ L}$
• $T=25.0^{\circ }\text{C}+273=298\text{ K}$
• $P_{\text{water vapor}}=23.8\text{ mmHg}$ (from table)

Step 2: Calculate dry gas pressure. $$P_{\text{dry }{H}_2}=P_{\text{total}}-P_{\text{water vapor}}=755\text{ mmHg}-23.8\text{ mmHg}=731.2\text{ mmHg}$$ We often convert pressure to atmospheres for use with $R=0.0821\frac{\text{Lcdotpatm}}{\text{molcdotpK}}$. $$P_{\text{dry }{H}_2}=731.2\text{ mmHg}\times \frac{1\text{ atm}}{760\text{ mmHg}}=0.9621\text{ atm}$$

Step 3: Apply the ideal gas law. $$PV=nRT$$ $$n=\frac{PV}{RT}=\frac{(0.9621\text{ atm})(0.455\text{ L})}{(0.0821\frac{\text{Lcdotpatm}}{\text{molcdotpK}})(298\text{ K})}$$ $$n\approx 0.0179\text{ moles of }{H}_2$$

An AP-style twist might ask for the molar mass of an unknown metal based on the gas produced. For instance, if this hydrogen came from reacting 0.500 g of magnesium with acid, you could now calculate experimental molar mass: moles of $H_2$ equals moles of Mg (from stoichiometry), so molar mass = mass / moles = $0.500\text{ g}/0.0179\text{ mol}\approx 27.9\text{ g/mol}$, close to magnesium's actual 24.3 g/mol, with discussion of error sources.

Common Pitfalls

• Forgetting the Subtraction: The most frequent error is using $P_{\text{total}}$ directly in the ideal gas law. Always remember: the gas collected is wet, and the ideal gas law requires the pressure of the specific gas you are modeling.
• Temperature Confusion: Using the wrong temperature for the vapor pressure lookup. The vapor pressure depends on the temperature of the water, which is usually the same as the gas temperature, but you must confirm this. Also, ensure $T$ is in Kelvin for the ideal gas law, but Celsius for the vapor pressure table.
• Unit Inconsistency: Mixing pressure units (e.g., subtracting mmHg from atm) without conversion. Standardize all pressures to one unit system before applying Dalton's Law or the ideal gas law.
• Misreading Tables: Vapor pressure tables can be dense. Double-check that you've selected the value for the correct temperature and note the units provided to avoid off-by-factor errors.

Summary

• Gas collected over water is mixed with water vapor, so the measured total pressure is the sum of the partial pressures of the dry gas and water vapor.
• Dalton's Law of Partial Pressures, $P_{\text{total}}=P_{\text{dry gas}}+P_{\text{water vapor}}$, is the essential tool for correcting this error.
• Water vapor pressure is temperature-dependent and must be obtained from a provided reference table; it cannot be assumed or calculated from the ideal gas law in this context.
• The core calculation is $P_{\text{dry gas}}=P_{\text{total}}-P_{\text{water vapor}}$, and this corrected pressure is used in the ideal gas law to find the true amount of gas.
• Always verify temperature units (Kelvin for ideal gas law, Celsius for vapor pressure) and ensure consistent pressure units throughout your calculations.
• This technique is vital for accurate stoichiometry and property determination in chemistry, with direct applications in environmental engineering and medical gas analysis.