Question

Hydrogen is collected over water at \(25^{\circ} \mathrm{C}\) and \(748 \mathrm{~mm} \mathrm{Hg}\) in a 250-mL (3 significant figures) flask. The vapor pressure of water at \(25^{\circ} \mathrm{C}\) is \(23.8 \mathrm{~mm} \mathrm{Hg}\) (a) What is the partial pressure of hydrogen? (b) How many moles of water are in the flask? (c) How many moles of dry gas are collected? (d) If \(0.0186 \mathrm{~g}\) of \(\mathrm{He}\) are added to the flask at the same temperature, what is the partial pressure of helium in the flask? (e) What is the total pressure in the flask after helium is added?

Short answer

Question: Calculate the total pressure in the flask after helium is added. Answer: The total pressure in the flask after helium is added is 1.3867 atm.

Step-by-step solution

(Step 1: Calculate the partial pressure of hydrogen gas)

We are given the total pressure and vapor pressure of water, and we can use these values to determine the partial pressure of hydrogen gas using the following formula: \(P_{H_2} = P_{total} - P_{H_2 O}\) Substituting the given values: \(P_{H_2} = 748\,\mathrm{mm\,Hg} - 23.8\,\mathrm{mm\,Hg} = 724.2\,\mathrm{mm\,Hg}\)

(Step 2: Calculate the moles of water in the flask)

We can find the moles of water using the Ideal Gas Law, \(PV = nRT\). First, we need to convert the given volume to liters and the pressure to atmospheres. Volume in liters: \(250\,\mathrm{mL}=\frac{250\,\mathrm{mL}}{1000}=0.250\,\mathrm{L}\) Pressure in atmospheres: \(P_{H_2 O} = \frac{23.8\,\mathrm{mm\,Hg}}{760} = 0.03132\,\mathrm{atm}\) Rearranging the Ideal Gas Law to solve for moles (n), we get: \(n=\frac{PV}{RT}\) Using the values for water vapor and the gas constant (R = 0.0821 L∙atm/mol∙K): \(n_{H_2 O} = \frac{0.03132\,\mathrm{atm} \times 0.250\,\mathrm{L}}{0.0821\,\mathrm{L\,atm\,mol^{-1}\,K^{-1}}\times(273.15 + 25)\,\mathrm{K}} = 0.001086\,\mathrm{mol}\)

(Step 3: Calculate the moles of dry hydrogen gas)

Using the Ideal Gas Law again, we can calculate the moles of dry hydrogen gas. Pressure in atmospheres: \(P_{H_2} = \frac{724.2\,\mathrm{mm\,Hg}}{760} = 0.9529\,\mathrm{atm}\) Using the values for hydrogen and the gas constant (R = 0.0821 L∙atm/mol∙K): \(n_{H_2}=\frac{0.9529\,\mathrm{atm}\times 0.250\,\mathrm{L}}{0.0821\,\mathrm{L\,atm\,mol^{-1}\,K^{-1}}\times (273.15 + 25)\,\mathrm{K}} = 0.008709\,\mathrm{mol}\)

(Step 4: Calculate the partial pressure of helium)

First, let's convert the mass of helium to moles: \(n_{He} = \frac{0.0186\,\mathrm{g}}{4.00\,\mathrm{g\,mol^{-1}}} = 0.00465\,\mathrm{mol}\) Now, we can use the Ideal Gas Law to find the partial pressure of helium in the flask. \(P_{He}=\frac{n_{He}RT}{V} = \frac{0.00465\,\mathrm{mol} \times 0.0821\,\mathrm{L\,atm\,mol^{-1}\,K^{-1}} \times (273.15 + 25)\,\mathrm{K}}{0.250\,\mathrm{L}} = 0.4025\,\mathrm{atm}\)

(Step 5: Calculate the total pressure in the flask)

To find the total pressure in the flask, add the partial pressures of each gas. \(P_{total} = P_{H_2} + P_{H_2 O} + P_{He}= 0.9529\,\mathrm{atm} + 0.03132\,\mathrm{atm} + 0.4025\,\mathrm{atm} = 1.3867\,\mathrm{atm}\)

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and amount (in moles) of an ideal gas. It is usually expressed as \(PV = nRT\), where \(P\) is the pressure of the gas, \(V\) is its volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. This law is especially useful in conditions where a gas behaves ideally, meaning its particles are point masses with no volume and no interactions, which generally occurs at high temperature and low pressure.

In the context of our exercise, we use this law to calculate the moles of water and dry hydrogen gas. These calculations rely on the assumption that gases behave ideally, which is typically valid for gases like hydrogen and helium under the described experiment conditions. It's important for students to convert all their measurements into the appropriate units, such as atmospheres for pressure and liters for volume, to ensure consistency with the gas constant \(R\) and temperature in Kelvin. In practice, corrections may be needed for real gases, but they are often neglected for simplicity in introductory chemistry problems.

Gas Collection Over Water

The technique of gas collection over water is a common laboratory method used to collect a gas that is not reactive with water. As a gas forms in a reaction, it displaces water in a container, usually an inverted measuring cylinder or flask submerged in water. The collected gas will be saturated with water vapor, meaning the gas will contain water vapor at the pressure equal to the vapor pressure at that specific temperature.

This vapor pressure must be accounted for when calculating the partial pressure of the dry collected gas. To determine the partial pressure of the gas collected, we subtract the vapor pressure of water from the total pressure inside the container. This is crucial since neglecting the vapor pressure would result in an overestimation of the gas's partial pressure and, consequently, its amount in moles. This adjustment is explicitly shown in step 1 of the exercise where we subtract the water vapor pressure from the total pressure to find the partial pressure of hydrogen gas.

Molar Mass of Gases

The molar mass of a gas is the weight of one mole (6.022 x 10²³ particles) of that gas. It is usually measured in grams per mole (g/mol) and plays a vital role in converting between the mass of a substance and the amount in moles. This conversion is especially important when dealing with gas collection experiments, as demonstrated in step 4 of our exercise, where the mass of helium added to the flask is converted to moles using the molar mass of helium, 4.00 g/mol.

Determining the molar mass of a gas can also enable us to derive the gas's density and to compare the properties of different gases. In the context of the given problem, knowing the molar mass of helium allows us to calculate its partial pressure once it is added to the flask. This understanding of molar mass is integral to performing accurate partial pressure calculations in a mixture of gases.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid (or solid) form at a given temperature. It is specific to each substance and increases with temperature because as more molecules have the energy to escape the liquid, the number of molecules transitioning to the vapor phase grows, raising the vapor pressure. The vapor pressure of water at a given temperature is a crucial reference point in many gas collection experiments.

In the exercise, we see that water vapor's pressure must be subtracted from the total atmospheric pressure to accurately calculate the partial pressure of the hydrogen gas collected over water. This is a pivotal step since ignoring the vapor pressure could lead to an incorrect determination of the amount of gas collected, as it would falsely include the water vapor in its calculation. Maintaining an accurate understanding of vapor pressure is key to grasping the nuances of gas behavior in varying experimental setups.