Question

The volume of hydrogen evolved during the course of a reaction is measured by the displacement of water as shown in the diagram below. Hydrogen is evolved in flask \(\mathrm{A}\) and displaces water from flask B into beaker C. If, during a particular run of this experiment in which atmospheric pressure is 765 torr and the water temperature is \(293.15^{\circ} \mathrm{K}\), \(65.0 \mathrm{ml}\) of water is displaced, how much water would be displaced at 760 torr and \(298.15^{\circ} \mathrm{K} ?\) (The equilibrium vapor pressure of water at \(293.15^{\circ} \mathrm{K}\) is \(17.5\) torr and at \(298.15^{\circ} \mathrm{K}\) is 23,8 torr.

Short answer

The volume of water displaced at the new conditions of 760 torr and \(298.15^{\circ} \mathrm{K}\) is approximately \(67.7\, \text{ml}\).

Step-by-step solution

Calculate the pressure of hydrogen at the initial conditions

\(P_{H_2} = P_{atm} - P_{H_2O}\) \(P_{H_2} = 765 \,\text{torr} - 17.5\, \text{torr}\) \(P_{H_2} = 747.5 \, \text{torr}\) Step 2: Apply the combined gas law To find the volume of water displaced at the new conditions, we need to apply the combined gas law. This law states that for a given amount of gas, the ratio of the initial pressure multiplied by the initial volume divided by the initial temperature is equal to the ratio of the final pressure multiplied by the final volume divided by the final temperature. The combined gas law can be written as: \(\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\) Since we are only interested in the final volume, we can solve for it as follows: \(V_2 = \frac{P_1V_1T_2}{T_1P_2}\) Step 3: Calculate the pressure of hydrogen at the new conditions Before we plug in the known values into the combined gas law, we need to find the pressure of hydrogen at the new conditions. We are given the equilibrium vapor pressure of water at \(298.15^\circ \mathrm{K}\) as \(23.8\) torr, and the new atmospheric pressure is \(760\) torr. By subtracting the pressure of water vapor from the new atmospheric pressure, we can find the pressure of hydrogen.

Calculate the pressure of hydrogen at the new conditions

\(P_{H_2}\, = P_{atm} - P_{H_2O}\) \(P_{H_2}\, = 760 \, \text{torr} - 23.8\, \text{torr}\) \(P_{H_2}\, = 736.2\, \text{torr}\) Step 4: Plug in the values into the combined gas law and solve for the final volume Now, we can input the values we have found into the combined gas law equation and solve for the volume at the new conditions:

Apply the combined gas law to find the new volume

\(V_2 = \frac{P_1V_1T_2}{T_1P_2}\) \(V_2 = \frac{(747.5\, \text{torr})(65.0\, \text{ml})(298.15\, \mathrm{K})}{(293.15\, \mathrm{K})(736.2\, \text{torr})}\) \(V_2 ≈ 67.7\, \text{ml}\) The volume of water displaced at the new conditions is approximately \(67.7\, \text{ml}\).

Key concepts

Gas Law Calculations

Understanding gas law calculations is crucial for students studying chemistry, as these calculations pertain to the behavior of gases under various conditions of pressure, volume, and temperature. The combined gas law integrates Boyle's Law, Charles's Law, and Gay-Lussac's Law, showing the relationship between the pressure (P), volume (V), and temperature (T) of a fixed amount of gas.

It's important to note that in these calculations, temperature must always be in Kelvin, and pressure should be consistent throughout the calculation. For example, in the given exercise, we see adjustments made for atmospheric pressure by accounting for water vapor pressure to find the true pressure exerted by the hydrogen gas. This pressure correction ensures the calculations are accurate.

When using the combined gas law formula, \[\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\], students must understand that P1, V1, and T1 represent initial conditions whereas P2, V2, and T2 represent final conditions. To solve for an unknown, algebraic manipulation is often required. This direct approach to solving the problem is an exercise improvement advice that helps students focus on the most crucial steps in gas law calculations.

Pressure-Volume-Temperature Relationships

The pressure-volume-temperature relationships, better known in scientific terms as the combined gas law, defines how gases change conditions in response to changes in pressure, volume, and temperature. This law tells us that if a gas's temperature increases, its volume tends to increase if the pressure remains constant. Similarly, if the pressure on a gas increases, its volume tends to decrease, provided the temperature is constant.

These relationships can be understood visually by imagining a syringe filled with air: when you pull the plunger out, you're increasing the volume and the pressure inside decreases if the temperature doesn't change. If you push the plunger in, you're decreasing the volume and the pressure increases.

In the context of the exercise, the volume of hydrogen gas evolved can be determined under different conditions using the combined gas law as a tool. Remembering to keep the units consistent and understanding that all gases behave similarly under these laws allows for smooth calculation and comprehension of these fundamental gas properties. Clarifying the influence of water vapor pressure on total pressure is crucial, as it is often a source of confusion but essential for accurate results.

Hydrogen Gas Evolution

Hydrogen gas evolution in a chemical reaction is a common laboratory procedure. When hydrogen is produced, it displaces water or another liquid, and the volume of the displaced liquid can be measured to determine the amount of hydrogen gas evolved.

During such an experiment, it's important to account for certain factors like atmospheric pressure, water vapor pressure, and temperature because they can affect the volume of hydrogen that is measured. For instance, the higher the atmospheric pressure, the more compressed the hydrogen gas is, and consequently, the less space it will take up.

The exercise shows this relationship by asking how much water would be displaced at a different temperature and pressure. Understanding hydrogen gas evolution is not only fundamental in chemistry for experimental purposes but also for industrial applications, including the production of cleaner fuel alternatives. As we move towards a hydrogen economy, knowledge of these reactions and calculations becomes even more pertinent.