Question

Three \(1.0-\mathrm{L}\) flasks, maintained at \(308 \mathrm{~K},\) are connected to each other with stopcocks. Initially, the stopcocks are closed. One of the flasks contains \(1.0 \mathrm{~atm}\) of \(\mathrm{N}_{2} ;\) the second, \(2.0 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{O} ;\) and the third, \(0.50 \mathrm{~g}\) of ethanol, \(\mathrm{C}_{2} \mathrm{H}_{6} \mathrm{O}\). The vapor pressure of \(\mathrm{H}_{2} \mathrm{O}\) at \(308 \mathrm{~K}\) is \(42 \mathrm{mmHg},\) and that of ethanol is \(102 \mathrm{mmHg} .\) The stopcocks are then opened and the contents mix freely. What is the pressure?

Short answer

After calculating the moles of H2O and C2H6O, and using their vapor pressures, the total pressure in the combined flasks is the sum of pressures of N2, H2O, and C2H6O.

Step-by-step solution

Convert Vapor Pressures to Atmospheres

Firstly, convert the vapor pressures of water (H2O) and ethanol (C2H6O) into atmospheres, since the pressure of N2 is given in atmospheres. Use the conversion factor 1 atm = 760 mmHg. For water, the pressure in atm is 42 mmHg * (1 atm / 760 mmHg). For ethanol, the pressure in atm is 102 mmHg * (1 atm / 760 mmHg).

Calculate the Number of Moles for H2O and C2H6O

Use the ideal gas law equation PV = nRT, where n = number of moles, R = 0.0821 L·atm/mol·K (ideal gas constant), and T = temperature in Kelvin, to calculate the moles. Since we want to find the number of moles without pressure, n = mass/MW (molecular weight). Calculate the number of moles for water (H2O) using its mass (2.0 g) and molecular weight (18 g/mol). Calculate the number of moles for ethanol (C2H6O) using its mass (0.50 g) and molecular weight (46 g/mol).

Calculate Partial Pressures of H2O and C2H6O

Using Dalton's Law of partial pressures, which states that total pressure of a gas mixture is the sum of the partial pressures of individual gases, calculate the partial pressures. For each, P = nRT/V, where n is number of moles from step 2, R is the ideal gas constant, T is temperature in Kelvin, and V is volume (1.0 L). Since we are at equilibrium, the partial pressure is also the vapor pressure we calculated in step 1.

Calculate Total Pressure in the Combined Flasks

Add the partial pressures of N2, H2O, and C2H6O to get the total pressure in the system. The pressure of N2 is already given as 1.0 atm. The partial pressures of H2O and C2H6O are obtained from step 3. Sum these values to find the total pressure.

Key concepts

Dalton's Law of Partial Pressures

When different gases are mixed together, each gas exerts pressure as if it were alone in the volume. This principle is the foundation of Dalton's Law of Partial Pressures, which states that the total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of each individual gas. The partial pressure of each gas is directly proportional to its mole fraction in the mixture.

To understand this with a practical example, consider a scenario where three different gases are in separate flasks with stopcocks. When these stopcocks are opened, the gases mix, and Dalton's Law can be used to predict the total pressure of the combined gaseous system by adding up the partial pressures of each gas. It's a straightforward yet powerful tool for predicting the behavior of gas mixtures in closed systems.

Ideal Gas Law

The Ideal Gas Law is a crucial equation in the study of thermodynamics and chemistry. It forms the relationship between pressure (P), volume (V), number of moles (n), and temperature (T) for an ideal gas. Expressed as the formula \(PV = nRT\), where R is the universal gas constant (0.0821 L·atm/mol·K), it enables us to calculate one of the four variables if the other three are known.

The Ideal Gas Law assumes that the particles of an ideal gas do not attract or repel each other and occupy no volume. Although real gases do not perfectly follow the assumptions, the law provides a good approximation under many conditions. For example, when a substance like water or ethanol is in a closed flask, and we are given its mass and the temperature, we can use the Ideal Gas Law to compute the number of moles of vapor present, which then helps to determine the vapor pressure of the substance in the flask.

Vapor Pressure

The pressure exerted by a vapor in equilibrium with its liquid or solid phase is called vapor pressure. It is a specific type of partial pressure and is critical in understanding the behavior of substances in their vapor phase. The vapor pressure of a substance depends on the temperature; as temperature increases, more molecules have enough energy to escape the liquid or solid phase, thereby increasing the vapor pressure.

The vapor pressures of water and ethanol are given in the exercise, indicating the pressure each vapor would exert if it were alone in the flask at a certain temperature. The ability to convert these values into atmospheres, understand them, and relate them to Dalton's Law is crucial. For instance, at a certain temperature, water has a specified vapor pressure, and when the flask is closed, that vapor pressure represents the partial pressure of water vapor in equilibrium with the liquid or solid phase of water inside the flask.