Question

Consider an apparatus in which \(A\) and B are two \(1.00-\mathrm{L}\) flasks joined by a stopcock \(\mathrm{C}\). The volume of the stopcock is negligible. Initially, \(\mathrm{A}\) and \(\mathrm{B}\) are evacuated, the stopcock \(\mathrm{C}\) is dosed, and \(1.50 \mathrm{~g}\) of diethyl ether, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OC}_{2} \mathrm{H}_{5}\), is introduced into flask A. The vapor pressure of diethyl ether is 57 Torr at \(-45^{\circ} \mathrm{C}\), 185 Torr at \(0 .{ }^{\circ} \mathrm{C}, 534\) Torr at \(25^{\circ} \mathrm{C}\), and negligible below \(-86^{\circ} \mathrm{C}\). (a) If the stopcock is left closed and the flask is brought to equilibrium at \(-45^{\circ} \mathrm{C}\), what will be the pressure of diethyl ether in flask A? (b) If the temperature is raised to \(25^{\circ} \mathrm{C}\), what will be the pressure of diethyl ether in the flask? (c) If the temperature of the assembly is returned to \(-45^{\circ} \mathrm{C}\) and the stopcock \(\mathrm{C}\) is opened, what will be the pressure of diethyl ether in the apparatus? (d) If flask \(\mathrm{A}\) is maintained at \(-45^{\circ} \mathrm{C}\) and flask B is cooled with liquid nitrogen (boiling point, \(-196^{\circ} \mathrm{C}\) ) with the stopcock open, what changes will take place in the apparatus? Assume ideal behavior.

Short answer

a) 57 Torr, b) 534 Torr, c) 57 Torr. d) Diethyl ether will condense in flask B; vapor pressure is determined by conditions in flask A at 57 Torr.

Step-by-step solution

Determine the Pressure of Diethyl Ether in Flask A at -45°C (Part a)

The pressure of diethyl ether in flask A at equilibrium at -45°C is equal to the vapor pressure of diethyl ether at that temperature. Therefore, the pressure in flask A will be 57 Torr.

Calculate the Pressure of Diethyl Ether in Flask A at 25°C (Part b)

Similarly to part a, the pressure of diethyl ether in flask A at 25°C will equal the vapor pressure of diethyl ether at 25°C. According to the given data, the pressure in flask A will be 534 Torr.

Determine the Pressure in the Apparatus After Opening C at -45°C (Part c)

After the stopcock C is opened at -45°C, the diethyl ether gas will distribute evenly throughout the entire 2.00 L apparatus (1.00 L from flask A and 1.00 L from flask B combined). Since the amount of gas and its temperature remain constant, by combining the volumes, the pressure will also remain constant. The pressure of diethyl ether in the apparatus will still be 57 Torr.

Describe Changes When Flask B is Cooled and Flask A is at -45°C (Part d)

When flask B is cooled with liquid nitrogen at -196°C and flask A is at -45°C with the stopcock open, the vapor pressure of diethyl ether becomes negligible at such a low temperature (-196°C). Therefore, the diethyl ether will condense in flask B, and essentially no vapor will be present in flask B. As a result, practically all the diethyl ether will be in flask A, where the pressure is determined by the vapor pressure at -45°C, which is 57 Torr.

Key concepts

Chemical Equilibrium

Chemical equilibrium plays a critical role when analyzing the vapor pressure of substances, such as diethyl ether in this exercise. It refers to the state reached when the rate of the forward reaction equals the rate of the reverse reaction, meaning the concentrations of reactants and products remain constant over time, but not necessarily equal. In the context of vapor pressure, equilibrium is the point at which the rate of evaporation of a liquid equals the rate of condensation of its vapor.

When diethyl ether is placed in a closed flask and allowed to come to equilibrium at -45°C, its vapor pressure is the equilibrium pressure exerted by the ether vapor above the liquid. This value is intrinsic to the substance at a given temperature and does not depend on the amount of liquid provided there is some liquid present. It is essential to understand this concept because it helps predict how a substance behaves when subjected to temperature changes or when it is allowed to move between containers with varying volumes.

Gas Laws

The gas laws are a set of laws that describe the behavior of gases under various conditions of temperature, volume, and pressure. These laws are crucial to solving problems like this, where the behavior of a gas—in this case, diethyl ether vapor—needs to be predicted under different conditions.

Boyle's Law, for instance, states that the pressure and volume of a gas have an inverse relationship when temperature is held constant. However, in our exercise, when the stopcock is opened, the volume changes but the pressure remains constant due to the temperature remaining constant and the amount of gas not changing, illustrating an application of Avogadro's Law, wherein equal volumes of gases at the same temperature and pressure contain the same number of particles. By understanding the gas laws, students can deduce the behavior of gases in the system without performing experimental trials.

Phase Changes

Phase changes are the processes by which a substance moves from one state of matter to another, such as from a solid to a liquid (melting) or from a liquid to a gas (evaporation/boiling). The exercise delves into phase changes when describing the behavior of diethyl ether as it is heated or cooled.

At low enough temperatures, the kinetic energy of the molecules in the diethyl ether vapor is insufficient to overcome intermolecular forces, and the ether condenses into a liquid. When flask B is cooled with liquid nitrogen, the diethyl ether will condense, highlighting the phase change from gas to liquid due to a significant decrease in temperature. This understanding of phase changes, combined with knowledge of the substance-specific vapor pressures, allows us to predict the physical state of the substance within the apparatus.

Temperature-Pressure Relationship

Understanding the temperature-pressure relationship is essential in predicting the behavior of vapors and gases. According to Gay-Lussac's Law, for a given mass and volume of a gas, the pressure is directly proportional to its temperature (in Kelvin), which implies that as the temperature increases, so does the pressure, if the volume is held constant.

In the exercise, this principle explains why the pressure of diethyl ether in flask A increases from 57 Torr at -45°C to 534 Torr at 25°C. The diethyl ether exhibits a higher vapor pressure at a higher temperature since the increased kinetic energy of molecules results in a greater number of molecules escaping from the liquid to the gas phase, raising the pressure. Consequently, temperature adjustments play a pivotal role when working with the vapor pressures of substances and in understanding the outcomes of the experiment.