Question

Assume that you have a liquid in a cylinder equipped with a movable piston. There is no air in the cylinder, the volume of space above the liquid is \(200 \mathrm{~mL}\), and the equilibrium vapor pressure above the liquid is \(28.0 \mathrm{~mm} \mathrm{Hg}\). What is the equilibrium pressure above the liquid when the volume of space is decreased from \(200 \mathrm{~mL}\) to \(100 \mathrm{~mL}\) at constant temperature?

Short answer

The equilibrium pressure remains 28.0 mm Hg.

Step-by-step solution

Understanding the Problem

We need to find the equilibrium pressure above a liquid in a piston-cylinder system when the volume decreases from \(200\, \text{mL}\) to \(100\, \text{mL}\). The equilibrium vapor pressure is given as \(28.0\, \text{mm Hg}\).

Apply the Concept of Vapor Pressure

The vapor pressure of a liquid is determined by the temperature and the nature of the liquid. At a constant temperature, reducing the volume above the liquid does not change the equilibrium vapor pressure; it remains \(28.0\, \text{mm Hg}\). This is because the vapor pressure is solely dependent on temperature for a given liquid.

Consider the Effects of Space Reduction

Although reducing the space from \(200\, \text{mL}\) to \(100\, \text{mL}\) would increase the concentration of vapor molecules, the vapor pressure does not increase with reduced volume in this case since the system is at equilibrium and temperature remains constant.

State the Final Equilibrium Pressure

The reduction in volume does not affect the equilibrium vapor pressure because the equilibrium is governed by the temperature, which is constant. Thus, the equilibrium pressure remains \(28.0\, \text{mm Hg}\).

Key concepts

Piston-Cylinder System

A piston-cylinder system is a common setup in chemistry and physics, particularly in experiments involving gases and vapors. Imagine a cylinder with a tight-fitting movable piston that can slide up and down. The piston can change the volume of the gas or vapor inside the cylinder by moving. This adjustment allows us to study how gases behave under different conditions.

In our exercise, the piston-cylinder system is closed off, containing a liquid and vapor without any air. This setup is crucial because it ensures that only the vapor pressure of the liquid is at play, making the scenario easier to analyze. The controlled environment helps us focus solely on the properties of the liquid and its vapor without interference from external factors.

This type of system is invaluable in understanding how vapors reach equilibrium, how pressures adjust, and how temperature influences these processes.

Constant Temperature

Maintaining a constant temperature is a key factor in studying vapor pressure, because vapor pressure is primarily determined by temperature. When a system like our piston-cylinder setup is at a constant temperature, the kinetic energy of the molecules remains the same.

This means that the rate at which molecules evaporate into the vapor phase and condense back into the liquid phase is unchanged. That's why in our exercise, even when the volume is reduced, the equilibrium vapor pressure remains unchanged at 28.0 mm Hg.

Keeping the temperature constant ensures that the vapor pressure reflects solely the intrinsic properties of the liquid, unaltered by external thermal variations. Being able to presume a constant temperature simplifies calculations and predictions about how the system will behave when other variables, such as volume, are altered.

Vapor Pressure and Volume

Vapor pressure is often misunderstood, particularly in relation to volume changes. A common misconception is that decreasing the volume in a closed system increases the vapor pressure. In fact, vapor pressure at equilibrium is independent of the volume of space the vapor occupies in a closed system.

In this case, though reducing the volume from 200 mL to 100 mL causes more vapor molecules to be packed in a smaller space, it doesn't increase the equilibrium vapor pressure. Why? Because vapor pressure depends on the temperature and the nature of the liquid, not the space they are in.

Therefore, regardless of how much you compress the space, the vapor pressure will remain steady as long as the temperature is constant. This principle is a crucial concept in thermodynamics and impacts how chemists and engineers design and interpret experiments relating to gas laws and vapor equilibrium.

Liquid-Vapor Equilibrium

Liquid-vapor equilibrium occurs when a liquid and its vapor are in a balanced state, meaning that the rate of evaporation equals the rate of condensation. When this equilibrium is achieved, the system's vapor pressure is determined by the temperature of the liquid.

In our piston-cylinder system, once equilibrium is established, any attempt to change the system's volume, such as compressing the vapor space, does not alter the equilibrium vapor pressure. This is because equilibrium vapor pressure solely depends on temperature.

Understanding liquid-vapor equilibrium is critical because it explains the stability of gas pressure in closed systems, even when the external conditions like volume are changed. Engineers use this concept to predict how gases behave in equipment like engines, refrigerators, and even human respiratory systems.