Problem 102 - Chemistry The Science in Context

Question

Why does the partial pressure of water increase as temperature increases?

Step-by-Step Solution

Answer

Answer: The partial pressure of water is related to temperature through the concept of vapor pressure, which increases exponentially with the increase in temperature. As the temperature increases, the kinetic energy of water molecules also increases, leading to an increased evaporation rate and a subsequent increase in vapor pressure. Since the partial pressure of water in a mixture of gases depends on the vapor pressure of water, the partial pressure of water also increases with an increase in temperature.

Step 1: Understanding Vapor Pressure

Vapor pressure is a measure of the pressure exerted by a gas above a liquid in thermodynamic equilibrium (the point at which the rate of evaporation equals the rate of condensation). In this case, we are concerned with the vapor pressure of water. The vapor pressure of a substance depends on its temperature; when the temperature increases, the kinetic energy of the molecules also increases, which leads to an increase in evaporation rate.

Step 2: Relationship between vapor pressure and temperature

According to the Clausius-Clapeyron equation, the relationship between vapor pressure and temperature can be mathematically expressed as: \[ln(P)=A-\frac{B}{T+C}\] where \(P\) is the vapor pressure, \(T\) is the temperature in degrees Celsius, and \(A\), \(B\), and \(C\) are constants specific to the substance being considered (in our case, water). The equation shows that the vapor pressure increases exponentially with the increase in temperature.

Step 3: Understanding Partial Pressure

The partial pressure of a gas in a mixture is the pressure exerted by that particular gas if it alone occupied the volume. In our case, the partial pressure of water refers to the pressure exerted by water vapor in a mixture of gases (e.g., in the air above a body of water). Since the vapor pressure of water increases with temperature, the partial pressure of water in a mixture of gases will also increase with temperature.

Step 4: Partial Pressure of Water and Temperature Increase

As temperature increases, the kinetic energy of water molecules also increases. When the kinetic energy is high enough, more water molecules can break free from the liquid surface and turn into vapor. This results in an increased evaporation rate and, subsequently, an increase in vapor pressure. Given that the partial pressure of water in a mixture of gases depends on the vapor pressure of water, the partial pressure of water also increases with an increase in temperature.

Key Concepts

Partial Pressure

Partial pressure represents the pressure exerted by a single type of gas within a mixture. It is computed as if the particular gas occupied the entire volume on its own.
For instance, when we talk about the partial pressure of water, we are referring to the gaseous water vapor pressure in a mixture of gases, such as the air.
The relationship between vapor pressure and partial pressure is key because as the vapor pressure of a liquid increases, so does its partial pressure in any gaseous mixture. This increase is primarily driven by temperature changes.

• Higher temperature leads to increased kinetic energy.
• This causes more molecules to escape into the gas phase.
• Thus, the partial pressure, or pressure due to the specific gas molecules, increases.

A good visual analogy is to imagine a room (closed container), where the more gaseous water releases inside, the more pressure builds up from water vapor alone.

Evaporation Rate

The evaporation rate is the speed at which a liquid turns into vapor. The transition happens when molecules at the surface of a liquid gain enough energy to overcome intermolecular forces binding them to the liquid.
Temperature is a crucial player here. With an increase in temperature, molecules at the liquid's surface gain more energy.

• Higher temperature means higher kinetic energy for molecules.
• Molecules can more easily overcome surface tension.
• This results in a higher number of molecules escaping and an increased evaporation rate.

As the temperature continues to rise, the rate of evaporation continues to increase. This automatic evaporation boost leads to an enhanced vapor pressure, contributing directly to changes in partial pressures in a gaseous mixture.

Clausius-Clapeyron Equation

The Clausius-Clapeyron equation offers a solid mathematical framework to understand the relationship between vapor pressure and temperature.
The equation is expressed as:
\[ln(P)=A-\frac{B}{T+C}\]
Where:

• \(P\) = vapor pressure of the substance
• \(T\) = temperature in degrees Celsius
• \(A, B, C\) = substance-specific constants

It shows that vapor pressure increases exponentially with temperature.
This is because as temperature climbs, not only kinetic energy but also the escaping potential of molecules increases, thus providing a mathematical description of why partial pressures rise with temperature as vapor pressure does.