# Vapor Pressure Formula & Example

## Vapor Pressure Definition

The **vapor pressure** definition is the pressure exerted by the vapors of a substance in a closed system at equilibrium and a fixed temperature. The pressure exerted by vapors of a substance that is at thermodynamic equilibrium with its solid or liquid is directly proportional to temperature; the vapor pressure increases as the temperature increases. The vapor pressure also decreases as a response to any decrease in the temperature. The reason behind this is related to the relationship between temperature and the kinetic energy of the molecules. A rise in the temperature of a liquid supplies its molecules with energy; this leads to an increase in the kinetic energy of the molecules and the frequency of molecular collisions. Consequently, the number of molecules entering the gas phase increases, which in turn increases the vapor pressure.

What does equilibrium mean? Take a bottle of water as an example of a closed system, where liquid water evaporates and turns into vapor over time. The water vapor trapped in the bottle condenses and returns to the liquid phase. The system is at equilibrium when the amount of water is evaporating is equal to the water that is condensing.

### Vapor Pressure Examples

A real-life vapor pressure example is the pressure cooker. It is a closed chamber that traps the vapor and prevents it from escaping to the atmosphere. The pressure cooker works by the following principle: the water in the cooker evaporates and accumulates in the closed chamber. The increased number of vapor molecules consequently increases the vapor. This speeds up the process of cooking the food. When the heat supplied to the cooker is terminated, the vapor condenses and returns to the liquid phase, decreasing the vapor pressure.

Boiling tea in a kettle is another **example of vapor pressure**. When the water in the kettle begins to boil, the number of water molecules transitioning to the vapor phase increase. The vapor molecules exert pressure on the kettle's lid. This can be seen by the moving of the kettle's lid.

## What Is Vapor Pressure?

Water can take the form of a liquid, a solid (ice), or a gas (water vapor). Ice melts to form water, and water evaporates to form water vapor. Think about a water bottle like the one shown here:

What's inside? You can see water and air, but what you may not have realized is that there is also water vapor. In a water bottle, a little bit of the water is constantly evaporating and becoming water vapor. At the same time, some of the water vapor is constantly condensing to become a liquid. At equilibrium, the amount of water evaporating is equal to the amount of water vapor condensing, so the amounts of water and water vapor are constant.

The water vapor exerts a pressure on the water bottle in the same way that air pumped into a tire exerts pressure on the tire. The pressure exerted by the water vapor is the **vapor pressure**. In more general terms, *vapor pressure is the pressure exerted by a gas in equilibrium with the same material in liquid or solid form.* Just as distance can be measured in a variety of units (miles, feet, kilometers, etc.), we can also measure vapor pressure with many different units (kPa, atm, bar, mm Hg, torr).

## Vapor Pressure Formula

The first **vapor pressure formula** is derived from Raoult's Law. It is as follows:

{eq}P_a = x_a P_{0,a} {/eq}

Where:

- {eq}P_a {/eq} is the vapor pressure of species a in a mixture. Alternatively, known as the partial pressure of species a.

- {eq}x_a {/eq} is the molar fraction of species a.

- {eq}P_{0,a} {/eq} is the vapor pressure of pure a.

The units of {eq}P_a {/eq} and {eq}P_{0,a} {/eq} can be any pressure units for long as the same unit is being used for both parameters. The second vapor pressure formula relates the vapor pressure of a pure substance at different temperatures:

{eq}ln\frac{P_1}{P_2}=\frac{H_v}{R}(\frac{1}{T_2}-\frac{1}{T_1}) {/eq}

Where:

- {eq}ln {/eq} is the natural log.

- {eq}P_1 {/eq} is the vapor pressure at the initial temperature {eq}T_1 {/eq}.

- {eq}P_2 {/eq} is the vapor pressure at the final temperature {eq}T_2 {/eq}.

- {eq}H_v {/eq} is the substance's enthalpy of vaporization. Its unit is in kJ/mol.

- R is the universal gas constant. This constant value differs from unit to unit. R = 0.008314 kJ/mol K will be the value used in this lesson.

Note that the temperature must always be in Kelvins. The unit of pressure in this equation will be presented in atm. This is preferable to avoid any unnecessary unit conversion.

### How to Calculate Vapor Pressure?

The **vapor pressure equation** is derived from Raoult's Law and is used when determining the vapor pressure of a certain species in an ideal liquid mixture. **Raoult's Law** states that the partial pressure of a species in a vapor is directly proportional to its molar fraction in the ideal mixture. This means that the more of the substance is present as a gas in the mixture, the higher the pressure this substance exerts. Considering a mixture made of 0.6 of species a and 0.4 of species b, the vapor pressure exerted by a is higher than that exerted by b. Because there is more a molecules than there are b molecules.

#### Example 1

What is the vapor pressure of a mixture that is made of 120 g of acetone (mwt = 58g/mol) and 800 g of propanol (mwt = 60g/mol)? Knowing that the vapor pressure of acetone and propanol are 30mmHg and 21mmHg respectively at 25{eq}^{\circ} {/eq}C.

Molar fractions calculations:

{eq}n_a = 120g/58{g/mol}= 2.1mol {/eq}

{eq}n_p = 800g/60{g/mol}= 13.3 mol {/eq}

{eq}n_{total} = n_a + n_p = 2.1 + 13.3 = 15.4 mol {/eq}

{eq}x_a= 2.1/15.4=0.136 {/eq}

{eq}x_p=13.3/15.4=0.863 {/eq}

Partial pressure of each component:

{eq}P_a = x_aP_{0,a} = 0.136*30=4.08mmHg {/eq}

{eq}P_p = x_pP_{0,p} = 0.863*21=18.12mmHg {/eq}

The pressure of the vapor mixture

{eq}P_{mixture} = P_a + P_p = 4.08 + 18.12 = 22.2 mmHg {/eq}

#### Example 2

A vapor mixture made of 0.2 methanol and 0.8 ethanol. What is the vapor pressure of the mixture provided that the pressures of methane ethane at 25{eq}^{\circ} {/eq}C are 94mmHg and 44mmHg, respectively?

Partial pressure of each component:

{eq}P_m = x_mP_{0,m} = 0.2*94=18.8mmHg {/eq}

{eq}P_e = x_eP_{0,e} = 0.8*44=35.2mmHg {/eq}

{eq}P_{mixture} = P_m + P_e = 18.8 + 35.2 = 54 mmHg {/eq}

### Vapor Pressure Equation with Temperature

The **Clausius-Clapeyron equation** is a differential equation used when evaluating the vapor pressure of a substance at different temperatures. This equation shows the direct proportionality between the vapor pressure and the temperature. An increase in temperature enhances the transition to the gas phase. Thus, increasing the gas molecules and increasing the pressure they exert.

{eq}ln\frac{P_1}{P_2}=\frac{H_v}{R}(\frac{1}{T_2}-\frac{1}{T_1}) {/eq}

This equation can be reformatted as the following:

{eq}P_1=P_2\:exp(\frac{H_v}{R}(\frac{1}{T_2}-\frac{1}{T_1})) {/eq}

This equation can be used for direct application.

#### Example 1

What is the vapor pressure of methanol at 90{eq}^{\circ} {/eq}C if the its vapor pressure is 0.123 atm at 25{eq}^{\circ} {/eq}C? Given that the enthalpy of vaporization of methanol is 35.21 kJ/mol.

{eq}T_1 = 25 + 273 = 298K {/eq}

{eq}T_2 = 90 + 273 = 363K {/eq}

{eq}P_1=P_2\:exp(\frac{H_v}{R}(\frac{1}{T_2}-\frac{1}{T_1})) {/eq}

{eq}0.123=P_2\:exp(\frac{35.21}{0.008314}(\frac{1}{363}-\frac{1}{298}) {/eq}

Solving for {eq}P_2 = 1.56 atm {/eq}.

#### Example 2

What is the vapor pressure of compound at 15{eq}^{\circ} {/eq}C if its vapor pressure at 100{eq}^{\circ} {/eq}C was 1.3 atm? Given that the enthalpy of vaporization of that compound was equal to 20 kJ/mol.

{eq}T_1 = 15 + 273 = 288K {/eq}

{eq}T_2 = 100 + 273 = 373K {/eq}

{eq}P_1=P_2\:exp(\frac{H_v}{R}(\frac{1}{T_2}-\frac{1}{T_1})) {/eq}

{eq}P_1=1.3*\:exp(\frac{20}{0.008314}(\frac{1}{373}-\frac{1}{288}) {/eq}

Solving for {eq}P_1 = 0.193 atm {/eq}.

## What is the Vapor Pressure of Water?

What is the vapor pressure of water? The normal vapor pressure of water (at 100{eq}^{\circ} {/eq}C and 1atm) is equal to 1 atm. The vapor pressure of water decreases as the temperature decreases; the vapor pressure of water at room temperature does not exceed 0.0313 atm. Water begins to boil when its vapor pressure is larger than the vapor pressure of the surrounding environment. It is a given fact that the pressure decreases at high elevations, which means that the boiling point of water is lower in places where the elevation is high. In other words, water's boiling point goes down because its vapor pressure is higher than the vapor pressure of an environment in an elevated area. Water's vapor pressure at different temperatures and pressures can be sought from many sources, including thermodynamics tables and steam tables. It can also be sought through the application of the Clausius-Clapeyron equation.

## Lesson Summary

The **vapor pressure** is the pressure exerted by a gas that is in equilibrium with its solid or liquid at a fixed temperature in a closed system. An **example of vapor pressure** is boiling tea in a kettle; as the water molecules transition to the gas phase, the pressure exerted by the vapor on the kettle's lid increases. The vapor pressure of an ideal liquid mixture can be sought using **Raoult's Law**, which shows that the total vapor pressure is equal to the summation of the vapor pressures of each species in an ideal liquid mixture. Raoult's **vapor pressure equation** is: {eq}P_a = x_a P_{0,a} {/eq}. Each component in a mixture has a different vapor pressure at a given temperature.

The **Clausius-Clapeyron equation** is used when determining the vapor pressure of a substance at different temperatures. It shows the direct proportionality between the vapor pressure and temperature. If the temperature of a water bottle at equilibrium decreases, the vapor pressure decreases because the amount of water vapor decreased. The **Clausius-Clapeyron equation** is {eq}P_1=P_2\:exp(\frac{H_v}{R}(\frac{1}{T_2}-\frac{1}{T_1})) {/eq}. Water boils when its vapor pressure is higher than the environment's pressure. The boiling point of water decreases in areas of high elevation; the air pressure decreases as the elevation increases.

## The Clausius-Clapeyron Equation

Let's consider what happens if the temperature of the water bottle increases a little. The water molecules will have more energy to evaporate, so there will be a little more water vapor and a little less water in the bottle at equilibrium at this higher temperature. The amount of pressure exerted by the water vapor will also increase, meaning that the vapor pressure will increase. The **Clausius-Clapeyron equation** describes how temperature affects vapor pressure.

In this equation, P1 and P2 are the vapor pressures of a material at temperatures T1 and T2, respectively. R is the ideal gas constant (8.314 J/(mol·K)) and Hv is the enthalpy of vaporization of the material, a number that you can look up for common materials. As expected, the equation shows that vapor pressure increases as temperature increases.

## Clausius-Clapeyron Equation Example

Let's look at an example using the Clausius-Clapeyron equation.

*Given that the vapor pressure of water is 1 atm at its boiling point, 100°C (373 K), and that the enthalpy of vaporization of water is 40,700 J/mol, use the Clausius-Clapeyron Equation to determine the vapor pressure of water at 80°C (353 K).*

The Clausius-Clapeyron equation can be rewritten in the following more-convenient form:

We know that T1 = 353 K, T2 = 373 K, P2 = 1 atm, and Hv = 40,700 J/mol. Plugging these values into the equation, we get the following formula:

Therefore, the vapor pressure of water at 80°C is 0.48 atm.

## Raoult's Law: Understanding Vapor Pressure in Mixtures

Sometimes we have to consider vapor pressure in a mixture of liquids. In the ideal case, the tendency of a molecule to escape will not change when it is mixed with another liquid and we can use **Raoult's Law** to determine the vapor pressure of the components.

Here, Pa is the vapor pressure of Liquid A in the mixture, xa is the mole fraction of Liquid A in the mixture, and P0,a is the vapor pressure of pure Liquid A. The mole fraction of Liquid A is equal to the number of molecules of Liquid A divided by the total number of molecules in the mixture. Raoult's Law, therefore, states that for ideal liquid mixtures, the vapor pressure of any one liquid in the mixture is equal to the vapor pressure of the pure liquid multiplied by the fraction of all the molecules in the mixture that are this liquid. The total vapor pressure of a mixture of ideal liquids is the sum of the vapor pressures of every component in the mixture.

## Raoult's Law Example

Let's look at an example using Raoult's Law.

*At room temperature, the vapor pressure of heptane is 5.33 kPa and the vapor pressure of hexane is 17.6 kPa. Use Raoult's Law to determine the vapor pressure of heptane, the vapor pressure of hexane, and the total vapor pressure for a room-temperature mixture of heptane and hexane with a mole fraction of heptane of 0.60.*

Let's break this down with Raoult's Law.

Easy, right?

## Lesson Summary

Some of the atoms or molecules in every solid and liquid have enough energy to break away and form a gas. The pressure exerted by this gas phase in equilibrium with its solid or liquid counterpart is known as vapor pressure. The two most important equations describing vapor pressure are the Clausius-Clapeyron equation and Raoult's Law.

The **Clausius-Clapeyron equation** describes how vapor pressure increases as temperature increases due to an increase in the amount of energy available for the atoms or molecules to form a gas. For example, the amount of water vapor will increase and the pressure will increase if a bottle of water is heated up.

The other equation we learned about, **Raoult's Law**, can be used to determine the vapor pressure of each liquid in an ideal mixture of liquids. It's basically the sum of the vapor pressures of the components that become mixed together. Knowledge of vapor pressure is important to our understanding of many things, like humidity and the process of boiling.

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## What Is Vapor Pressure?

Water can take the form of a liquid, a solid (ice), or a gas (water vapor). Ice melts to form water, and water evaporates to form water vapor. Think about a water bottle like the one shown here:

What's inside? You can see water and air, but what you may not have realized is that there is also water vapor. In a water bottle, a little bit of the water is constantly evaporating and becoming water vapor. At the same time, some of the water vapor is constantly condensing to become a liquid. At equilibrium, the amount of water evaporating is equal to the amount of water vapor condensing, so the amounts of water and water vapor are constant.

The water vapor exerts a pressure on the water bottle in the same way that air pumped into a tire exerts pressure on the tire. The pressure exerted by the water vapor is the **vapor pressure**. In more general terms, *vapor pressure is the pressure exerted by a gas in equilibrium with the same material in liquid or solid form.* Just as distance can be measured in a variety of units (miles, feet, kilometers, etc.), we can also measure vapor pressure with many different units (kPa, atm, bar, mm Hg, torr).

## The Clausius-Clapeyron Equation

Let's consider what happens if the temperature of the water bottle increases a little. The water molecules will have more energy to evaporate, so there will be a little more water vapor and a little less water in the bottle at equilibrium at this higher temperature. The amount of pressure exerted by the water vapor will also increase, meaning that the vapor pressure will increase. The **Clausius-Clapeyron equation** describes how temperature affects vapor pressure.

In this equation, P1 and P2 are the vapor pressures of a material at temperatures T1 and T2, respectively. R is the ideal gas constant (8.314 J/(mol·K)) and Hv is the enthalpy of vaporization of the material, a number that you can look up for common materials. As expected, the equation shows that vapor pressure increases as temperature increases.

## Clausius-Clapeyron Equation Example

Let's look at an example using the Clausius-Clapeyron equation.

*Given that the vapor pressure of water is 1 atm at its boiling point, 100°C (373 K), and that the enthalpy of vaporization of water is 40,700 J/mol, use the Clausius-Clapeyron Equation to determine the vapor pressure of water at 80°C (353 K).*

The Clausius-Clapeyron equation can be rewritten in the following more-convenient form:

We know that T1 = 353 K, T2 = 373 K, P2 = 1 atm, and Hv = 40,700 J/mol. Plugging these values into the equation, we get the following formula:

Therefore, the vapor pressure of water at 80°C is 0.48 atm.

## Raoult's Law: Understanding Vapor Pressure in Mixtures

Sometimes we have to consider vapor pressure in a mixture of liquids. In the ideal case, the tendency of a molecule to escape will not change when it is mixed with another liquid and we can use **Raoult's Law** to determine the vapor pressure of the components.

Here, Pa is the vapor pressure of Liquid A in the mixture, xa is the mole fraction of Liquid A in the mixture, and P0,a is the vapor pressure of pure Liquid A. The mole fraction of Liquid A is equal to the number of molecules of Liquid A divided by the total number of molecules in the mixture. Raoult's Law, therefore, states that for ideal liquid mixtures, the vapor pressure of any one liquid in the mixture is equal to the vapor pressure of the pure liquid multiplied by the fraction of all the molecules in the mixture that are this liquid. The total vapor pressure of a mixture of ideal liquids is the sum of the vapor pressures of every component in the mixture.

## Raoult's Law Example

Let's look at an example using Raoult's Law.

*At room temperature, the vapor pressure of heptane is 5.33 kPa and the vapor pressure of hexane is 17.6 kPa. Use Raoult's Law to determine the vapor pressure of heptane, the vapor pressure of hexane, and the total vapor pressure for a room-temperature mixture of heptane and hexane with a mole fraction of heptane of 0.60.*

Let's break this down with Raoult's Law.

Easy, right?

## Lesson Summary

Some of the atoms or molecules in every solid and liquid have enough energy to break away and form a gas. The pressure exerted by this gas phase in equilibrium with its solid or liquid counterpart is known as vapor pressure. The two most important equations describing vapor pressure are the Clausius-Clapeyron equation and Raoult's Law.

The **Clausius-Clapeyron equation** describes how vapor pressure increases as temperature increases due to an increase in the amount of energy available for the atoms or molecules to form a gas. For example, the amount of water vapor will increase and the pressure will increase if a bottle of water is heated up.

The other equation we learned about, **Raoult's Law**, can be used to determine the vapor pressure of each liquid in an ideal mixture of liquids. It's basically the sum of the vapor pressures of the components that become mixed together. Knowledge of vapor pressure is important to our understanding of many things, like humidity and the process of boiling.

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#### How is vapor pressure used in everyday life?

The pressure exerted by the vapor is utilized in pressure cookers, which are closed containers that trap the vapor in. Pressure cookers speed up the cooking of food by increasing the vapor pressure. It is done by sealing the cooker and letting vapors accumulate inside.

#### How to calculate vapor pressure?

The vapor pressure is the pressure exerted by vapor at equilibrium with its solid or liquid in a closed chamber. It can be calculated either by using Clausius-Clapeyron Equation or Raoult's Law.

- Raoult's Law is used when determining the vapor pressure of a component in a mixture.
- Clausius-Clapeyron Equation is used when determining the vapor pressure of a substance at different temperatures.

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