Question

A quantity of liquid water comes into equilibrium with the air in a closed container, without completely evaporating, at a temperature of \(25.0^{\circ} \mathrm{C} .\) How many grams of water vapor does a liter of the air contain in this situation? The vapor pressure of water at \(25.0^{\circ} \mathrm{C}\) is \(3.1690 \mathrm{kPa}\).

Short answer

Approximately 0.0231 grams of water vapor are present in a liter of the air at equilibrium with liquid water at a temperature of 25.0°C.

Step-by-step solution

Write down known values

We are given the temperature, \(T = 25.0^\circ \mathrm{C}\) and the vapor pressure of liquid water, \(P = 3.1690 \mathrm{kPa}\). First, we need to convert the temperature to Kelvin: \(T = 25.0 + 273.15 = 298.15 \;\text{K}\).

Convert vapor pressure to appropriate units

The vapor pressure of water is given in \(\text{kPa}\), but we need it in \(\text{Pa}\), so we perform the conversion: \(P = 3.1690 \times 1000 = 3169 \; \text{Pa}\).

Apply the Ideal Gas Law

The Ideal Gas Law can be written as: $$ PV = nRT $$ where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles of the gas, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. The volume of the air is given as \(1\;\text{L}\) which we need to convert to \(\text{m}^3\) by: $$ V = 1 \times 10^{-3}\;\text{m}^3 $$ Next, we will use the Ideal Gas Law to solve for the number of moles (n) of water vapor in the container: $$ n = \frac{PV}{RT} $$

Calculate the amount of water vapor present

Now, plugging in the known values, we can solve for \(n\): $$ n = \frac{3169\text{ Pa}\times 1\times 10^{-3}\text{ m}^3}{8.314\text{ J/mol K}\times 298.15\text{ K}} $$ which gives us \(n\approx 0.00128\;\text{moles}\).

Convert moles to grams

Finally, we need to convert the amount of water vapor present from moles to grams using the molar mass of water \(18.0153 \;\text{g/mol}\): $$ m = n \times M_h $$ where \(m\) is the mass of water vapor in grams, \(n\) is the number of moles, and \(M_h\) is the molar mass of water. So, $$ m = 0.00128\;\text{moles} \times 18.0153\;\text{g/mol} \approx 0.0231 \;\text{g} $$ There are approximately \(0.0231 \;\text{grams}\) of water vapor in a liter of the air at equilibrium with liquid water at a temperature of \(25.0^\circ\text{C}\).

Key concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics that relates the pressure (P), volume (V), and temperature (T) of an ideal gas to the amount of substance present, measured in moles (n). The law is represented as \(PV = nRT\), where R is a constant known as the ideal gas constant, with a value of 8.314 \(\text{J/mol K}\).

The beauty of this law lies in its ability to predict the behavior of gases under different conditions. For instance, when calculating the amount of water vapor in the air, as in our exercise, we start by ensuring that the temperature is in Kelvin and the pressure in Pascals. We then rearrange the equation to solve for n, the only unknown: \(n=\frac{PV}{RT}\).

Real-life Application

In practice, this law can be applied to scenarios such as calculating the amount of gas required to inflate a balloon to a particular size at room temperature, or determining the pressure change when the volume of a gas is increased. It's essential for understanding how gases react under various environmental conditions.

Vapor Pressure and Equilibrium States

Vapor pressure is the pressure exerted by a vapor in equilibrium with its liquid or solid phase at a given temperature. This concept is vital for understanding the behavior of a substance as it transitions between phases. When a liquid is in a closed container, some molecules will evaporate and others condense back into the liquid. Equilibrium is reached when the rates of evaporation and condensation are equal, resulting in a constant vapor pressure.

Vapor pressure can be influenced by the temperature. As the temperature increases, the vapor pressure rises because more molecules have enough energy to escape from the liquid. Consequently, in the exam problem, the vapor pressure of water is a pivotal value that represents the dynamic balance between the liquid water and its vapor at 25°C.

Practical Implications

In real-world applications, vapor pressure is critical in fields such as meteorology, where it's used to understand humidity and predict weather changes, and in cooking, where it's important for processes such as boiling or canning food.

Molar Mass Conversion Made Simple

Molar mass is the weight of one mole of a substance and is expressed in grams per mole (g/mol). It is a bridge between the microscopic world of atoms and molecules and the macroscopic world we measure in grams and kilograms. To convert moles to grams, we multiply the number of moles by the molar mass of the substance: \(m = n \times M_h\), where m is the mass, n is the moles, and \(M_h\) is the molar mass.

For the given problem, once we know the number of moles of water vapor, we simply multiply by water's molar mass to convert to grams - this is crucial for connecting the molecular-level insight to tangible quantities we can measure and understand.

Why it Matters

This conversion is not just academic; it's used pharmaceutically to measure drug dosages, in cooking for recipes that require precise chemical quantities, and in environmental science to gauge pollutant concentrations.

Equilibrium Thermodynamics: Balancing Act of Molecules

Equilibrium thermodynamics is a branch of thermodynamics that studies systems in a state of balance—where there is no net change over time. The concept is vital in understanding the balance between different phases of a substance, like the liquid and vapor states of water in the container from our exercise.

At equilibrium, the rates of forward (evaporating) and reverse (condensing) reactions are equal. For vapor pressure, this means that the number of water molecules moving from the liquid phase to the gas phase per unit time is the same as those condensing back into the liquid.

Everyday Examples

This concept is seen in everyday life, such as when a puddle of water evaporates over time, or dew forms on grass in the morning. Understanding equilibrium thermodynamics helps us to predict and control physical and chemical processes, making it crucial in designing chemical reactors, creating sustainable energy systems, and even brewing the perfect cup of coffee.