Problem 122 - Chemistry The Molecular Science

Question

At \(25^{\circ} \mathrm{C}\) the vapor pressure of water is \(0.03126 \mathrm{~atm}\). (a) Calculate \(K_{\mathrm{p}}\) and \(K_{\mathrm{c}}\) for $$ \mathrm{H}_{2} \mathrm{O}(\ell) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) $$ (b) Calculate the value of \(K_{\mathrm{p}}\) for this same system at \(100 .{ }^{\circ} \mathrm{C}\) (c) Suggest a general rule for calculating \(K_{\mathrm{p}}\) for any liquid in equilibrium with its vapor at its normal boiling point.

Step-by-Step Solution

Answer

(a) \( K_{\mathrm{p}} = 0.03126 \text{ atm}, K_{\mathrm{c}} = 0.00128 \text{ M} \) at 25°C; (b) \( K_{\mathrm{p}} = 1 \text{ atm} \) at 100°C; (c) \( K_{\mathrm{p}} = 1 \text{ atm} \) at boiling point.

Step 1: Determine the Reaction Observables

For the reaction \( \mathrm{H}_{2}\mathrm{O}(\ell) \rightleftharpoons \mathrm{H}_{2}\mathrm{O}(\mathrm{g}) \), the equilibrium involves liquid water and its vapor. At equilibrium, the vapor pressure of water is given as 0.03126 atm at 25\( ^{\circ} \mathrm{C} \). This can be used as the partial pressure of water in the gas phase for the purpose of calculating \( K_{\mathrm{p}} \).

Step 2: Calculate \( K_{\mathrm{p}} \) at 25°C

Since \( \mathrm{H}_{2}\mathrm{O}(\ell) \) is a pure liquid, its activity is 1 and does not appear in the expression for \( K_{\mathrm{p}} \). Thus, \( K_{\mathrm{p}} \) is equal to the equilibrium pressure of water vapor:\[K_{\mathrm{p}} = P_{\mathrm{H}_2\mathrm{O(g)}} = 0.03126 \text{ atm}\]

Step 3: Convert \( K_{\mathrm{p}} \) to \( K_{\mathrm{c}} \) at 25°C

Use the relationship between \( K_{\mathrm{p}} \) and \( K_{\mathrm{c}} \):\[K_{\mathrm{p}} = K_{\mathrm{c}} (RT)^{\Delta n}\]For this reaction, \( \Delta n = 1 \) (1 mole of gas is formed), so:\[K_{\mathrm{c}} = \frac{K_{\mathrm{p}}}{RT}\]Using \( R = 0.0821 \frac{\text{L atm}}{\text{K mol}} \) and \( T = 298 \text{ K} \):\[K_{\mathrm{c}} = \frac{0.03126}{0.0821 \times 298} = 0.00128 \text{ M}\]

Step 4: Calculate \( K_{\mathrm{p}} \) at 100°C

At the boiling point of water (100\( ^{\circ} \mathrm{C} \)), the vapor pressure is 1 atm. Therefore, at this temperature, \( K_{\mathrm{p}} = 1 \text{ atm} \).

Step 5: Suggest a Rule for \( K_{\mathrm{p}} \) at Boiling Point

The value of \( K_{\mathrm{p}} \) for any liquid in equilibrium with its vapor at its normal boiling point is 1 atm, since the vapor pressure equals atmospheric pressure.

Key Concepts

Equilibrium Constant

In chemical reactions, the equilibrium constant is key to understanding the balance between products and reactants at equilibrium. In simple terms, the equilibrium constant for a reaction quantifies the ratio of concentrations of products to reactants when the reaction is at a state of balance. For the vaporization reaction of water, \( \mathrm{H}_{2}\mathrm{O}(\ell) \rightleftharpoons \mathrm{H}_{2}\mathrm{O}(\mathrm{g}) \), the equilibrium constant can be expressed in terms of partial pressures \( K_{\mathrm{p}} \) or concentrations \( K_{\mathrm{c}} \).

Here's how it works:

• \(K_{\mathrm{p}}\) is used when dealing with gases and is calculated using the partial pressures of the gaseous species involved.
• \(K_{\mathrm{c}}\) involves the molar concentrations of substances in a reaction.

For this water vaporization reaction, since liquid water is a pure substance and its activity is 1, the \( K_{\mathrm{p}} \) is determined by the vapor pressure. Just remember, to convert between \( K_{\mathrm{p}} \) and \( K_{\mathrm{c}} \), you employ the relationship \( K_{\mathrm{p}} = K_{\mathrm{c}} (RT)^{\Delta n} \).

Phase Equilibria

Phase equilibria is the study of the balances found in a system where different states (or phases) coexist. Water, for example, can exist as a liquid and its vapor simultaneously at equilibrium. This happens in a closed system where the rate of liquid water evaporating equals the rate of water vapor condensing. This balance is essential for understanding phase changes

Such equilibrium is critical for determining conditions like vapor pressure, where the pressure exerted by the vapor phase at equilibrium with its liquid matches the vapor pressure. At specific temperatures like 25°C, you can observe a stable vapor pressure, defining the equilibrium between the liquid and vapor phases. This concept becomes particularly important when studying boiling points, where the vapor pressure equals the atmospheric pressure.

Vapor-Liquid Equilibrium

Vapor-liquid equilibrium pertains to the specific equilibrium condition where liquid and vapor phases of a substance coexist without net mass transfer between phases. It's incredibly significant in understanding how a liquid evaporates while it maintains the correct conditions to remain stable.

Take water at 25°C as an example. At this temperature, water vaporizes to a point where the pressure of vapor formed is exactly 0.03126 atm, establishing equilibrium with the liquid form. This equilibrium pressure signifies the vapor pressure of water at this temperature.
When heating occurs, such as reaching 100°C, the vapor pressure rises to 1 atm. This is why water boils; the vapor pressure at this point coincides with atmospheric pressure, marking the vapor-liquid equilibrium.

• For students, understanding this balance is crucial when discussing distillation or cooling processes.

Chemical Thermodynamics

Chemical thermodynamics involves the study of energy transformations and reactions. It allows us to predict the extent of reactions, state constants like equilibrium constant, and understand heat exchange involved during phase transitions.

For the vapor-liquid equilibrium of water, thermodynamics explains how increased temperature impacts \( K_{\mathrm{p}} \) and vapor pressures. Essentially, it shows us that at a higher temperature, more energy is available to facilitate vaporization, thus increasing the vapor pressure until it equals operative environmental pressure (like at 100°C, where boiling occurs).

• In essence, thermodynamics provides a framework to understand the shifts in energies and phases when the temperature changes.

By mastering these principles, students can better appreciate how substances move between phases and how energy dictates those movements.