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Chapter 12: Problem 32

The vapor pressure of water at \(80 .{ }^{\circ} \mathrm{C}\) is \(0.467 \mathrm{~atm} .\) Determine the value of \(K_{\mathrm{c}}\) for the process $$ \mathrm{H}_{2} \mathrm{O}(\ell) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) $$ at this temperature.

Short Answer

Expert verified
\( K_c \approx 0.0159\, \mathrm{mol/L} \) at \(80^{\circ} \mathrm{C}\).

Step by step solution

01

Understand the equilibrium expression

In the given process \( \text{H}_2\text{O}(\ell) \rightleftharpoons \text{H}_2\text{O}(\text{g}) \), the equilibrium involves the conversion of liquid water to gaseous water (vapor). The equilibrium constant \( K_c \) is defined for a reaction in terms of concentrations of products and reactants. However, since water's liquid phase concentration remains constant, \( K_c \) depends only on the concentration of the gaseous water.
02

Relate vapor pressure to concentration

The vapor pressure given is \(0.467\, \mathrm{atm}\) at \(80^{\circ} \mathrm{C}\). For gases, concentration \( [\text{H}_2\text{O}(\text{g})] \) is calculated using the ideal gas law as \( \frac{P}{RT} \), where \( P \) is the pressure, \( R \) is the ideal gas constant \(0.0821\, \mathrm{L \cdot atm \cdot mol^{-1} \cdot K^{-1}}\), and \( T \) is the temperature in Kelvin.
03

Convert temperature to Kelvin

First, convert the temperature from Celsius to Kelvin: \( T = 80 + 273.15 = 353.15\, \mathrm{K} \).
04

Calculate concentration of gaseous water

Using the ideal gas relationship \([\text{H}_2\text{O}(\text{g})] = \frac{P}{RT}\), substitute the values: \( P = 0.467\, \mathrm{atm} \), \( R = 0.0821\, \mathrm{L \cdot atm \cdot mol^{-1} \cdot K^{-1}} \), and \( T = 353.15\, \mathrm{K}\).\[ [\text{H}_2\text{O}(\text{g})] = \frac{0.467}{0.0821 \times 353.15} \]
05

Calculate \( K_c \)

Simplify the equation from the previous step to find the concentration:\[ [\text{H}_2\text{O}(\text{g})] = \frac{0.467}{0.0821 \times 353.15} \approx 0.0159\, \mathrm{mol/L} \]This concentration is the value of \( K_c \) because it represents the equilibrium concentration of \( \text{H}_2\text{O}(\text{g}) \) when the system is at equilibrium under the given conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vapor Pressure
Vapor pressure is an important concept in understanding how liquids transition to gas. It refers to the pressure exerted by the vapor present above a liquid in a closed system when both the liquid and vapor are in dynamic equilibrium. This means the rate of evaporation of the liquid into a gas phase equals the rate of condensation back into the liquid phase.
  • At a given temperature, a substance's vapor pressure is stable and specific, acting as a unique characteristic of that substance.
  • In the equilibrium process involving water, as described in the exercise, the vapor pressure is the partial pressure of water vapor at the surface of the liquid water.
Understanding vapor pressure is necessary for determining equilibrium constants such as in phase transition reactions.
Ideal Gas Law
The Ideal Gas Law is a key principle used to relate various properties of gases, and it is given as \( PV = nRT \). This equation connects pressure \( P \), volume \( V \), and temperature \( T \) with the gas's amount in moles \( n \), while \( R \) is the universal gas constant. In the context of the exercise, the Ideal Gas Law helps convert vapor pressure into concentration:
  • This conversion is crucial because the equilibrium constant \( K_c \) is calculated using concentrations instead of pressures.
  • By rearranging the Ideal Gas Law to \( n/V = P/RT \), you can solve for concentration \( [ ext{H}_2O(g)] \).
Understanding these relationships allows for accurate calculation of chemical equilibria involving gases.
Equilibrium Constant
In the scope of chemistry, an equilibrium constant \( K_c \) expresses the state of balance between products and reactants in a reaction at equilibrium. For the phase transition of water into vapor:
  • The equilibrium constant is formulated considering the concentration of gaseous water, as the liquid鈥檚 concentration does not change in meaningful ways for \( K_c \).
  • The relationship \( K_c = [ ext{H}_2O(g)]_{equilibrium} \) shows that at equilibrium, the concentration of gaseous water directly reflects the value of \( K_c \).
  • An understanding of \( K_c \) provides insights into how far a reaction will proceed before reaching equilibrium.
It's a central tool in predicting and understanding chemical reactions, especially in closed systems.
Phase Equilibrium
Phase equilibrium refers to the state where different phases of a substance coexist at equilibrium, such as ice floating in water at \(0^{\circ}C\) or vapor above liquid at constant temperature and pressure. The phase equilibrium concept in the exercise involves balancing liquid water and its vapor:
  • In a closed container, liquid water reaches an equilibrium where molecules constantly move between the liquid and vapor phases.
  • This equilibrium determines the vapor pressure and allows prediction of system behavior when conditions like temperature change.
  • Phase equilibrium is essential not just in chemistry, but also in engineering and material sciences where phase transitions occur.
This understanding is vital in predicting how substances will behave in various practical and experimental setups.

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Most popular questions from this chapter

The equilibrium constant \(K_{\mathrm{c}}\) for this reaction is 0.16 at \(25^{\circ} \mathrm{C},\) and the standard reaction enthalpy is \(16.1 \mathrm{~kJ}\). $$ 2 \mathrm{NOBr}(\mathrm{g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\ell) $$ Predict the effect of each of these changes on the position of the equilibrium; that is, state which way the equilibrium will shift (left, right, or no change) when each of these changes is made for a constant-volume system. (a) Adding more \(\mathrm{Br}_{2}\) (b) Removing some \(\mathrm{NOBr}\) c). Lowering the temnerature

Indicate whether each statement below is true or false. If a statement is false, rewrite it to produce a closely related statement that is true. (a) For a given reaction, the magnitude of the equilibrium constant is independent of temperature. (b) If there is an increase in entropy and a decrease in enthalpy when reactants in their standard states are converted to products in their standard states, the equilibrium constant for the reaction must be negative. (c) The equilibrium constant for the reverse of a reaction is the reciprocal of the equilibrium constant for the reaction itself. (d) For the reaction $$ \mathrm{H}_{2} \mathrm{O}_{2}(\ell) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(\ell)+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) $$ the equilibrium constant is one half the magnitude of the equilibrium constant for the reaction $$ 2 \mathrm{H}_{2} \mathrm{O}_{2}(\ell) \rightleftharpoons 2 \mathrm{H}_{2} \mathrm{O}(\ell)+\mathrm{O}_{2}(\mathrm{~g}) $$

Consider these two equilibria involving \(\mathrm{SO}_{2}(\mathrm{~g})\) and their corresponding equilibrium constants. $$ \begin{array}{cl} \mathrm{SO}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{SO}_{3}(\mathrm{~g}) & K_{\mathrm{c}_{1}} \\ 2 \mathrm{SO}_{3}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) & K_{\mathrm{c}_{2}} \end{array} $$ Which of these expressions correctly relates \(K_{c_{1}}\) to \(K_{c_{2}} ?\) (a) \(K_{\mathrm{c}_{2}}=K_{\mathrm{c}_{1}}^{2}\) (b) \(K_{\mathrm{c}_{2}}^{2}=K_{\mathrm{c}_{1}}\) (c) \(K_{\mathrm{c}_{2}}=1 / K_{\mathrm{c}_{1}}\) (d) \(K_{\mathrm{c}_{2}}=K_{\mathrm{c}_{1}}\) (e) \(K_{\mathrm{c},}=1 / K_{\mathrm{c}}^{2}\)

Write the equilibrium constant expression for each of these heterogeneous systems. (a) \(\mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g})+\mathrm{O}_{3}(\mathrm{~g}) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{5}(\mathrm{~s})+\mathrm{O}_{2}(\mathrm{~g})\) (b) \(\mathrm{C}(\mathrm{s})+2 \mathrm{~N}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons \mathrm{CO}_{2}(\mathrm{~g})+2 \mathrm{~N}_{2}(\mathrm{~g})\) (c) \(\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(\mathrm{g})\)

For each of these reactions at \(25^{\circ} \mathrm{C}\), indicate whether the entropy effect, the energy effect, both, or neither favors the reaction. (a) \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{~F}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NF}_{3}(\mathrm{~g}) \quad \Delta_{1} H^{\circ}=-249 \mathrm{~kJ} / \mathrm{mol}\) (b) \(\mathrm{N}_{2} \mathrm{~F}_{4}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NF}_{2}(\mathrm{~g})\) \(\Delta_{t} H^{\circ}=93.3 \mathrm{~kJ} / \mathrm{mol}\) (c) \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NCl}_{3}(\mathrm{~g}) \quad \Delta_{1} H^{\circ}=460 \mathrm{~kJ} / \mathrm{mol}\)
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