Question

A humidity sensor consists of a cardboard square that is colored blue in dry weather and red in humid weather. The color change is due to the reaction: $$\mathrm{CoCl}_{2}(s)+6 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right] \mathrm{Cl}_{2}(s)$$ Blue For this reaction at \(25^{\circ} \mathrm{C}, \Delta H^{\circ}=-352 \mathrm{~kJ} / \mathrm{mol}\) and \(\Delta S^{\circ}=\) \(-899 \mathrm{~J} /(\mathrm{K} \cdot \mathrm{mol})\). Assuming that \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are independent of temperature, what is the vapor pressure of water (in \(\mathrm{mm} \mathrm{Hg}\) ) at equilibrium for the above reaction at \(35^{\circ} \mathrm{C}\) on a hot summer day?

Short answer

The vapor pressure of water is approximately 42.9 mm Hg at 35°C.

Step-by-step solution

Understand the Reaction

The given chemical reaction is a reversible one involving cobalt chloride and water vapor. The reaction:\[\text{CoCl}_2(s) + 6 \text{H}_2O(g) \rightleftharpoons [\text{Co}(\text{H}_2O)_6] \text{Cl}_2(s)\]indicates the equilibrium between blue (anhydrous) and red (hydrated) forms, determined by humidity.

Calculate ΔG at 35°C

We calculate the change in Gibbs free energy, \(\Delta G^{\circ}\), at \(35^{\circ} \text{C}\) using the Gibbs Free Energy equation:\[\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ}\]Where:- \(\Delta H^{\circ} = -352\, \text{kJ/mol} = -352,000\, \text{J/mol}\)- \(\Delta S^{\circ} = -899\, \text{J/K mol}\)- Temperature (T) in Kelvin is \(308\, \text{K}\) (since \(35^{\circ} \text{C} = 273 + 35\))Substitute these values into the equation to find \(\Delta G^{\circ}\).

Calculate the Equilibrium Constant (K)

Use the relationship between the Gibbs free energy and the equilibrium constant:\[\Delta G^{\circ} = -RT \ln K\]where: - \(R = 8.314\, \text{J/mol K}\) is the universal gas constant.Rearrange to solve for \(K\):\[ K = e^{-\Delta G^{\circ} / (RT)} \]Substitute the values for \(\Delta G^{\circ}\), \(R\), and \(T\) to calculate \(K\).

Use Henry's Law for Vapor Pressure

The equilibrium constant \(K\) is related to the pressure of the water vapor \(P_{water}\) by the stoichiometry of the reaction. Since 6 moles of water are involved:\[K = P_{water}^{6}\]Therefore, the vapor pressure \(P_{water}\) can be calculated by:\[ P_{water} = K^{1/6} \]This gives the pressure in atmospheres. Convert atmospheres to mm Hg:1 atm = 760 mm Hg.

Calculate Final Vapor Pressure in mm Hg

Convert the vapor pressure from atmospheres to mm Hg using the conversion factor:\[P_{water} (\text{mm Hg}) = P_{water} (\text{atm}) \times 760\]Substitute \(P_{water} (\text{atm}) = K^{1/6}\) into this equation to get the final answer for vapor pressure in mm Hg.

Key concepts

Equilibrium Constant

In chemical reactions, the equilibrium constant, denoted as \(K\), is a crucial concept as it provides insight into the position of equilibrium during a reversible chemical reaction. It relates to the proportions of reactants and products at equilibrium, indicating whether the reactants or products are favored. The value of \(K\) is derived from the equation \(\Delta G^{\circ} = -RT \ln K\), connecting it to Gibbs free energy and allowing us to calculate \(K\) when \(\Delta G^{\circ}\) is known. This enables us to predict the reaction behavior under different conditions.

• If \(K\) is much greater than 1, products are favored at equilibrium.
• If \(K\) is much less than 1, reactants are favored at equilibrium.

In the exercise, after calculating \(\Delta G^{\circ}\) for the reaction at the given temperature, \(K\) is determined, helping in subsequent calculations for vapor pressure.

Vapor Pressure

Vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases at a given temperature in a closed system. For the given reaction, it refers to the pressure of water vapor that can achieve equilibrium with the dehydrated cobalt chloride. By using the equilibrium constant \(K\), hydrogen's vapor pressure can be calculated using the relationship \(K = P_{\text{water}}^6\). This setup implies that the equilibrium position heavily depends upon the pressure exerted by water vapor.
To find \(P_{\text{water}}\):

• Calculate \(K\) from \(\Delta G^{\circ}\).
• Determine \(P_{\text{water}} = K^{1/6}\).
• Convert the pressure from atmospheres to mm Hg, knowing that 1 atm equals 760 mm Hg.

Understanding vapor pressure and its calculation is essential in predicting how reactants and products distribute between phases.

Temperature Dependence

Temperature plays a significant role in chemical reactions, influencing both the equilibrium position and reaction rates. For reactions involving Gibbs free energy, the relationship \(\Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ}\) emphasizes how changes in temperature can affect \(\Delta G^{\circ}\) and thus the equilibrium constant \(K\). As temperature varies:

• An increase in temperature can shift the equilibrium, favoring endothermic reactions.
• A decrease in temperature typically shifts the equilibrium towards exothermic reactions.

In the provided exercise, a temperature shift from 25°C to 35°C requires recalculating \(\Delta G^{\circ}\) and subsequently \(K\). This temperature dependence is critical for understanding how vapor pressure and other aspects of the reaction change under different thermal conditions, such as a hot summer day.