Problem 97 - Chemistry: The Central Science

Question

The relationship between the temperature of a reaction, its standard enthalpy change, and the equilibrium constant at that temperature can be expressed as the following linear equation: $$ \ln K=\frac{-\Delta H^{\circ}}{R T}+\text { constant } $$ (a) Explain how this equation can be used to determine \(\Delta H^{\circ}\) experimentally from the equilibrium constants at several different temperatures. (b) Derive the preceding equation using relationships given in this chapter. To what is the constant equal?

Step-by-Step Solution

Answer

To determine \(\Delta H^{\circ}\) experimentally, perform an experiment to obtain the equilibrium constants \(K\) at different temperatures and then find their natural logarithm values, \(\ln K\). Plot a graph of \(\ln K\) against \(1/T\), and calculate the slope of the linear graph. The value of \(\Delta H^{\circ}\) can be calculated using the equation \(\Delta H^{\circ} = -R \times \text{slope}\). To derive the equation, use the van't Hoff equation and Gibbs free energy relationships to obtain the desired equation, and conclude that the constant in the given equation is equal to \(\frac{\Delta S^{\circ}}{R}\).

Step 1: Understand the given equation

In the given equation, \(\ln K\) represents the natural logarithm of the equilibrium constant, \(\Delta H^{\circ}\) is the standard enthalpy change, \(R\) is the gas constant, \(T\) is the absolute temperature in Kelvin, and "constant" is a constant value for a specific reaction. Our task is to understand how this equation can be used to determine \(\Delta H^{\circ}\) experimentally from the equilibrium constants at several different temperatures.

Step 2: Calculate values of \(\ln K\) at different temperatures

Perform the experiment and obtain the values of the equilibrium constant \(K\) at several different temperatures. Then, calculate the corresponding values of \(\ln K\) using the natural logarithm.

Step 3: Plot the values of \(\ln K\) versus \(1 / T\)

With the obtained values of \(\ln K\) and the corresponding temperatures, create a graph where \(\ln K\) is on the vertical axis and \(1 / T\) is on the horizontal axis.

Step 4: Obtain the slope of the plotted graph

Observe the plotted graph and calculate the slope of the graph. The graph should be linear as given in the problem statement.

Step 5: Determine the value of \(\Delta H^{\circ}\)

Since the slope of the graph obtained in Step 4 is equal to \(\frac{-\Delta H^{\circ}}{R}\), the value of \(\Delta H^{\circ}\) can be calculated using the equation: $$ \Delta H^{\circ} = -R \times \text{slope} $$ Now you have successfully determined the value of \(\Delta H^{\circ}\) using the experimentally obtained values of \(K\) at different temperatures. Part (b)

Step 1: Derive the relationship using the van't Hoff equation

The van't Hoff equation connects equilibrium constant (\(K\)) with temperature (\(T\)), enthalpy change (\(\Delta H\)), and entropy change (\(\Delta S\)). $$ \ln K = \frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} $$ This equation is derived from the relationship between Gibbs free energy change (\(\Delta G^{\circ}\)) and the equilibrium constant, which is given by: $$ \Delta G^{\circ} = -R T \ln K $$

Step 2: Use Gibbs free energy relationships

We also know that Gibbs free energy change (\(\Delta G^{\circ}\)) is related to the enthalpy change (\(\Delta H^{\circ}\)) and entropy change (\(\Delta S^{\circ}\)) as: $$ \Delta G^{\circ} = \Delta H^{\circ} - T \Delta S^{\circ} $$

Step 3: Obtain the final equation disregarding the constant

Now, we combine the equations derived in Step 1 and Step 2 to eliminate the entropy change term and get the desired equation: $$ -R T \ln K = \Delta H^{\circ} - T \Delta S^{\circ} $$ Rearrange the equation to get \(\ln K\) on one side and the others on the other side: $$ \ln K = \frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} $$ To find the constant, recall from the equation obtained in the previous step: $$ \text{constant} = \frac{\Delta S^{\circ}}{R} $$ As demonstrated, the constant in the equation is equal to \(\frac{\Delta S^{\circ}}{R}\), where \(\Delta S^{\circ}\) is the standard entropy change for the reaction and \(R\) is the gas constant.

Key Concepts

Equilibrium Constant

The equilibrium constant, represented by the symbol \( K \), is a quantitative measure of the position of equilibrium in a chemical reaction. It is a unitless value that provides us with an idea of the extent to which reactants are converted to products under equilibrium conditions.

For a general reaction where reactants \( A \) and \( B \) react to form products \( C \) and \( D \), the equilibrium constant expression can be written as:
\[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \]
where \( [A] \), \( [B] \), \( [C] \), and \( [D] \) represent the molar concentrations of the reactants and products, and \( a \), \( b \), \( c \), and \( d \) are their respective coefficients in the balanced chemical equation.

The value of \( K \) is temperature-dependent and can be determined experimentally. The Van't Hoff equation which relates the equilibrium constant to the temperature and the standard enthalpy change is central in understanding the impact of temperature on equilibrium.

Using the Van't Hoff Equation

By measuring equilibrium constants at different temperatures and plotting \( \ln K \) against \( 1/T \), the standard enthalpy change \( \Delta H^\circ \) for the reaction can be calculated from the slope of the resulting line. This technique allows chemists to understand the energetics of reactions and predict how changes in temperature will affect equilibrium positions.

Standard Enthalpy Change

The standard enthalpy change, denoted by \( \Delta H^\circ \), is the heat released or absorbed when a reaction occurs at standard conditions (1 bar pressure). It's an essential thermodynamic quantity that offers information on the energy changes associated with a chemical process.

Positive values of \( \Delta H^\circ \) indicate endothermic reactions where heat is absorbed from the surroundings, while negative values are indicative of exothermic reactions where heat is released to the surroundings. The standard enthalpy change can be procedurely determined through calorimetry or can also be evaluated indirectly by using the Van't Hoff equation.

Importance in Equilibrium

The Van't Hoff equation offers an experimental method for determining \( \Delta H^\circ \) using equilibrium constants obtained at various temperatures. Plotting \( \ln K \) versus \( 1/T \) yields a straight line whose slope is \( -\Delta H^\circ/R \). Hence, the interaction between equilibrium constants and temperature provides a gateway to deducing enthalpy changes, furthering our grasp of the thermodynamic landscape of chemical reactions.

Gibbs Free Energy

Gibbs free energy, symbolized as \( \Delta G \), is a thermodynamic quantity that predicts the direction of a chemical process and determines whether a reaction can occur spontaneously under constant pressure and temperature. A negative \( \Delta G \) indicates a spontaneous process, whereas a positive value suggests the reaction is non-spontaneous.

In the context of equilibrium, the Gibbs free energy of a reaction is tied to the equilibrium constant through the relationship: \[ \Delta G^\circ = -RT \ln K \]where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( K \) is the equilibrium constant.

Relationship with Enthalpy and Entropy

Gibbs free energy also relates to the standard enthalpy change \( \Delta H^\circ \) and the standard entropy change \( \Delta S^\circ \) via the equation: \[ \Delta G^\circ = \Delta H^\circ - T \Delta S^\circ \]This key equation bridges the concepts of energy, disorder, and equilibrium, allowing chemists to dissect the balance of forces driving reactions. It also reflects that a reaction at equilibrium has no net change in Gibbs free energy, making it a cornerstone for understanding and manipulating chemical processes.