The subunits of the hemocyanin of the shrimp Callianassa can...

The subunits of the hemocyanin of the shrimp Callianassa can undergo the following association reactions: $$ \begin{array}{r} 2 \mathrm{P} \Leftrightarrow \mathrm{P}_2 \quad K_{12}=\frac{\left[\mathrm{P}_2\right]}{[\mathrm{P}]^2} \\ 2 \mathrm{P}_2 \Leftrightarrow \mathrm{P}_4 \quad K_{24}=\frac{\left[\mathrm{P}_4\right]}{\left[\mathrm{P}_2\right]^2} \\ K_{14}=\frac{\left[\mathrm{P}_4\right]}{[\mathrm{P}]^4} \end{array} $$ or $$ K_{14}=\frac{\left[\mathrm{P}_4\right]}{[\mathrm{P}]^4} $$ The equilibrium constants have been measured as a function of $T$. $$ \begin{array}{rcc} \left.\hline \boldsymbol{T}{ }^{\circ} \mathbf{C}\right) & \boldsymbol{K}_{\mathbf{1 2}} & \boldsymbol{K}_{\mathbf{1 4}} \\ \hline 4 & 1.3 \times 10^7 & 4.5 \times 10^{20} \\ 10 & 1.2 \times 10^7 & 1.0 \times 10^{21} \\ 15 & 1.5 \times 10^7 & 3.0 \times 10^{21} \\ 30 & - & 4.0 \times 10^{22} \\ \hline \end{array} $$ All the concentrations are expressed in moles per liter. Estimate $\Delta H^{\circ}, \Delta S^0$, and $\Delta G^{\circ}$ at $20^{\circ} \mathrm{C}$ for the reactions.

The subunits of the hemocyanin of the shrimp Callianassa can undergo the following association reactions:
$$
\begin{array}{r}
2 \mathrm{P} \Leftrightarrow \mathrm{P}_2 \quad K_{12}=\frac{\left[\mathrm{P}_2\right]}{[\mathrm{P}]^2} \\
2 \mathrm{P}_2 \Leftrightarrow \mathrm{P}_4 \quad K_{24}=\frac{\left[\mathrm{P}_4\right]}{\left[\mathrm{P}_2\right]^2} \\
K_{14}=\frac{\left[\mathrm{P}_4\right]}{[\mathrm{P}]^4}
\end{array}
$$
or
$$
K_{14}=\frac{\left[\mathrm{P}_4\right]}{[\mathrm{P}]^4}
$$

The equilibrium constants have been measured as a function of $T$.
$$
\begin{array}{rcc}
\left.\hline \boldsymbol{T}{ }^{\circ} \mathbf{C}\right) & \boldsymbol{K}_{\mathbf{1 2}} & \boldsymbol{K}_{\mathbf{1 4}} \\
\hline 4 & 1.3 \times 10^7 & 4.5 \times 10^{20} \\
10 & 1.2 \times 10^7 & 1.0 \times 10^{21} \\
15 & 1.5 \times 10^7 & 3.0 \times 10^{21} \\
30 & - & 4.0 \times 10^{22} \\
\hline
\end{array}
$$
All the concentrations are expressed in moles per liter. Estimate $\Delta H^{\circ}, \Delta S^0$, and $\Delta G^{\circ}$ at $20^{\circ} \mathrm{C}$ for the reactions.

Key Concepts

Chemical Equilibrium

Chemical equilibrium refers to the state of a reversible reaction in which the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products over time. Understanding chemical equilibrium is crucial for analyzing how reaction conditions such as concentration and temperature can affect the system's behavior.

Equilibrium Constant (K)

The equilibrium constant (K) quantitatively expresses the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of their respective stoichiometric coefficients. It is a fundamental parameter that links the macroscopic observation of reaction extents to the underlying molecular interactions.

Standard Free Energy Change (?G°)

The standard free energy change, ?G°, is a thermodynamic quantity that indicates the spontaneity of a reaction under standard conditions. It is related to the equilibrium constant through the equation ?G° = -RT ln K, where R is the gas constant and T is the temperature in Kelvin. A negative ?G° signifies a spontaneous process under given conditions.

Van't Hoff Equation

The van't Hoff equation describes the temperature dependence of the equilibrium constant. In its integrated form, ln K = (-?H°/R)(1/T) + (?S°/R), it provides a linear relationship between ln K and 1/T, allowing for the extraction of the standard enthalpy (?H°) and entropy (?S°) changes of the reaction from experimental data over a range of temperatures.

Standard Enthalpy (?H°) and Entropy (?S°) Changes

The standard enthalpy change (?H°) represents the heat absorbed or released during a reaction under constant pressure, while the standard entropy change (?S°) measures the change in disorder or randomness. These parameters are fundamental in determining the position of equilibrium and the temperature dependence of the reaction as they contribute to the overall free energy change.

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