The standard reaction Gibbs energy, Δ_rG°, is related to the equilibrium constant (K) and the standard reaction enthalpy (Δ_rH°) through fundamental thermodynamic equations. The given equation is:

$\mathrm{\Delta }{r}_{}{G}^{°}=A-BT$

where A and B are non-zero constants.

Step 1: Relate Δ_rG° to the Equilibrium Constant (K)
The van't Hoff equation provides a crucial connection: $\mathrm{\Delta }{r}_{}{G}^{°}=-RT \ln K$ where R is the universal gas constant.

Step 2: Relate Δ_rG° to Δ_rH° (The Key to Exo/Endothermicity)
The Gibbs-Helmholtz equation is: $\left[\frac{\partial \left(\frac{\mathrm{\Delta }{r}_{}{G}^{°}}{T}\right)}{\partial T}\right]p=-\frac{\mathrm{\Delta }{r}_{}{H}^{°}}{T^2}$ This equation directly connects the temperature dependence of Δ_rG° to the standard reaction enthalpy, Δ_rH°, which tells us if a reaction is exothermic (Δ_rH° < 0) or endothermic (Δ_rH° > 0).

Step 3: Substitute the Given Equation into the Gibbs-Helmholtz Equation
First, we find the expression for Δ_rG°/T from our given equation: $\frac{\mathrm{\Delta }{r}_{}{G}^{°}}{T}=\frac{A}{T}-B$
Now, we take the partial derivative of this expression with respect to temperature (T) at constant pressure (p): $\frac{\partial }{\partial T}(\frac{A}{T}-B)=-\frac{A}{T^2}-0=-\frac{A}{T^2}$
According to the Gibbs-Helmholtz equation, this result must equal -Δ_rH°/T²: $-\frac{A}{T^2}=-\frac{\mathrm{\Delta }{r}_{}{H}^{°}}{T^2}$
Multiplying both sides by -T² gives us the final, critical relationship: $A=\mathrm{\Delta }{r}_{}{H}^{°}$

Step 4: Analyze the Result and Evaluate the Options
We have found that the constant A is numerically equal to the standard reaction enthalpy, Δ_rH°.

• A reaction is endothermic if Δ_rH° > 0. Since A = Δ_rH°, this means the reaction is endothermic if A > 0.
• A reaction is exothermic if Δ_rH° < 0. Since A = Δ_rH°, this means the reaction is exothermic if A < 0.

Notice that the constant B does not appear in this result for the enthalpy. B is related to the reaction entropy (Δ_rS°), but it is not needed to determine if the reaction is exothermic or endothermic.

Conclusion: The correct statement is "Endothermic if A > 0". The sign of the constant B has no effect on whether the reaction is exothermic or endothermic.

Related Concepts & Formulae

Gibbs Free Energy: ΔG = ΔH - TΔS. This is the central equation linking free energy (spontaneity), enthalpy (heat), and entropy (disorder).

Standard Reaction Gibbs Energy: $\mathrm{\Delta }{r}_{}{G}^{°}=-RT \ln K$. It defines the relationship between the standard free energy change and the equilibrium constant.

Gibbs-Helmholtz Equation: $\left[\frac{\partial \left(\frac{\mathrm{\Delta }G}{T}\right)}{\partial T}\right]p=-\frac{\mathrm{\Delta }H}{T^2}$. This is the differential form used to extract enthalpy data from how free energy changes with temperature.