Which of the following lines correctly show the temperature dependence of equilibrium constant, K, for an exothermic reaction?

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Which of the following lines correctly show the temperature dependence of equilibrium constant, K, for an exothermic reaction?

Engineering

Chemistry

Equilibrium Constant and Mass Action Law

Calculation of Various Parameters at Equilibrium

Arrhenius Equation

Question

Which of the following lines correctly show the temperature dependence of equilibrium constant, K, for an exothermic reaction?

A

A and D

B

B and C

C

C and D

D

A and B

ln (K) = $\frac{- ΔH °}{RT} + \frac{- ΔS °}{R}$

For exothermic reaction $∆$ H° < 0

so slope will be +ve

A and B are correct

The equilibrium constant (K) for a reaction is related to temperature by the van't Hoff equation:

\frac{d (\ln K)}{d T} = \frac{Δ H^{\circ}}{R T^{2}}

For an exothermic reaction, the standard enthalpy change $Δ H^{\circ}$ is negative. This means:

\frac{d (\ln K)}{d T} < 0

This inequality tells us that ln K decreases as temperature T increases. Since ln K is a monotonically increasing function of K, this also means the equilibrium constant K itself decreases as temperature increases.

Now let's analyze the graph options. The x-axis is temperature (T) and the y-axis is the equilibrium constant (K).

Step 1: Identify the decreasing curves
Since K must decrease with increasing T for an exothermic reaction, we are looking for curves that slope downward from left to right. In the image, curves A and D are decreasing.

Step 2: Verify the mathematical behavior
The van't Hoff equation also provides the integrated form for a constant $Δ H^{\circ}$ : $\ln K = - \frac{Δ H^{\circ}}{R} \frac{1}{T} + \frac{Δ S^{\circ}}{R}$ This equation has the form y = mx + c, where y = ln K and x = 1/T. For an exothermic reaction (negative slope m), a plot of ln K vs 1/T is a straight line with a positive slope (because of the negative sign in front of the negative $Δ H^{\circ}$ ). However, the given graph is K vs T, not ln K vs 1/T. The relationship between K and T is not linear but follows the form above, resulting in a curve that decreases, like A and D.

Final Answer: The lines that correctly show the temperature dependence for an exothermic reaction are A and D.

Related Topics & Formulae

van't Hoff Equation: This equation is fundamental for understanding how equilibrium constants change with temperature. It is derived from the relationship between the Gibbs free energy change and the equilibrium constant: $Δ G^{\circ} = - R T \ln K$ and the Gibbs-Helmholtz equation: $(\frac{&partial (Δ G^{\circ} / T)}{&partial T}) p = - \frac{Δ H^{\circ}}{T^{2}}$ .

Le Chatelier's Principle: This qualitative principle predicts that for an exothermic reaction (which releases heat), increasing the temperature will shift the equilibrium position to the left, towards the reactants. This shift results in a decrease in the equilibrium constant K, which is a quantitative measure of the position of equilibrium. This principle is consistent with the mathematical result from the van't Hoff equation.