Cyclohexane \((C)\) and methylcyclopentane \((M)\) are isomers with the chemical formula \(\mathrm{C}_{6} \mathrm{H}_{12}\). The equilibrium constant for the rearrangement \(\mathrm{C} \rightleftharpoons \mathrm{M}\) in solution is \(0.140\) at \(25^{\circ} \mathrm{C}\). (a) \(\mathrm{A}\) solution of \(0.0200 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) cyclohexane and \(0.100 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) methylcyclopentane is prepared. Is the system at equilibrium? If not, will it will form more reactants or more products? (b) What are the concentrations of cyclohexane and methylcyclohexane at equilibrium? (c) If the temperature is raised to \(50 .{ }^{\circ} \mathrm{C}\), the concentration of cyclohexane becomes \(0.100 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) when equilibrium is re-established. Calculate the new equilibrium constant. (d) Is the reaction exothermic or endothermic at \(25^{\circ} \mathrm{C}\) ? Explain your conclusion.

The equilibrium constant, represented by the symbol \(K\), is a critical aspect of chemical equilibrium. It reflects the ratio of the concentration of products to reactants at equilibrium for a reversible chemical reaction. In essence, the value of \(K\) determines the extent of the reaction; a larger \(K\) suggests the equilibrium favors the formation of products, whereas a smaller \(K\) indicates a predilection towards the reactants.

Taking the textbook exercise as an example, the equilibrium constant for the cyclohexane and methylcyclopentane rearrangement is \(0.140\) at \(25^\circ C\). This value is used to assess whether the provided concentrations of reactants and products are at equilibrium or if the reaction will shift to balance itself. By comparing \(K\) with the calculated reaction quotient \(Q\), which is based on the current concentrations, one can anticipate the direction in which the reaction will proceed to achieve equilibrium.

The reaction quotient, \(Q\), serves as a 'snapshot' of a reaction's composition at any moment and can be compared against the equilibrium constant, \(K\), to predict the direction of shift towards equilibrium. It is calculated using the same formula as \(K\), but with the current concentrations of reactants and products rather than those at equilibrium.

In our cyclohexane and methylcyclopentane scenario, by calculating \(Q = \frac{[M]}{[C]} = \frac{0.100}{0.0200} = 5\) and observing that \(Q > K\), we deduce that the reaction mixture is not at equilibrium and will move towards the formation of more cyclohexane, the reactant, to lower the value of \(Q\) to match \(K\).

Le Chatelier's principle provides insight on how a system at equilibrium responds to external stresses, such as changes in concentration, pressure, or temperature. It posits that a system will adjust to counteract the effect of the disturbance and re-establish equilibrium. This principle is fundamental when predicting the outcome of perturbations to the equilibrium state.

For instance, increasing the temperature of an endothermic reaction will shift the equilibrium to the right towards the products to absorb the extra heat, as seen in the cyclohexane conversion in our exercise when the temperature is raised from \(25^\circ C\) to \(50^\circ C\). Conversely, cooling an exothermic reaction will prompt the equilibrium to shift to the right to release heat and restore balance.

Chemical reactions can either release energy (exothermic) or absorb it (endothermic), with these energy changes influencing equilibrium. An exothermic reaction, which releases energy in the form of heat, will have an increased rate of the reverse reaction when heated, resulting in a decrease in the value of the equilibrium constant \(K\). On the other hand, an endothermic reaction absorbs energy, and as such, an increase in temperature will cause an increase in the value of \(K\) and the formation of more products.

Through our example—wherein the equilibrium constant rises from 0.140 at \(25^\circ C\) to 0.800 at \(50^\circ C\)—we can deduce the reaction is endothermic. The system absorbs additional heat upon heating, favoring the formation of products and reflecting a higher \(K\) value at the elevated temperature.