\(2 \mathrm{HI}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g)+\mathrm{I}_{2}(g)+\) energy A gaseous reaction occurs and comes to equilibrium, as shown above. Which of the following changes to the system will serve to increase the number of moles of \(\mathrm{I}_{2}\) present at equilibrium? (A) Increasing the volume at constant temperature (B) Decreasing the volume at constant temperature (C) Increasing the temperature at constant volume (D) Decreasing the temperature at constant volume

First, we can understand and define the system. We have a gaseous reaction 2HI(g) + Cl2(g) ⇌ 2HCl(g) + I2(g) + energy that is happening at equilibrium.

Next, we use Le Chatelier's Principle. It predicts how the equilibrium will shift when changes are made to a chemical system at equilibrium. If an external stress is applied to the system such as changes in temperature, pressure, or concentration of the components, the system will react in a way to minimize the effect of this stress.

Next, we apply Le Chatelier's Principle to the proposed changes. Let's look at the options: A) Increasing the volume at constant temperature will result in the system moving to the side with more moles of gas. In our equation, that’s on the reactants side, but we want to increase the product of I2, so this option is incorrect. B) Decreasing the volume at constant temperature will make the system move to the side with fewer moles of gas. Again, this is on the reactants side, so this option is also incorrect. C) Increasing the temperature at constant volume will cause the system to absorb the heat and shift the equilibrium to the right towards the products side to produce more I2 and HCl. Therefore, this option is correct. D) Decreasing the temperature at constant volume is incorrect since it will shift the system towards the reactants side (the side with heat as product) to produce heat.

Finally, we conclude that only an increase in temperature causes an increase in I2 production at equilibrium, as it shifts the equilibrium towards the products side to absorb the heat.