Question

A 1.00-L flask was filled with \(2.00\) mol gaseous \(\mathrm{SO}_{2}\) and \(2.00 \mathrm{~mol}\) gaseous \(\mathrm{NO}_{2}\) and heated. After equilibrium was reached, it was found that \(1.30\) mol gaseous NO was present. Assume that the reaction $$ \mathrm{SO}_{2}(g)+\mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{SO}_{3}(g)+\mathrm{NO}(g) $$ occurs under these conditions. Calculate the value of the equilibrium constant, \(K\), for this reaction.

Short answer

The equilibrium constant for the reaction \( \mathrm{SO}_{2}(g) + \mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{SO}_{3}(g) + \mathrm{NO}(g) \) under the given conditions is approximately 3.42.

Step-by-step solution

Set up the ICE table

An ICE table is used to organize the initial concentrations, changes, and equilibrium concentrations of each species in the reaction. The table looks like this: | | SO2 | NO2 | SO3 | NO | |:---------:|:---:|:---:|:---:|:---:| | Initial | 2 | 2 | 0 | 0 | | Change | -x | -x | +x | +x | | Equilibrium|2-x | 2-x | x | x | Notice how the changes in the concentration of SO2 and NO2 are negative (since they are being used up in the reaction) and the changes in the concentration of SO3 and NO are positive (since they are being produced).

Calculate the equilibrium concentrations

We are given the amount of NO at equilibrium, which is 1.30 mol. Since the volume of the flask is 1.00 L, we can find the concentration of NO at equilibrium: \[ \mathrm{[NO]}_{eq}= \frac{1.3 \mol}{1.0 \mathrm{L}} = 1.3 \mathrm{M} \] Since the change in concentration for NO is +x, and the equilibrium concentration is 1.3 M, we can find the value of x: \( x = 1.3 \mathrm{M} \) Now we can find the equilibrium concentrations of the other species by plugging in the value of x into the equilibrium row of our ICE table: \[ [\mathrm{SO}_2]_{eq} = 2 - x = 2 - 1.3 = 0.7 \mathrm{M} \] \[ [\mathrm{NO}_2]_{eq} = 2 - x = 2 - 1.3 = 0.7 \mathrm{M} \] \[ [\mathrm{SO}_3]_{eq} = x = 1.3 \mathrm{M} \]

Calculate the equilibrium constant, K

Now that we have the equilibrium concentrations of all species, we can calculate the equilibrium constant, K, using this equation: \[ K = \frac{[\mathrm{SO}_3][\mathrm{NO}]}{[\mathrm{SO}_2][\mathrm{NO}_2]} \] Plug in the equilibrium concentrations: \[ K = \frac{(1.3)(1.3)}{(0.7)(0.7)} \] Now, compute the equilibrium constant, K: \[ K = \approx 3.42 \] So, the equilibrium constant for this reaction is approximately 3.42.

Key concepts

ICE Table

An ICE table is a handy tool for visualizing what's happening in a chemical reaction as it progresses to equilibrium.
It helps you organize the initial concentrations, the changes that occur, and the final equilibrium concentrations of all the reactants and products involved in the reaction.
"ICE" stands for Initial, Change, and Equilibrium, which are the three main components you'll enter into the table.

• Initial: Start with the initial concentrations of your reactants and products before the reaction begins. For reactants, you’ll likely have some starting concentrations, while products typically start at zero.


• Change: This represents the shift in concentration as the reaction proceeds towards equilibrium. This change is indicated by a variable, often represented as "+x" or "-x," which helps keep track of how much each species increases or decreases.


• Equilibrium: This is the concentration of each species when the reaction reaches a steady state. It is calculated by adding the initial concentration and the change in concentration.

Using the ICE table properly allows you to solve for unknowns, helping predict the behavior of a chemical reaction as it approaches equilibrium.

Chemical Equilibrium

Chemical equilibrium is a state in a reversible reaction where the rate of the forward reaction equals the rate of the reverse reaction.
At this point, the concentrations of reactants and products remain constant over time, even though both reactions continue to occur. This doesn’t mean both sides have equal concentrations, but rather that the system reaches a point where there’s no net change in their quantities. For example:

• The reaction \[ \mathrm{SO}_{2}(g) + \mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{SO}_{3}(g) + \mathrm{NO}(g) \] involves sulfur dioxide and nitrogen dioxide converting into sulfur trioxide and nitric oxide, and vice versa.


• When the system reaches equilibrium, the concentrations of \(\mathrm{SO}_{2}, \mathrm{NO}_{2}, \mathrm{SO}_{3},\) and \(\mathrm{NO}\) stop changing because the formation and decomposition reactions are occurring at the same rate.

Understanding chemical equilibrium is crucial since it allows chemists to predict the concentrations of different chemicals involved in a reaction under given conditions.

Concentration

Concentration measures how much solute is dissolved in a given volume of solvent.
In the context of chemical reactions, concentration is often expressed in moles per liter, or molarity (M).
The higher the concentration, the more particles of a substance are present in a liter of solution. In terms of reactions in equilibrium:

• Initial concentrations refer to how much reactants and products are present when the reaction starts.


• Equilibrium concentrations are determined by the changes that occur over the course of the reaction, as captured in an ICE table.


• It is essential to understand that changes in concentration drive reactions.
  For instance, increasing the concentration of reactants typically speeds up the rate of the reaction because more molecules collide and react.

In our example, knowing the concentration of NO at equilibrium allowed us to find the equilibrium constant and assess the overall balance of the reaction.

Reaction Stoichiometry

Reaction stoichiometry involves using the balanced chemical equation to find the ratio in which reactants transform to products.
It’s about examining the quantities of reactants and products involved, ensuring they comply with the law of conservation of mass.

• In the equation \[ \mathrm{SO}_{2}(g) + \mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{SO}_{3}(g) + \mathrm{NO}(g) \], the stoichiometric coefficients indicate that one mole of \(\mathrm{SO}_{2}\) reacts with one mole of \(\mathrm{NO}_{2}\) to produce one mole of \(\mathrm{SO}_{3}\) and one mole of \(\mathrm{NO}\).


• This ratio is essential for setting up the ICE table and understanding the changes in concentrations during the reaction.
  If the stoichiometry was different, the changes in concentration reflected in the ICE table would also differ.

Grasping reaction stoichiometry is vital for predicting the amounts of substances used and formed during a chemical reaction, providing insight into how and why reactions occur.