Question

At a particular temperature, \(K=2.0 \times 10^{-6}\) for the reaction $$ 2 \mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g) $$ If \(2.0 \mathrm{~mol} \mathrm{CO}_{2}\) is initially placed into a 5.0-L vessel, calculate the equilibrium concentrations of all species.

Short answer

The equilibrium concentrations of \(CO_2\), \(CO\), and \(O_2\) are approximately 0.395 M, \(5.04 \times 10^{-3}\,M\), and \(2.52 \times 10^{-3}\,M\), respectively.

Step-by-step solution

Write the equilibrium expression

For the reaction: \[2 CO_{2}(g) \rightleftharpoons 2 CO(g) + O_{2}(g)\] the equilibrium constant expression can be written as: \[K = \frac{[CO]^2[O_2]}{[CO_2]^2}\]

Set up an ICE table

We set up an ICE table to keep track of the concentrations of the species during the reaction. The initial concentration of CO2 can be calculated using its initial moles and volume of the vessel: Initial concentration of CO2 = \(\frac{2.0\,\text{mol}}{5.0\,\text{L}} = 0.4\,M\) Since initially CO and O2 are not present, their concentrations are 0. The ICE table looks like this: | Species | Initial | Change | Equilibrium | | ------------- | ------- | ------ | ----------- | | \(CO_{2}\) | 0.4 | -2x | 0.4-2x | | \(CO\) | 0 | +2x | 2x | | \(O_{2}\) | 0 | +x | x | where x represents the change in the concentration of the species from their initial to equilibrium states.

Substitute the equilibrium concentrations into the equilibrium expression

Now, we can substitute the equilibrium concentrations expressed in terms of x into the equilibrium constant expression: \[K = \frac{(2x)^2(x)}{(0.4-2x)^2}\] Given that \(K = 2.0 \times 10^{-6}\), we have: \[2.0 \times 10^{-6} = \frac{(2x)^2(x)}{(0.4-2x)^2}\]

Solve for x

Solving for x in the above equation can be a little tedious with algebraic manipulation. Instead, we make the assumption that x is small compared to 0.4, so the change in CO2 is negligible. This simplifies the equation to: \[2.0 \times 10^{-6} = \frac{(2x)^2(x)}{(0.4)^2}\] Now, we can solve for x: \[x = \sqrt[3]{2.0 \times 10^{-6} \times (0.4)^2 / 4}\] \[x \approx 2.52 \times 10^{-3}\]

Calculate the equilibrium concentrations

Now that we have x, we can find equilibrium concentrations for each species: Equilibrium concentration of \(CO_2\) = \(0.4 - 2x \approx 0.4 - 2(2.52 \times 10^{-3}) \approx 0.395\,M\) Equilibrium concentration of \(CO\) = \(2x \approx 2(2.52 \times 10^{-3}) \approx 5.04 \times 10^{-3}\,M\) Equilibrium concentration of \(O_2\) = \(x \approx 2.52 \times 10^{-3}\,M\) So the equilibrium concentrations of CO2, CO, and O2 are approximately 0.395 M, 5.04 x 10^(-3) M, and 2.52 x 10^(-3) M, respectively.

Key concepts

Equilibrium Constant Expression

Understanding the equilibrium constant expression is crucial when studying chemical reactions at equilibrium. This expression quantifies the concentrations of the products and reactants when they have reached a state of balance. It is represented by the symbol 'K' and varies with temperature.

For a generic chemical reaction where the reactants A and B convert into products C and D, with their respective stoichiometric coefficients a, b, c, and d: \[aA + bB \rightleftharpoons cC + dD\] The equilibrium constant expression is written as: \[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\] Here, the concentrations of the products are raised to the power of their coefficients and multiplied together. The same is done for the reactants, but instead, their concentrations are used in the denominator. Pure solids and liquids are omitted as they have constant concentrations.
This expression can be related directly to the example given in the exercise, where the equilibrium constant for the reaction involving carbon dioxide, carbon monoxide, and oxygen has to be used in order to calculate the equilibrium concentrations of these species.

ICE Table Method

The ICE table method is a systematic approach used to determine the changes in concentration of reactants and products as they shift towards equilibrium. ICE stands for Initial, Change, and Equilibrium. It's an invaluable tool in calculations involving chemical equilibrium.

First, we list the initial concentrations of all species involved in the reaction. If any product is not present initially, its concentration starts at zero. Next, we express how these concentrations change as the reaction progresses. The changes are typically represented by a variable, often 'x', which is connected to the stoichiometry of the reaction. Finally, we describe the equilibrium concentrations of all species in terms of this variable.

Once the ICE table is complete, we can plug the expressions for equilibrium concentrations into the equilibrium constant expression to solve for 'x'. After finding 'x', we can easily calculate the equilibrium concentrations of all species. For reactions with small equilibrium constants, we might assume that the change in concentration of reactants is negligible compared to their initial concentration, simplifying the calculations. This method allows us to systematically track the concentrations and is particularly useful when dealing with complex equilibrium systems.

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rates of the forward and reverse reactions are equal, leading to no net change in the concentrations of reactants and products over time. It is a dynamic process, which means that while the concentrations remain constant, the reactants continuously convert to products and vice versa.

Achieving equilibrium does not necessarily mean that the reactants and products are present in equal amounts but rather that their ratios at this point will remain steady. The equilibrium constant 'K' helps to determine the direction in which the reaction prefers to lie under certain conditions. For example, a large 'K' implies more products at equilibrium, whereas a small 'K' indicates that the reactants are favored.

Chemical equilibrium is an essential concept in many areas of chemistry and is especially important when predicting the outcome of reactions, synthesizing materials, and understanding natural processes. By mastering the calculation of equilibrium concentrations, students can better predict the behavior of chemical systems under various conditions, which is a vital skill in the study of chemistry.