Question

The equilibrium constant $K_c$ for $\mathrm{C}(s)+{\mathrm{CO}}_2(g)\rightleftharpoons$ $2\mathrm{CO}(g)$ is $1.9$ at $1000\mathrm{K}$ and $0.133$ at $298\mathrm{K}$. (a) If excess $\mathrm{C}$ is allowed to react with $25.0\mathrm{g}$ of ${\mathrm{CO}}_2$ in a $3.00$ - $\mathrm{L}$ vessel at 1000 $\mathrm{K}$, how many grams of $\mathrm{CO}$ are produced? (b) How many grams of $\mathrm{C}$ are consumed? (c) If a smaller vessel is used for the reaction, will the yield of CO be greater or smaller? (d) Is the reaction endothermic or exothermic?

Short answer

In summary, for the given reaction C(s) + CO2(g) ⇌ 2 CO(g) at 1000 K with excess C and 25.0 g of CO2 in a 3.00 L vessel: (a) approximately 22.3 g of CO are produced, (b) approximately 4.78 g of C are consumed, (c) the yield of CO will be smaller in a smaller vessel, and (d) the reaction is endothermic.

Step-by-step solution

(a) Calculate initial moles of CO2 and C

First, we need to find the initial moles of CO2 and C in the 3.00 L vessel. We are given the mass of CO2 (25.0 g) and the volume of the vessel (3.00 L), so we can use the molar mass of CO2 (44.01 g/mol) to calculate the number of moles of CO2: moles of CO2 = mass / molar mass = 25.0 g / 44.01 g/mol ≈ 0.568 mol We are also given that there is an excess of C, so we do not need to worry about limiting reactants.

(b) Set up the ICE table

Next, we set up an ICE (Initial, Change, Equilibrium) table to track the change in moles of each substance as the reaction proceeds. As there is no information about CO, we assume there is no CO initially. The table looks like this: | | C(s) | CO2(g) | 2 CO(g) | |---------|:-----:|:------:|:--------:| | Initial | - | 0.568 | 0 | | Change | -x | -x | +2x | | Equilib.| - | 0.568-x| 2x | Note that the initial moles of C are not given, as it is in excess, and we are only concerned about its consumption.

(c) Calculate the equilibrium concentrations and moles of CO

We are given Kc at 1000 K (1.9) and we know the volume of the vessel (3.00 L). We can use these to find the equilibrium concentrations of CO2 and CO, and set up the equilibrium constant expression: $K_c=\frac{[CO{]}^2}{[CO2]}=\frac{(2x{)}^2}{0.568-x}$ Now we can solve for x: $1.9=\frac{(2x{)}^2}{0.568-x}$ On solving for x, we get x ≈ 0.398. This means that 2x ≈ 0.796 moles of CO are present at equilibrium. We can then convert the moles of CO into mass: mass of CO = moles × molar mass = 0.796 mol × 28.01 g/mol ≈ 22.3 g So, at 1000 K, approximately 22.3 g of CO are produced.

(d) Calculate the mass of C consumed

From the ICE table, we know that for every mole of CO2 reacted, one mole of C is consumed. So the moles of C consumed is equal to the moles of CO2 reacted (x). We can calculate the mass of C consumed: mass of C = moles × molar mass = 0.398 mol × 12.01 g/mol ≈ 4.78 g Hence, at 1000 K, approximately 4.78 g of C are consumed.

(c) Effect of vessel size on CO yield

According to Le Châtelier's principle, if we increase the pressure (or use a smaller vessel size for the same number of moles), the reaction will shift in the direction that reduces the number of moles of gas to alleviate the increased pressure. In this case, there are fewer moles of gas on the left side of the reaction (one mole of CO2) compared to the right side (two moles of CO). So, if we use a smaller vessel, the reaction will shift towards the left, reducing the yield of CO.

(d) Determine whether the reaction is endothermic or exothermic

We are given Kc at two different temperatures (1000 K and 298 K). We know that if Kc increases with increasing temperature, the reaction is endothermic, whereas if Kc decreases with increasing temperature, the reaction is exothermic. Comparing the given Kc values, we can see that Kc at 1000 K (1.9) is larger than Kc at 298 K (0.133). This indicates that Kc increases as temperature increases. Therefore, the reaction is endothermic. In conclusion, (a) approximately 22.3 g of CO are produced, (b) approximately 4.78 g of C are consumed, (c) the yield of CO will be smaller in a smaller vessel, and (d) the reaction is endothermic.

Key concepts

Equilibrium Constant

In chemical reactions, the extbf{equilibrium constant}, denoted as $K_c$, is a numerical value that shows the ratio of concentrations of products to reactants at equilibrium. For a general reaction: $$\text{aA}+\text{bB}\rightleftharpoons \text{cC}+\text{dD}$$ the equilibrium constant is expressed as:$$K_c=\frac{[C{]}^c[D{]}^d}{[A{]}^a[B{]}^b}$$ Where $[C]$ and $[D]$ are the concentrations of the products, and $[A]$ and $[B]$ are the concentrations of the reactants. The exponents are their respective coefficients in the balanced equation.

• A higher $K_c$ value indicates that, at equilibrium, the products are favored.
• A lower $K_c$ value means the reactants are favored.

In the exercise, the different $K_c$ values at two temperatures also tell us about the reaction's temperature dependency, which we explore through thermodynamics.

Le Châtelier's Principle

Le Châtelier's Principle describes how, if a system at equilibrium experiences a change in concentration, temperature, or pressure, the system will shift in such a way as to counteract this change and restore a new equilibrium. This is particularly useful to predict the behavior of reactions under various conditions. For example:

• Pressure increase: The equilibrium will shift toward the side with fewer moles of gas.
• Concentration change: Adding a reactant or product will shift the equilibrium away from the added substance.
• Temperature change: Increasing temperature favors the endothermic direction, while decreasing temperature favors the exothermic direction.

In this exercise, using a smaller vessel (increasing pressure) predicts that less CO will be formed because the reaction shifts to the left, where there are fewer moles of gas (fewer moles of product).

Endothermic and Exothermic Reactions

Chemical reactions can either absorb heat, termed as endothermic, or release heat, known as exothermic.

• Endothermic: Heat is absorbed, $\mathrm{\Delta }H>0$. The equilibrium constant $K_c$ will increase with increasing temperature, as the system requires heat to proceed to the products.
• Exothermic: Heat is released, $\mathrm{\Delta }H<0$. Here, $K_c$ decreases with an increase in temperature since the excess heat pushes the equilibrium back to the reactants.

In the exercise, since $K_c$ increases as the temperature goes from 298 K to 1000 K, it indicates that our reaction is absorbing heat, thus proving it to be endothermic.

ICE Table in Chemistry

The ICE (Initial, Change, Equilibrium) table is a key tool used to analyze the concentrations of reactants and products during a chemical equilibrium process. It helps in visualizing the changes as the reaction moves from start to equilibrium.To set up an ICE table:

• Initial: Record the initial concentrations or moles of reactants and products.
• Change: Note the changes in concentrations as the reaction proceeds, typically in terms of $x$, where $x$ is the amount that changes.
• Equilibrium: Calculate the concentrations at equilibrium by adding the initial and change values.

In the exercise, the ICE table helps determine how many moles of each substance will be present at equilibrium. Solving for $x$ using the equilibrium constant equation gives us the equilibrium shifts related to the initial and final concentrations.