Video: Calculating the Concentration of CO from Equilibrium Constant

Carbon is reacted with water vapor at 1000°C, establishing the shown equilibrium. C(s) + H₂O(g) ⇌ CO(g) + H₂(g), 𝐾_c = 0.200 at 1 000°C. What is the concentration of CO in the equilibrium mixture if [H₂O] = 0.750 M?

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Video Transcript

Carbon is reacted with water vapor at 1000 degrees Celsius, establishing the shown equilibrium C solid + H₂O gas in equilibrium with CO gas plus H₂ gas, with the equilibrium constant equal to 0.200 at 1000 degrees Celsius. What is the concentration of CO in the equilibrium mixture if the concentration of H₂O is equal to 0.750 molar?

To answer this question, we need to identify what 𝐾 c is. It’s the equilibrium constant. For a simple equilibrium where quantities of 𝐴 and 𝐵 are in equilibrium with 𝐶 and 𝐷, this is the equilibrium constant expression. The presence of any solids or liquids is ignored. So what does the equilibrium constant expression look like for this equilibrium? For the products, we have a concentration of carbon monoxide multiplied by the concentration of hydrogen. And for reactants, we ignore the presence of carbon because it’s a solid and only include the concentration of water.

Here, the water is being heated with 1000 degrees Celsius. So it’s in gas form and therefore does appear in the equilibrium constant expression. So how do we use this information to work out the concentration of carbon monoxide in an equilibrium mixture with the concentration of water vapor at 0.750 molar? The first difficulty is we’re not told the concentration of hydrogen. However, the question tells us that we only generated carbon monoxide and hydrogen after we reacted carbon with water vapor. So we didn’t have any to start with.

Since carbon monoxide and hydrogen are in a one-to-one ratio at all times, the carbon monoxide concentration will be equal to the hydrogen concentration. If we substitute this into our 𝐾 c expression, we’ll be able to figure out the carbon monoxide concentration at equilibrium. The first thing we want to do is rearrange the equation in terms of the carbon monoxide concentration. If we multiply through by the concentration of water, we end up with the concentration of carbon monoxide squared equal to 𝐾 c times the concentration of water. Taking the square root of both sides gives us the carbon monoxide concentration as the square root of 𝐾 c times the concentration of water.

Before we go any further, we should figure out the units for 𝐾 c. 𝐾 c is often given without units. But we’re given 𝐾 c in terms of concentrations. So we can work out the units. The units for the concentration of carbon monoxide squared will be molar squared. And the units for the concentration of water will be molars. So the units for 𝐾 c in this case will be molars. We can now substitute it into the equation and respect the units.

The concentration of carbon monoxide is equal to the square root of 0.200 molar times 0.750 molar. This gives us the square root of 0.15 molar squared which is equal to 0.387298 molar. All the values in our calculation are given to three significant figures. So we should give our answer to the same precision. So our final answer for the concentration of carbon monoxide in an equilibrium mixture where the concentration of water is 0.750 molar is 0.387 molar.

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