The equilibrium constant for the reaction CO(g) + H2O(g) ⇔ CO2(g) + H2(g) is 3 at 500º C.In a 2 litre vessel 60 g of water gad and 90 g steam . Find the number of moles of H2 at equilibrium

To determine the number of moles of hydrogen gas (H2) at equilibrium for the reaction CO(g) + H2O(g) ⇔ CO2(g) + H2(g), we need to start by calculating the initial amounts of each substance involved in the reaction and then apply the equilibrium constant (K) to find the equilibrium concentrations. Let's break this down step by step.

Step 1: Calculate Initial Moles of Reactants

First, we need to convert the masses of water and steam into moles. The molar mass of water (H2O) is approximately 18 g/mol.

• For water (H2O):

  Initial mass = 60 g

  Initial moles = 60 g / 18 g/mol = 3.33 moles

• For steam (H2O):

  Initial mass = 90 g

  Initial moles = 90 g / 18 g/mol = 5 moles

Since both water and steam are forms of H2O, we can combine these amounts. Therefore, the total initial moles of H2O is:

Total moles of H2O = 3.33 moles + 5 moles = 8.33 moles

Step 2: Set Up the Reaction Table

Next, we can set up an ICE (Initial, Change, Equilibrium) table to track the changes in moles as the reaction reaches equilibrium. Let’s denote the change in moles of H2O that reacts as x.

Step 3: Apply the Equilibrium Constant

The equilibrium constant (K) for the reaction is given as 3 at 500º C. The expression for K in terms of the equilibrium concentrations is:

K = [CO2][H2] / [CO][H2O]

Substituting the equilibrium values from our table into the K expression gives:

3 = (x)(x) / (x)(8.33 - x)

This simplifies to:

3 = x² / (8.33 - x)

Step 4: Solve for x

Now, we can rearrange the equation to solve for x:

3(8.33 - x) = x²

Expanding this gives:

24.99 - 3x = x²

Rearranging leads to:

x² + 3x - 24.99 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Here, a = 1, b = 3, and c = -24.99:

x = [-3 ± √(3² - 4(1)(-24.99))] / 2(1)

x = [-3 ± √(9 + 99.96)] / 2

x = [-3 ± √108.96] / 2

x = [-3 ± 10.44] / 2

This gives us two potential solutions for x:

x = 3.72 (taking the positive root) or x = -6.72 (not physically meaningful)

Step 5: Calculate Moles of H2 at Equilibrium

At equilibrium, the number of moles of H2 is equal to x. Therefore, we find:

Moles of H2 = x = 3.72 moles

Final Thoughts

Thus, at equilibrium, there will be approximately 3.72 moles of hydrogen gas (H2) present in the vessel. This methodical approach, using the ICE table and the equilibrium constant, allows us to systematically find the equilibrium concentrations of all species involved in the reaction.