The value of Kp for the reaction, CO2 (g) + C (s) ? 2CO (g) is 3.0 at 1000 K. If initially 2 = 0.48COp bar and CO p = 0 bar and pure graphite is present, calculate the equilibrium partial pressures of CO and CO2. ?

To solve the problem of finding the equilibrium partial pressures of CO and CO2 in the reaction CO2 (g) + C (s) ⇌ 2CO (g) with a given Kp value, we can use the principles of chemical equilibrium. Let's break this down step by step.

Understanding the Reaction

The reaction involves carbon dioxide (CO2) reacting with solid carbon (C) to produce carbon monoxide (CO). Since carbon is a solid, it does not appear in the equilibrium expression. The equilibrium constant Kp for this reaction is given as 3.0 at 1000 K.

Setting Up the Equilibrium Expression

The equilibrium constant expression for this reaction can be written as:

Kp = (P_CO)^2 / (P_CO2)

Where P_CO is the partial pressure of carbon monoxide and P_CO2 is the partial pressure of carbon dioxide.

Initial Conditions

Initially, we have:

• P_CO2 = 0.48 bar
• P_CO = 0 bar

Since we start with pure graphite, we do not include its concentration in the equilibrium expression.

Change in Concentration

As the reaction progresses towards equilibrium, let’s denote the change in the partial pressure of CO2 that reacts as x. Therefore, the changes in partial pressures can be expressed as:

• P_CO2 at equilibrium = 0.48 - x
• P_CO at equilibrium = 2x

Substituting into the Kp Expression

Now we can substitute these expressions into the Kp equation:

3.0 = (2x)^2 / (0.48 - x)

Solving for x

Expanding this gives:

3.0 = 4x² / (0.48 - x)

Cross-multiplying to eliminate the fraction results in:

3.0(0.48 - x) = 4x²

Distributing the 3.0 leads to:

1.44 - 3.0x = 4x²

Rearranging this into standard quadratic form gives:

4x² + 3.0x - 1.44 = 0

Applying the Quadratic Formula

We can solve this quadratic equation using the quadratic formula, x = [-b ± √(b² - 4ac)] / 2a, where:

• a = 4
• b = 3.0
• c = -1.44

Calculating the discriminant:

b² - 4ac = (3.0)² - 4(4)(-1.44) = 9 + 23.04 = 32.04

Now substituting into the formula:

x = [-3.0 ± √32.04] / 8

Calculating the square root gives approximately 5.66, so:

x = [-3.0 ± 5.66] / 8

This results in two possible values for x:

• x ≈ 0.33 (taking the positive root)
• x ≈ -1.06 (not physically meaningful)

Finding Equilibrium Partial Pressures

Now, substituting x back into our expressions for the equilibrium partial pressures:

• P_CO2 = 0.48 - 0.33 = 0.15 bar
• P_CO = 2(0.33) = 0.66 bar

Final Results

At equilibrium, the partial pressures are:

• P_CO2 = 0.15 bar
• P_CO = 0.66 bar

This calculation shows how the initial conditions and the equilibrium constant work together to determine the final state of the system. If you have any further questions about this process or need clarification on any steps, feel free to ask!