The equilibrium constant for the following reaction is 1.6 ×105 at 1024K H2(g) + Br2(g) = 2HBr(g) Find the equilibrium pressure of all gases if 10.0 bar of HBr is introduced into a sealed container at 1024K.

To find the equilibrium pressures of all gases in the reaction H2(g) + Br2(g) ⇌ 2HBr(g) when starting with 10.0 bar of HBr, we can use the equilibrium constant (K) and the initial conditions to set up an expression. The equilibrium constant for this reaction at 1024 K is given as 1.6 × 10^5.

Setting Up the Reaction

First, let's define the initial conditions and the changes that occur as the system reaches equilibrium. We start with:

• Initial pressure of HBr = 10.0 bar
• Initial pressure of H2 = 0 bar
• Initial pressure of Br2 = 0 bar

Change in Pressures

As the reaction proceeds towards equilibrium, some of the HBr will dissociate into H2 and Br2. Let's denote the change in pressure of HBr that dissociates as 'x'. Therefore, at equilibrium, we have:

• Pressure of HBr = 10.0 - 2x
• Pressure of H2 = x
• Pressure of Br2 = x

Equilibrium Expression

The equilibrium constant expression for this reaction is given by:

K = (P_HBr)^2 / (P_H2 * P_Br2)

Substituting the equilibrium pressures into this expression, we get:

1.6 × 10^5 = (10.0 - 2x)^2 / (x * x)

Solving for x

Now, let's rearrange the equation to solve for x:

1.6 × 10^5 = (10.0 - 2x)^2 / x^2

Cross-multiplying gives:

1.6 × 10^5 * x^2 = (10.0 - 2x)^2

Expanding the right side:

1.6 × 10^5 * x^2 = 100 - 40x + 4x^2

Rearranging this into a standard quadratic form:

(1.6 × 10^5 - 4)x^2 + 40x - 100 = 0

This simplifies to:

1.59996 × 10^5 x^2 + 40x - 100 = 0

Using the Quadratic Formula

To solve for x, we can apply the quadratic formula, x = [-b ± √(b² - 4ac)] / 2a, where:

• a = 1.59996 × 10^5
• b = 40
• c = -100

Calculating the discriminant:

b² - 4ac = 40² - 4 * (1.59996 × 10^5) * (-100)

Now, substituting these values into the quadratic formula will yield the value of x.

Finding Equilibrium Pressures

Once you find the value of x, substitute it back into the expressions for the equilibrium pressures:

• Pressure of HBr = 10.0 - 2x
• Pressure of H2 = x
• Pressure of Br2 = x

Final Thoughts

By following these steps, you can determine the equilibrium pressures of all gases involved in the reaction. This approach not only helps in solving this specific problem but also reinforces your understanding of equilibrium concepts and the application of the equilibrium constant in chemical reactions.