in an equillibrium a+bc+d A and B are mixed in a vessel at temparature T. the initial concentration of A was twice the initial concentration of B . after equillibrium has established , concentration ofC was thrice the equillibrium concentration of B .calculate kc

To solve this problem, we need to analyze the equilibrium situation involving the chemical reaction represented by the equation A + B ⇌ C + D. Given the initial concentrations and the relationships between them, we can derive the equilibrium constant (Kc) for the reaction. Let’s break this down step by step.

Understanding Initial Concentrations

We start with the information that the initial concentration of A is twice that of B. Let’s denote the initial concentration of B as [B]₀. Therefore, the initial concentration of A can be expressed as:

• [A]₀ = 2[B]₀

Setting Up the Equilibrium Expression

At equilibrium, we denote the changes in concentrations as follows:

• Let x be the change in concentration of B that reacts to form C and D.
• Then, the change in concentration of A will also be x, since one mole of A reacts with one mole of B.
• The concentration of C formed will be x, and the concentration of D formed will also be x.

Now we can express the equilibrium concentrations:

• [A] = [A]₀ - x = 2[B]₀ - x
• [B] = [B]₀ - x
• [C] = x
• [D] = x

Using Given Relationships

According to the problem, the concentration of C at equilibrium is three times the equilibrium concentration of B:

• [C] = 3[B]

Substituting the expressions we derived:

• x = 3([B]₀ - x)

Solving for x

Now we can solve for x. Expanding the equation gives:

• x = 3[B]₀ - 3x

Rearranging this, we get:

• x + 3x = 3[B]₀
• 4x = 3[B]₀
• x = (3/4)[B]₀

Finding Equilibrium Concentrations

Now that we have x, we can find the equilibrium concentrations:

• [A] = 2[B]₀ - (3/4)[B]₀ = (8/4)[B]₀ - (3/4)[B]₀ = (5/4)[B]₀
• [B] = [B]₀ - (3/4)[B]₀ = (1/4)[B]₀
• [C] = (3/4)[B]₀
• [D] = (3/4)[B]₀

Calculating Kc

The equilibrium constant expression for the reaction is given by:

• Kc = [C][D] / ([A][B])

Substituting the equilibrium concentrations into the expression:

• Kc = ((3/4)[B]₀)((3/4)[B]₀) / (((5/4)[B]₀)((1/4)[B]₀))

Now simplifying this:

• Kc = (9/16)[B]₀² / (5/16)[B]₀²
• Kc = 9/5

Final Result

The equilibrium constant Kc for the reaction A + B ⇌ C + D, given the initial conditions and relationships, is:

• Kc = 1.8

This value indicates the ratio of the concentrations of products to reactants at equilibrium, providing insight into the extent of the reaction under the specified conditions.