This problem involves calculating the equilibrium concentration of D in a reversible reaction where the equilibrium constant K is given as 100 at 298 K. The reaction is A + B ⇌ C + D, and all initial concentrations are 1 M.

Step 1: Write the reaction and the equilibrium constant expression.

The reaction is: $A+B\rightleftharpoons C+D$

The equilibrium constant K for this reaction is given by the ratio of the product of the concentrations of the products to the product of the concentrations of the reactants, each raised to the power of their stoichiometric coefficients. Since all coefficients are 1, the expression is:

$K=\frac{\left[C\right]\left[D\right]}{\left[A\right]\left[B\right]}=100$

Step 2: Define the change in concentration.

Let the amount of A and B that react to reach equilibrium be x mol/L. Since the reaction produces one mole of C and D for each mole of A and B consumed, the changes in concentrations are:

Change in [A] = -x

Change in [B] = -x

Change in [C] = +x

Change in [D] = +x

Step 3: Write the equilibrium concentrations in terms of x.

Initial concentrations: [A]₀ = 1 M, [B]₀ = 1 M, [C]₀ = 1 M, [D]₀ = 1 M

Equilibrium concentrations:

[A] = 1 - x

[B] = 1 - x

[C] = 1 + x

[D] = 1 + x

Step 4: Substitute the equilibrium concentrations into the K expression and solve for x.

$K=\frac{(1+x)(1+x)}{(1-x)(1-x)}=100$

This simplifies to:

${\left(\frac{1+x}{1-x}\right)}^2=100$

Taking the square root of both sides:

$\frac{1+x}{1-x}=10$

Now, solve for x:

$1+x=10(1-x)$

$1+x=10-10x$

$x+10x=10-1$

$11x=9$

$x=\frac{9}{11}\approx 0.818$

Step 5: Find the equilibrium concentration of D.

[D] = 1 + x = 1 + 0.818 = 1.818 M

Final Answer: The equilibrium concentration of D is 1.818 mol L⁻¹.

Related Topics and Formulae

Equilibrium Constant (K): For a general reaction aA + bB ⇌ cC + dD, the equilibrium constant K is defined as K = ([C]^c [D]^d) / ([A]^a [B]^b), where the concentrations are those at equilibrium. K is constant at a given temperature.

Reaction Quotient (Q): Q has the same form as K but uses the initial concentrations. Comparing Q to K predicts the direction in which the reaction will proceed to reach equilibrium.

Le Chatelier's Principle: If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. This principle helps predict the effect of changes in concentration, pressure, or temperature on a system at equilibrium.