a) Use the relationship between K_p and K_c to show that Dn = 0 when K_p = K_c:

K_p = K_c (RT)^Dn

Since R and T are constant, Dn must be 0 for K_p = K_c

where Dn = moles of gaseous product – moles gaseous reactant

With this information determine the minimum integer values of a, b, c, and d:

Dn = 0 = (b + c + d) – a

a = b + c + d

since 1 is the smallest positive integer

a = 1 + 1 + 1 = 3

a = 3, b = c = d = 1

b) Set up an icebox and use the known value of K_c and initial concentration of A to calculate the equilibrium concentrations of all reactants and products:

3A <=> B + C + D

i) y 0 0 0

c) -3x x x x

e) y-3x x x x

K_c = [B][C][D]/[A]³

1.00x10^-15 = x³/(y-3x)³ where y = [A]_initial

To solve this cubic equation assume that 3x is small compared to the magnitude of y since K_c << 10^-5:

1.00x10^-15 = x³/y³

which is straightforward to solve. Check your assumption and use successive approximation if needed. Once x converges and no longer changes, substitute back into the icebox to calculate the equilibrium concentrations of A, B, C and D.

Another way to solve the equation:

1.00x10^-15 = x³/(y-3x)³

is to simply take the cube root of both sides of the equation and solve for x. This avoids making any assumptions and using the method of successive approximation.