At 100 °C the reaction:
SO2Cl2(g) ⇄ SO2(g) + Cl2(g)
has Kp = 2.4.
Calculate the equilibrium composition of the mixture when 1.2 atm of SO2Cl2 are mixed with 0.8 atm of Cl2 at 100 °C.
.......SO2Cl2(g) ⇄ SO2(g) + Cl2(g)
I......1.2..........0........0.8'
C.......-x..........x.........x
E.....1.2-x.........x......0.8+x
Kp = 2.4 = (x)(0.8+x)/(1.2-x)
Solve for x and evaluate each of the components.
To calculate the equilibrium composition of the mixture, we need to use the Kp value and the initial pressures of the reactants.
Given:
Kp = 2.4
Initial pressure of SO2Cl2 (P(SO2Cl2)) = 1.2 atm
Initial pressure of Cl2 (P(Cl2)) = 0.8 atm
Let's assume the change in pressures for SO2, Cl2, and SO2Cl2 as x atm. Therefore, the equilibrium pressures will be:
P(SO2) = P(SO2Cl2) - x
P(Cl2) = P(Cl2) - x
Using the ideal gas law, we can relate the pressures to the mole fractions:
P(SO2) = n(SO2) * R * T / V
P(Cl2) = n(Cl2) * R * T / V
P(SO2Cl2) = n(SO2Cl2) * R * T / V
Here,
n(SO2) = moles of SO2
n(Cl2) = moles of Cl2
n(SO2Cl2) = moles of SO2Cl2
R = gas constant
T = temperature
V = volume
Let's assume the volume is constant, and R = 0.0821 L.atm/(mol.K) (gas constant)
We can now write the expression for the Kp value:
Kp = (P(SO2) * P(Cl2)) / (P(SO2Cl2))
Substituting the equation for the pressures derived from the ideal gas law:
Kp = [(n(SO2) * R * T / V) * (n(Cl2) * R * T / V)] / [(n(SO2Cl2) * R * T / V)]
Canceling out the V from both sides of the equation, we get:
Kp = (n(SO2) * n(Cl2)) / n(SO2Cl2)
Rearranging the equation:
(n(SO2) * n(Cl2)) / n(SO2Cl2) = Kp
Let's substitute the given values:
(1.2 - x) * (0.8 - x) / x = 2.4
Simplifying the equation:
(0.96 - 2.0x + x^2) / x = 2.4
Expanding the equation:
0.96/x - 2x/x + x^2/x = 2.4
Simplifying further:
0.96/x - 2 + x = 2.4
Multiplying the equation by x to eliminate the denominator:
0.96 - 2x + x^2 = 2.4x
Rearranging the equation:
x^2 + 2.4x - 2.4 = 0
Solving the quadratic equation, we find:
x = (-2.4 +/- sqrt(2.4^2 - 4 * 1 * (-2.4))) / (2 * 1)
Using the quadratic formula:
x = (-2.4 +/- sqrt(9.36 + 19.2)) / 2
x = (-2.4 +/- sqrt(28.56)) / 2
x = (-2.4 +/- 5.34) / 2
Considering the positive root:
x = (-2.4 + 5.34) / 2
x = 1.44 / 2
x = 0.72 atm
Now we can calculate the equilibrium pressures:
P(SO2) = 1.2 - 0.72 = 0.48 atm
P(Cl2) = 0.8 - 0.72 = 0.08 atm
Therefore, the equilibrium composition of the mixture when 1.2 atm of SO2Cl2 are mixed with 0.8 atm of Cl2 at 100 °C is approximately:
P(SO2) = 0.48 atm
P(Cl2) = 0.08 atm
P(SO2Cl2) = 1.2 atm
To calculate the equilibrium composition of the mixture, we need to use the concept of equilibrium constant and the given partial pressures of the reactants.
The equilibrium constant (Kp) of a reaction is defined as the ratio of the partial pressures of the products to the partial pressures of the reactants, each raised to the power of their stoichiometric coefficients in the balanced chemical equation.
In this case, the balanced chemical equation for the reaction is:
SO2Cl2(g) ⇄ SO2(g) + Cl2(g)
Given that Kp = 2.4 and the initial partial pressures of SO2Cl2 and Cl2 are 1.2 atm and 0.8 atm, respectively, we can set up an ice table to determine the change in partial pressures at equilibrium.
Let's denote the change in partial pressure of SO2Cl2 as 'x'. Since there is a 1:1 stoichiometric ratio between SO2Cl2 and Cl2, the change in partial pressure of Cl2 will also be 'x'.
ICE Table:
SO2Cl2(g) ⇄ SO2(g) + Cl2(g)
Initial: 1.2 atm 0 atm 0.8 atm
Change: -x atm +x atm +x atm
Equilibrium: (1.2 - x) atm x atm (0.8 + x) atm
Now, we can substitute these values into the equilibrium constant expression:
Kp = (partial pressure of SO2)(partial pressure of Cl2) / (partial pressure of SO2Cl2)
Kp = (x)(0.8 + x) / (1.2 - x)
Next, we can rearrange the equation to solve for 'x'. Cross multiply and simplify the equation:
2.4(1.2 - x) = x(0.8 + x)
2.88 - 2.4x = 0.8x + x^2
Rearrange the equation so that it equals to zero:
x^2 + 3.2x - 2.88 = 0
Now we can solve this quadratic equation for 'x' using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation:
a = 1, b = 3.2, c = -2.88
We can substitute these values into the quadratic formula:
x = (-3.2 ± √(3.2^2 - 4(1)(-2.88))) / (2(1))
Simplifying this expression will give us two possible values for 'x'. Plugging the values into the quadratic formula, we get:
x1 ≈ 0.47 atm
x2 ≈ -3.07 atm
Since partial pressures cannot be negative, we discard the negative value and only consider the positive one:
x ≈ 0.47 atm
Now, we can substitute this value back into our ice table to find the equilibrium partial pressures:
Equilibrium: (1.2 - 0.47) atm (0.47) atm (0.8 + 0.47) atm
Simplifying these values, we get:
Equilibrium: 0.73 atm 0.47 atm 1.27 atm
Therefore, at equilibrium, the composition of the mixture is approximately:
SO2Cl2: 0.73 atm
SO2: 0.47 atm
Cl2: 1.27 atm