55.3=(2x)^2/(1-x)^2
solve for x Use the quadratic equation
solve for x Use the quadratic equation
The equilibrium expression is
Kp = reactants/products=[2HI]^2/[H2][I2]
Since you have 2 moles of HI for every mole of H2 or I2, you square the products.
You start off with 1 mole of H2 and I2, and some, not all of the reaction produces x amount of HI. You don't know how much, but you know the molar ratio of HI will be twice as much as H2 or I2, so that is why he told you (2x)^2 for the product. Since the amount of HI is subtracted from the initial concentrations of H2 and I2 to make HI, the amount subtracted is 1-X for each. Looking at the equation that I gave you above, you will see that reactants are multiplied by each other to produce (1-x)^2. The rest is just using algebra to solve for x, and the only way that you can do that is to use the quadratic equation.
I hope this helps.
First, write down the balanced equation for the reaction:
H2 + I2 ⇌ 2HI
Next, let's analyze the initial moles of each substance. Since 1 mole of H2 and 1 mole of I2 are placed in the flask, the initial moles of H2 (nH2) and I2 (nI2) are both 1 mole.
Now, we need to set up an expression for the equilibrium concentrations of H2 (CH2), I2 (CI2), and HI (CHI) using their mole fractions.
Let x be the mole fraction of HI, so the mole fractions of H2 and I2 are (1 - x) and (1 - x) respectively.
Since mole fraction is defined as moles of a component divided by the total moles of all components, we can write:
CH2 = (moles of H2 / total moles) = (1 - x) / (1 + 1 + 1) = (1 - x) / 3
CI2 = (moles of I2 / total moles) = (1 - x) / (1 + 1 + 1) = (1 - x) / 3
CHI = (moles of HI / total moles) = x / (1 + 1 + 1) = x / 3
Now, we can substitute these expressions into the equilibrium constant expression (Kp) and solve for x.
Kp = (CHI^2) / (CH2 * CI2^2)
Plugging in the values:
55.5 = [(x/3)^2] / [((1 - x) / 3) * ((1 - x) / 3)^2]
To solve this equation, you can rearrange it and solve for x. One way to do this is to cross-multiply and solve the resulting quadratic equation for x.
Multiply both sides of the equation by [(1 - x) / 3]^3:
55.5 * [(1 - x) / 3]^3 = (x/3)^2
Now, simplify and expand:
55.5 * (1 - x)^3 = x^2
Expand the left side:
55.5 * (1 - 3x + 3x^2 - x^3) = x^2
Distribute:
55.5 - 166.5x + 166.5x^2 - 55.5x^3 = x^2
Rearrange to form a cubic equation:
55.5x^3 + x^2 - 166.5x^2 + 166.5x - 55.5 = 0
Now, you can use numerical methods or a calculator to solve this cubic equation for x. Once you find the value of x, you can calculate the mole fraction of HI by using x / 3.