To tackle this problem, we need to analyze the equilibrium reaction between nitrogen and oxygen to form nitric oxide (NO). The reaction can be represented as follows:
Nitrogen and Oxygen Reaction
N2(g) + O2(g) → 2NO(g)
Given the equilibrium constant (K) at 1500 K is 1.0 x 10-5, we can use this information along with the initial concentrations of the reactants to find the equilibrium concentrations of all species involved.
Initial Conditions
Before any reaction occurs, the concentrations are:
- [N2] = 0.8 mol L-1
- [O2] = 0.2 mol L-1
- [NO] = 0 mol L-1
Setting Up the Equilibrium Expression
The equilibrium constant expression for this reaction is given by:
K = \(\frac{[NO]^2}{[N_2][O_2]}\)
Substituting the known value of K, we have:
1.0 x 10-5 = \(\frac{[NO]^2}{[N_2][O_2]}\)
Change in Concentrations
Let’s denote the change in concentration of N2 and O2 that reacts to form NO as x. Therefore, at equilibrium, the concentrations will be:
- [N2] = 0.8 - x
- [O2] = 0.2 - x
- [NO] = 2x
Substituting into the Equilibrium Expression
Now we can substitute these expressions into the equilibrium constant equation:
1.0 x 10-5 = \(\frac{(2x)^2}{(0.8 - x)(0.2 - x)}\)
Solving for x
This simplifies to:
1.0 x 10-5 = \(\frac{4x^2}{(0.8 - x)(0.2 - x)}\)
To solve for x, we can cross-multiply:
1.0 x 10-5 * (0.8 - x)(0.2 - x) = 4x2
Expanding the left side:
1.0 x 10-5 * (0.16 - 1.0x + x2) = 4x2
Distributing gives:
1.6 x 10-6 - 1.0 x 10-5x + 1.0 x 10-5x2 = 4x2
Rearranging this into a standard quadratic form:
(1.0 x 10-5 - 4)x2 - 1.0 x 10-5x + 1.6 x 10-6 = 0
Using the Quadratic Formula
Now we can apply the quadratic formula, x = \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where:
- a = (1.0 x 10-5 - 4)
- b = -1.0 x 10-5
- c = 1.6 x 10-6
Calculating the discriminant and solving will yield the value of x. However, given the small value of K, we can anticipate that x will be very small compared to the initial concentrations, allowing us to simplify our calculations further.
Final Equilibrium Concentrations
Once we find x, we can substitute it back into our expressions for [N2], [O2], and [NO] to find the equilibrium concentrations:
- [N2] = 0.8 - x
- [O2] = 0.2 - x
- [NO] = 2x
This approach will give you the equilibrium concentrations of the reactants and product after the mixture has been heated to 1500 K. If you need help with the calculations or further clarification, feel free to ask!
