At a particular temperature, K = 3.75 for the following reaction.

SO2(g) + NO2(g)= SO3(g) + NO(g)

If all four gases had initial concentrations of 0.840 M, calculate the equilibrium concentrations of the gases.

K = [SO3][NO / [SO2][NO2]] = 3.75

Let the moles of each reactant = x
[0.840+x][0.840+x] / [0.840-x][0.840-x] = 3.75
a) Solve for x.
b) To get the concentration of each reactant, subtract the value of x from 0.840M.
c) To get the concentration of each product, add the value of x to 0.840M.

Well, let's take a step back. You know what they say, when life gives you gases, make balloon animals!

Now, to solve this problem, we can use the expression for the equilibrium constant, K, which is given as K = [SO3][NO]/[SO2][NO2]. Since we have the value of K, we can set up the following equation:

3.75 = ([SO3][NO]) / ([SO2][NO2])

Now, let's assign the variable 'x' to the equilibrium concentration of any of the gases. Since all four gases have the same initial concentration of 0.840 M, their equilibrium concentrations can be written as:

[SO2] = [NO2] = 0.840 - x
[SO3] = [NO] = x

Plugging these values into the equation, we get:

3.75 = (x * x) / ((0.840 - x) * (0.840 - x))

Now, we'll solve this equation to find the value of 'x' which represents the equilibrium concentration of the gases.

Just give me a moment, I need to put on my mathematical clown shoes for this one... *Honk honk*

After some calculations, I find that the equilibrium concentration of the gases can be approximated as follows:

[SO2] = [NO2] ≈ 0.211 M
[SO3] = [NO] ≈ 0.629 M

But remember, these are just approximations! It's always useful to double-check your calculations and consult your chemistry textbooks, because as a clown, I often find myself juggling numbers instead of balloons.

To calculate the equilibrium concentrations of the gases, we can use the expression for the equilibrium constant:

K = [NO][SO3] / [SO2][NO2]

Given that K = 3.75, and all four gases have initial concentrations of 0.840 M, we can assume that at equilibrium, the concentrations of SO2, NO2, SO3, and NO are represented by the variables x.

So, the equilibrium concentrations can be represented as follows:

[SO2] = 0.840 - x
[NO2] = 0.840 - x
[SO3] = x
[NO] = x

Plugging these values into the equilibrium constant expression, we get:

3.75 = [(x)(x)] / [(0.840 - x)(0.840 - x)]

Simplifying this equation, we have:

3.75 = x^2 / (0.840 - x)^2

Cross-multiplying:

3.75(0.840 - x)^2 = x^2

Expanding and re-arranging terms:

3.15 - 6.30x + 3.75x^2 = x^2

2.75x^2 + 6.30x - 3.15 = 0

Now, let's solve this quadratic equation for x using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 2.75, b = 6.30, and c = -3.15. Substituting these values into the quadratic formula:

x = (-6.30 ± √(6.30^2 - 4(2.75)(-3.15))) / (2(2.75))

Simplifying the equation under the square root:

x = (-6.30 ± √(39.69 + 34.65)) / 5.5

x = (-6.30 ± √(74.34)) / 5.5

x = (-6.30 ± 8.62) / 5.5

Now, we have two possible values for x:

1) x = (-6.30 + 8.62) / 5.5 = 2.32 / 5.5 ≈ 0.42
2) x = (-6.30 - 8.62) / 5.5 = -14.92 / 5.5 ≈ -2.71

Since concentrations cannot be negative, we can disregard the second value of x (-2.71). Therefore, the equilibrium concentration of all four gases is approximately:

[SO2] ≈ 0.840 - 0.42 = 0.42 M
[NO2] ≈ 0.840 - 0.42 = 0.42 M
[SO3] ≈ 0.42 M
[NO] ≈ 0.42 M

Note: The concentrations are rounded to two decimal places.

To calculate the equilibrium concentrations of the gases, we need to use the concept of the equilibrium constant and the given value of K.

The equilibrium constant expression for the given reaction is:

K = [SO3] * [NO] / [SO2] * [NO2]

Where [SO3], [NO], [SO2], and [NO2] represent the concentrations of the respective gases at equilibrium.

Given that the initial concentrations of all four gases are 0.840 M, we can assume that the concentration change will be represented as -x for reactants and +x for products.

Let's set up a table to keep track of concentrations:

Reactants:
SO2: 0.840 M - x
NO2: 0.840 M - x

Products:
SO3: 0.840 M + x
NO: 0.840 M + x

Now we can substitute these values into the equilibrium constant expression:

3.75 = (0.840 M + x) * (0.840 M + x) / (0.840 M - x) * (0.840 M - x)

Simplify and rearrange the equation to solve for x:

3.75 = (0.840^2 + 2 * 0.840x + x^2) / (0.840^2 - x^2)

Cross multiply:

3.75 * (0.840^2 - x^2) = 0.840^2 + 2 * 0.840x + x^2

Expand and rearrange the equation:

3.75 * 0.840^2 - 3.75x^2 = 0.840^2 + 2 * 0.840x + x^2

Multiply and simplify:

3.75 * 0.840^2 - 2 * 0.840x - 3.75x^2 = 0.840^2

Now we have a quadratic equation. Let's solve it to find the value of x using either factoring, quadratic formula, or numerical methods like Newton-Raphson.

Once you find the value of x, substitute it back into the expressions for the reactant and product concentrations to find the equilibrium concentrations of the gases.

K= (.84+x)^2 /(.84-x)^2

take the square root of each side, then

sqrtK=(.84+x)/(.84-x)
and you can multipy both sides by .84-x and then solve for x.
Knowing x, you can then find the concentration of each product (either add or subtract x from .84)
check my thinking.