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.