For The Following Equilibrium Expression: (0.35 + X) 2000 (0.15 – X)(0.05 – 2x) Find The Boundary Conditions (Show Your Work) And Roots For This Equation
The Correct Answer and Explanation is :
To analyze the equilibrium expression and find its roots, let’s first set up the equation clearly. We interpret your given expression as follows:
[
(0.35 + x) \cdot 2000 = (0.15 – x)(0.05 – 2x)
]
Step 1: Expand and Simplify
First, we expand both sides of the equation:
[
2000 \cdot (0.35 + x) = (0.15 – x)(0.05 – 2x)
]
[
700 + 2000x = (0.15 – x)(0.05 – 2x)
]
Now, expanding the right-hand side:
[
0.15 \times 0.05 – 0.15 \times 2x – 0.05 \times x + x \times 2x
]
[
0.0075 – 0.30x – 0.05x + 2x^2
]
[
2x^2 – 0.35x + 0.0075
]
Equating both sides:
[
700 + 2000x = 2x^2 – 0.35x + 0.0075
]
Bringing all terms to one side gives a standard form quadratic equation:
[
2x^2 – 0.35x – 2000x + 0.0075 – 700 = 0
]
[
2x^2 – 2000.35x – 699.9925 = 0
]
Step 2: Boundary Conditions
Boundary conditions are determined by the physical or chemical constraints of the problem. Since (x) represents a change in concentration, (x) must be such that all concentrations remain non-negative:
- (0.35 + x \geq 0)
- (0.15 – x \geq 0)
- (0.05 – 2x \geq 0)
From (0.35 + x \geq 0), we get (x \geq -0.35).
From (0.15 – x \geq 0), we get (x \leq 0.15).
From (0.05 – 2x \geq 0), we get (x \leq 0.025).
Therefore, the boundary conditions are (x \geq -0.35) and (x \leq 0.025) since 0.025 is the most restrictive upper boundary.
Step 3: Finding Roots
The roots of the quadratic equation can be found using the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
where (a = 2), (b = -2000.35), and (c = -699.9925). Let’s solve this to find the roots.
import numpy as np
a = 2
b = -2000.35
c = -699.9925
discriminant = b**2 - 4*a*c
x1 = (-b + np.sqrt(discriminant)) / (2*a)
x2 = (-b - np.sqrt(discriminant)) / (2*a)
x1, x2These roots must be checked against the boundary conditions to determine if they are physically meaningful.
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