At 5000 K and 1.000 atm, 83.00% of the oxygen molecules in a sample have dissociated to atomic oxygen. At what pressure will 99.0% of the molecules dissociate at this temperature?
To solve this problem, we need to use the concept of the equilibrium constant (K_eq) for the dissociation reaction:
O2 ⇌ 2O
The equilibrium constant, K_eq, can be calculated by taking the ratio of the concentration of the products to the concentration of the reactant. In this case, we are given the percentage of dissociated oxygen molecules, which we can interpret as the concentration of the products. We will use this value to find K_eq.
Given:
- Temperature (T) = 5000 K
- Pressure (P) = 1.000 atm
- Percentage of dissociation = 83.00% = 0.830 (as decimal)
To calculate K_eq, we need to use the ideal gas law, which relates pressure, volume, and temperature:
PV = nRT
Where:
- P is pressure
- V is volume
- n is the number of moles of gas
- R is the ideal gas constant (0.0821 L•atm/mol•K)
- T is temperature
Let's start by considering an initial state where no dissociation has occurred. At this state, the concentration of O2 is equal to the total pressure, P, and the concentration of atomic oxygen (O) is zero.
Let x be the number of moles dissociated per mole of O2. Since we have two O atoms for every O2 molecule dissociated, we would have a total of 2x moles of O for every mole of O2 dissociated.
Using the ideal gas law, we can express the concentrations as:
[O2] = P (Initial concentration of O2)
[O] = 2x (Concentration of O formed from dissociation of O2)
Now, we can substitute these concentrations into the equilibrium constant expression:
K_eq = [O]^2 / [O2]
K_eq = (2x)^2 / (P - 2x)^2 (as [O2] = P - 2x)
Given that 83.00% of the O2 dissociates, we have:
0.830 = (2x)^2 / (P - 2x)^2
Simplifying the equation, we get:
0.830(P - 2x)^2 = (2x)^2
0.830P^2 - 3.320Px + 3.320x^2 = 4x^2
0.830P^2 - 3.320Px + 3.320x^2 - 4x^2 = 0
Simplifying further:
0.830P^2 - 3.320Px - x^2 = 0
Now, we can solve this quadratic equation for x using the quadratic formula:
x = [-(-3.320P) ± √((-3.320P)^2 - 4(0.830)(-x^2))] / (2 * 0.830)
This equation provides two possible values for x. To find the correct value, we need to consider the physical implication of the dissociation reaction. Since we cannot have a negative value for moles, we can discard the negative root. Therefore, we consider only the positive root:
x = (-(-3.320P) + √((-3.320P)^2 - 4(0.830)(-x^2))) / (2 * 0.830)
Simplifying inside the square root:
x = (-(-3.320P) + √(10.9744P^2 + 3.464x^2)) / 1.660
Now, we can substitute the value of x in the equation for the dissociation percentage:
0.830 = 2x / (P - 2x)
Substituting x, we get:
0.830 = 2[(-(-3.320P) + √(10.9744P^2 + 3.464x^2)) / 1.660] / [P - 2((-(-3.320P) + √(10.9744P^2 + 3.464x^2)) / 1.660)]
Simplifying further, we get:
0.830 = {(-(-6.640P) + 2√(10.9744P^2 + 3.464x^2)) / 1.660} / {P - [(-6.640P) + 2√(10.9744P^2 + 3.464x^2)] / 1.660}
Now, we can simplify this equation and solve for P. However, the equation is quite complex and cannot be easily solved algebraically. Therefore, we need to use numerical methods or software to solve this equation.
One approach is to use a solver function in a programming language or a mathematical software package like Python, MATLAB, or Mathematica to find the value of P.