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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 consider the reaction equilibrium equation and the concept of the equilibrium constant.

First, let's write down the reaction equation for the dissociation of oxygen molecules into atomic oxygen:

2O₂ ⇌ 4O

The equilibrium constant, K, for this reaction can be expressed as follows:

K = [O]⁴ / [O₂]²

Given that 83.00% of the oxygen molecules have dissociated, we can say that:

[O] / [O₂] = 0.83

We are asked to find the pressure at which 99.0% of the molecules dissociate. Let's assume that x represents the fraction of oxygen molecules that have dissociated, so we can say that:

[O] / [O₂] = x

Now, let's substitute these values into the equilibrium constant expression:

0.83 = x² / (1 - x)²

Next, let's solve this equation for x:

0.83 = (x²) / (1 - 2x + x²)

0.83 - 0.83x² = x²

1.83x² - 0.83 = x²

0.83x² = 0.83

x² = 1

x = 1

This means that when 99.0% of the oxygen molecules have dissociated, x = 1.

Now, let's substitute this value back into the equation for [O] / [O₂]:

1 = [O] / [O₂]

We know that [O] + [O₂] = 1, so we substitute this equation into the previous one:

1 = [O] / (1 - [O])

1 - [O] = [O]

2[O] = 1

[O] = 1 / 2

This means that the atomic oxygen concentration is 1/2, and the remaining oxygen concentration is also 1/2.

Finally, we can calculate the pressure at which 99.0% of the molecules dissociate:

Total pressure = [O] + [O₂] = 1/2 + 1/2 = 1.

Therefore, the pressure at which 99.0% of the molecules dissociate at this temperature is 1.000 atm.

To find the pressure at which 99.0% of the oxygen molecules dissociate at a temperature of 5000 K, we can use the concept of equilibrium constant for dissociation reaction.

The dissociation of oxygen molecules can be represented by the equation:
O2 ⇌ 2O

The equilibrium constant for this reaction is given by:
K = [O]^2 / [O2]

Where [O] represents the concentration of atomic oxygen and [O2] represents the concentration of oxygen molecules.

Given that 83.00% of the oxygen molecules have dissociated at 5000 K and 1.000 atm, we can write the following expressions for the concentrations:

[O] = 2 * (83.00% of initial concentration of O2)
[O2] = (100% - 83.00%) of the initial concentration of O2

Now, let's substitute the values into the equation for the equilibrium constant:

K = ([O]^2) / [O2]
K = [2 * (83.00% of initial concentration of O2)]^2 / [(100% - 83.00%) of the initial concentration of O2]

We know that K is a constant at a given temperature. Since we want to find the pressure at which 99.0% of the molecules dissociate, we can set up a proportion:

(K1) = ([O1]^2) / [O2]
(K2) = ([O2]^2) / [O3]

where:
(K1) represents the equilibrium constant at the initial condition
([O1]) represents the concentration of atomic oxygen at the initial condition
([O2]) represents the concentration of atomic oxygen at the desired condition
(K2) represents the equilibrium constant at the desired condition
([O3]) represents the concentration of oxygen molecules at the desired condition

We know that the percent dissociation at the initial condition is 83.00%, so [O1] can be obtained using the same equation as before:
[O1] = 2 * (83.00% of the initial concentration of O2)

Substituting these values into the proportion equation:
(K1) = [O1]^2 / [O2]
(K2) = [O2]^2 / [O3]

We want to find the pressure at which 99.0% of the molecules dissociate, so we set the desired condition of [O2] and [O3] as follows:
[O2] = 2 * (99.0% of the initial concentration of O2)
[O3] = (100% - 99.0%) of the initial concentration of O2

Now, substitute the values into the proportion equation:
(K1) = [O1]^2 / [O2]
(K2) = [O2]^2 / [O3]

Solve for K2 by rearranging the equation:
K2 = [O2]^2 / [O3]

Given that K1 is constant, we can rewrite the equations as:
(K1) = [O1]^2 / [O2]
(K1) = [O2]^2 / [O3]

Comparing both equations, we can conclude that K1 = K2, and thus:
[O1]^2 / [O2] = [O2]^2 / [O3]

Now rearrange the equation to solve for [O2]:
[O2]^2 = ([O1]^2 * [O3]) / [O2]
[O2]^3 = [O1]^2 * [O3]

Substitute the values and solve for [O2]:
[O2]^3 = ([2 * (83.00% of the initial concentration of O2)]^2 * [(100% - 99.0%) of the initial concentration of O2])

Now that we know the concentration of atomic oxygen at the desired condition, we can find the concentration of oxygen molecules:
[O3] = (100% - 99.0%) of the initial concentration of O2

Finally, substitute the values of [O2] and [O3] into the ideal gas law to find the pressure at which 99.0% of the molecules dissociate at 5000 K:
P = (nRT) / V

where:
P is the pressure in atm
n is the number of moles (based on the concentration of oxygen molecules)
R is the ideal gas constant (0.0821 L·atm/(mol·K))
T is the temperature in Kelvin (5000 K)
V is the volume of the gas sample (assuming it is constant)

By following these steps and plugging in the appropriate values, you can calculate the pressure at which 99.0% of the oxygen molecules dissociate at a temperature of 5000 K.