Two gases X and Y are found in the atmosphere in only trace amounts because they decompose quickly. When exposed to ultraviolet light the half-life of X is 1.25 h, while that of Y is 150 min.
Suppose an atmospheric scientist studying these decompositions fills a transparent 20.0 L flask with X and Y and exposes the flask to UV light. Initially, the partial pressure of X is 25.0% greater than the partial pressure of Y.
1. Will the partial pressure of X ever fall below the partial pressure of Y? (yes or no)
2. If yes, calculate the time it take the partial pressure of X to fall below the partial pressure of Y.

1. Yes

2. After 68.1 mins, pX < pY.

Explanation:

Assuming the total gas pressure is 1 atm, let the partial pressure of Y be y, partial pressure of X will be 0.25y + y = 1.25y

1.25y + y = 1atm

2.25y = 1 atm

y = 1 atm / 2.25 = 0.44 atm

Then partial pressure of X = 0.56 atm

The partial pressure of a gas in a mixture of gases is directly proportional to the mole fraction of the gas.

Therefore the mole fraction of X and Y is 0.56 and 0.44 respectively.

The partial pressures of X and Y becomes half of their original values at 1.25 h = 85 min and Y at 150 min respectively.

The partial pressure after some time can be calculated from the half-life equation :

m = m⁰ *  1/2ⁿ

Where m = the remaining mass, m⁰ = initial mass, and n is number of half-lives undergone.

Let partial pressures represent the mass, and n for X and Y be a and b respectively:

pX  = 0.56/2ᵃ

pY = 0.44/2ᵇ

We then determine when Partial pressure of X, pX = Partial pressure of Y, pY

0.56/2ᵃ = 0.44/2ᵇ

2ᵃ/2ᵇ = 0.56/0.44

2ᵃ/2ᵇ = 1.27

2ᵃ⁻ᵇ = 2⁰°³⁴⁵

a - b = 0.345

Let this time be t, therefore,

For X; t = 85a and For Y: t = 150b

85a = 150b

then, a = 1.76b

1.76b - b = 0.345

0.760b = 0.345

b = 0.454 and,

a = 0.345 + 0.454 = 0.799

So,  X goes through 0.779 half-lives while Y goes through 0.454 half-lives Then, the time for both X and Y to have the same amount is:

t = 150 * 0.454 = 68.1 min

After 68.1 mins, pX < pY.

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