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23 March, 21:15

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.

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  1. 24 March, 00:20
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    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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