Question

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 1.75 h, while that of Y is 120. min. Suppose an atmospheric scientist studying these decompositions fills a transparent 10.0 L flask with X and Y and exposes the flask to UV light. Initially, the partial pressure of X is 4.0 times greater than the partial pressure of Y. 1. As both gases decompose, will the partial pressure of X ever all yes below the partial pressure of Y? A. Yes.B. No.2. If you said yes, calculate the time it takes the partial pressure of X to fall below the partlal pressure of Y. Round your answer to 2 significant digits. =___________h.

Answer 1

Answer:

1. Yes

2. 28 h

Explanation:

The gases are being decomposed, so their partial pressure is decreasing. The partial pressure is proportional to the molar fraction, so when the half-life passes, the partial pressure decreases by half too. Let's call the partial pressure of X pX and the partial pressure of Y pY.

1. Let's assume a total pressure as 1.00 atm. At the beginning

pX = 4pY

pX + pY = 1

5pY = 1

pY = 0.2 atm

pX = 0.8 atm

X loses half of it at 1.75h = 105 min and Y at 120 min. So X is disappearing faster than Y, then the partial pressure of X will be below the partial pressure of Y at some time.

2. The partial pressure after some time can be calculated by the equation of half-life:

m = m0/2ⁿ

Where m is the mass, m0 is the initial mass, and n is how much half-lives have passed. So, let's substitute the mass for partial pressures, and n for a, for X, and b, for Y:

X: pX = 0.8/2ᵃ

Y: pY = 0.2/2ᵇ

Let's first identify how long it will take for the partial pressures be the same:

0.8/2ᵃ = 0.2/2ᵇ

2ᵃ/2ᵇ = 0.8/0.2

2ᵃ/2ᵇ = 4

$2^{a-b}$ = 2²

a - b = 2

Let's call t the time that it happens, so:

X: t = 105a

Y: t = 120b

105a = 120b

a = 1.143b

1.143b - b = 2

0.143b = 2

b = 14

So, it has passed 14 half-lives for Y, and 12 for X, then, the time they have the same amount is:

t = 120*14 = 1680 min

t = 28 h

After that, pX < pY.