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
[tex]2^{a-b}[/tex] = 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.
