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 0.75h, while that of Y is 90.min. Suppose an atmospheric scientist studying these decompositions fills a transparent 5.0L flask with X and Y and exposes the flask to UV light. Initially, the partial pressure of X is 3.0 times greater than the partial pressure of Y. Will the partial pressure of x ever be lower than y. If so at what time will it be lower?

Answer 1

Half life (t_1/2) is the time required by a substance to get decomposed or reduced to half of its original amount. Now here given

Half life (t_1/2) for X = 0.75h = 45 min

Half life (t_1/2) for Y = 90 min

That means half life of Y is greater than half life of X. So X will be reduces first or it will take just half the time of Y to get reduced.

Now given initially the partial pressure of X is 3.0 times greater than the partial pressure of Y. Now we know partial pressure of a gas is directly proportional to its number of moles. So we can say initially moles of X is 3 times greater than Y

So lets say we have initially 3x moles of X and x moles of Y

Now once they get start decomposing,

After 45 mins,

X will be reduced to 3x/2 = 1.5x moles

And Y will be reduced by 1/4 (because Y takes 90 min to reduced to 1/2. So in the 1st 45 min, only 14 of it will decomposed). So we can say, after 45 mins, moles of Y left = x-1/4x = 3/4x = 0.75‬ moles

So here after 45 mins, moles of Y is still less than moles of X.

So partial pressure of X is greater than partial pressure of Y

After 90 mins,

X will be reduced to = 1.5x *1/2 = 0.75X

And Y will be reduced to = 1/2 = 0.5 moles (as half life of Y is 90 min)

So here after 90 mins, moles of Y is still less than moles of X.

So partial pressure of X is greater than partial pressure of Y

Now at 180 mins,

amount of X left = 0.75x * 1/4 = 0.1875‬x

But moles of Y left = x/2 * 1/2 = 0.25x.

Now here mols of Y left is more than mols of X left.

So after 180 mins. partial pressure of X will be lower than partial pressure of Y