Question

Consider two reactions. Reaction (1) has a constant half-life, whereas reaction (2) has a half-life that gets longer as the reaction proceeds.What can you conclude about the rate laws of these reactions from these observations?a.The first reaction should have a first or a second order, the second one should have an order greater than second.b.The first reaction is a first order reaction, the second one has a second order.c.The first reaction should have a first or a second order, the second one should have an order not less than second.d.The first reaction is a first order reaction, the second one should have an order greater than first.

Answer 1

• choice d. The first reaction is a first order reaction, the second one should have an order greater than first.

Step-by-step explanation:

1) Reaction 1:

It is stated that reaction 2 has a constant half-life time.

Constant half-life time is a characteristic of first order reactions.

This fact is derived from:

• r = k [A]

• - d[A] / dt = k[A]

• - d[A] / [A] = k dt

• - ln [A/Ao] = k t

• Half-life ⇒ [A / Ao] = (1/2)

• - ln (1/2) = k t (half - life) ⇒ t (half-life) = ln (2) / k

• Since, ln (2) / k is a constant, t (jalf-life) is a constant.

So, the first conclusion is that the first reactions is a first order reaction.

2) Reaction 2.

It is stated that reaction 2 has a half-life that gets longer as the reaction proceeds.

• You can suppose r = k [A]ⁿ, where n = 2 (second order reaction)

• Then, - d[A] / A² = k t

• 1 / [A] = k t + 1 / [Ao]

• When you solve of t (half-life) you get: t (half-life) = = 1 / ( k [Ao] )

• This shows that for second order reactions, as the reaction proceeds ([Ao] decreases) the half-life increases.

When you do the same analysis for n > 2 (third or higher order reaction), you get the initial concenration raised to a power greater than two in the denominator, driving to the same conclusion, that as the reaction proceeds the half-life gets longer.

So, your second conclusion is that the seond reaction has an order greater than first.

Summarizing the two general conclusions:

• the first reactions is a first order reaction.

• the second one should have an order greater than first.

That is the choice d. The first reaction is a first order reaction, the second one should have an order greater than first.