# Question 9439b

Mar 6, 2016

You are indeed dealing with a first-order reaction.

#### Explanation:

The idea here is that you can determine the order of the reaction by using the units of the rate constant, $k$.

As you know, the rate of a reaction tells you how the concentrations of the reactants or of the products that are taking part in a chemical reaction change as the reaction proceeds.

For all intended purposes, the decomposition of sulfuryl chloride, ${\text{SO"_2"Cl}}_{2}$, can be written like this

$\textcolor{b l u e}{\text{SO"_2"Cl"_2 -> "products}}$

Now, the rate of this reaction will tell you how the concentration of sulfuryl chloride changes per unit of time. This means that you can write

color(blue)("rate" = -(Delta["SO"_2"Cl"_2])/(Deltat))

The minus sign is used here because the concentration of a product decreases during a chemical reaction.

Since the rate of a reaction expresses change in concentration per unit of time, its units will usually be given as "mol L"^(-1)color(red)("s"^(-1)). Keep this in mind.

For a given chemical reaction, the differential rate law tells you how the concentration of a reactant affects the rate of the reaction.

In simple terms, the differential rate law establishes a relationship between the rate of the reaction and the concentration of the reactant by using a proportionality factor, i.e. rate constant, $k$.

Let's assume that this reaction is $n$ order. The differential rate law will take the form

color(blue)("rate" = k * ["SO"_2"Cl"_2]^n)

Rearrange the above equation to get

k = "rate"/["SO"_2"Cl"_2]^n

Now focus on the units of the rate constant and of the rate of the reaction. The rate constant is said to be equal to

$k = 4.68 \cdot {10}^{- 5} \textcolor{red}{{\text{s}}^{- 1}}$

Since concentration is expressed in ${\text{mol L}}^{- 1}$, you can say, using only units, that

$\textcolor{red}{{\text{s"^(-1)) = ("mol L"^(-1))/("mol L"^(-1))^n * color(red)("s}}^{- 1}}$

In order to have a valid equality, you need to have

${\left({\text{mol L"^(-1))/("mol L}}^{- 1}\right)}^{n} = 1$

This can only happen if $n = 1$, since

("mol L"^(-1))/("mol L"^(-1))^1 = color(red)(cancel(color(black)("mol L"^(-1))))/color(red)(cancel(color(black)("mol L"^(-1)))) = 1

Since $n$ represents the order of the reaction, the decomposition of sulfuryl chloride will indeed be a first-order reaction, which means that the rate of the reaction depends linearly on the concentration of the product.

"rate" = k * ["SO"_2"Cl"_2]#