To determine which of the given options correctly represents the reaction rate for the chemical reaction
[tex]\[2 N_2 O_{5(g)} \rightarrow 4 NO_{2(g)} + O_{2(g)},\][/tex]
we need to analyze the reaction rates in terms of the change in concentration of each species per unit time.
First, recall that the rate of a chemical reaction for the general reaction:
[tex]\[aA + bB \rightarrow cC + dD\][/tex]
is given by:
[tex]\[ \text{rate} = -\frac{1}{a} \frac{\Delta [A]}{\Delta t} = -\frac{1}{b} \frac{\Delta [B]}{\Delta t} = \frac{1}{c} \frac{\Delta [C]}{\Delta t} = \frac{1}{d} \frac{\Delta [D]}{\Delta t}. \][/tex]
For the reaction in question:
[tex]\[2 N_2 O_{5(g)} \rightarrow 4 NO_{2(g)} + O_{2(g)},\][/tex]
we can express the rate of reaction as follows:
1. For [tex]\(N_2 O_5\)[/tex]:
[tex]\[ \text{rate} = -\frac{1}{2} \frac{\Delta [N_2 O_5]}{\Delta t}. \][/tex]
2. For [tex]\(NO_2\)[/tex]:
[tex]\[ \text{rate} = \frac{1}{4} \frac{\Delta [NO_2]}{\Delta t}. \][/tex]
3. For [tex]\(O_2\)[/tex]:
[tex]\[ \text{rate} = \frac{\Delta [O_2]}{\Delta t}. \][/tex]
Now let's compare each of the given options with the correct expressions for the reaction rate:
a. [tex]\(-\left(\frac{\Delta [O_2]}{\Delta t}\right)\)[/tex]
- This suggests that the rate is the negative change in the concentration of [tex]\(O_2\)[/tex] over time. However, the rate for [tex]\(O_2\)[/tex] production should be positive and is [tex]\(\frac{\Delta [O_2]}{\Delta t}\)[/tex].
b. [tex]\(4\left(\frac{\Delta [NO_2]}{\Delta t}\right)\)[/tex]
- This implies multiplying the rate of change of [tex]\(NO_2\)[/tex] concentration by 4. The correct relationship should have [tex]\(\frac{1}{4}\)[/tex] as a factor, not 4.
c. [tex]\(\frac{1}{4}\left(\frac{\Delta [NO_2]}{\Delta t}\right)\)[/tex]
- This correctly reflects the rate expression for [tex]\(NO_2\)[/tex]:
[tex]\[ \text{rate} = \frac{1}{4} \frac{\Delta [NO_2]}{\Delta t}. \][/tex]
d. [tex]\(-\left(\frac{\Delta [N_2 O_5]}{\Delta t}\right)\)[/tex]
- This suggests that the rate is the negative change in the concentration of [tex]\(N_2 O_5\)[/tex] over time. The correct expression uses a [tex]\(\frac{1}{2}\)[/tex] factor:
[tex]\[ \text{rate} = -\frac{1}{2} \frac{\Delta [N_2 O_5]}{\Delta t}. \][/tex]
From the above analysis, it is clear that:
- Option (d) is incorrect because it lacks the [tex]\(\frac{1}{2}\)[/tex] factor.
- Option (a) is incorrect because it wrongly represents the rate for [tex]\(O_2\)[/tex].
- Option (b) is incorrect because it incorrectly multiplies the rate for [tex]\(NO_2\)[/tex] by 4.
Thus, the correct representation for the reaction rate is:
[tex]\[ \boxed{\frac{1}{4}\left(\frac{\Delta [NO_2]}{\Delta t}\right)} \][/tex]
Therefore, the correct answer is option (c).
