Determine the rate law and the value of k for the following reaction using the data provided.



CO(g) + Cl2(g) → COCl2(g)

[CO]i (M)

[Cl2]i (M)

Initial Rate

​

0.25

0.40

0.696

​

0.25

0.80

1.97

​

0.50

0.80

3.94


To determine the rate law and the rate constant \( k \) for the reaction:

\[
\text{CO(g)} + \text{Cl}_2\text{(g)} \rightarrow \text{COCl}_2\text{(g)}
\]

we need to analyze the given dataTo determine the rate law and the rate constant \( k \) for the reaction:

\[
\text{CO(g)} + \text{Cl}_2\text{(g)} \rightarrow \text{COCl}_2\text{(g)}
\]

we need to analyze the given 

| [CO] (M)  | [Cl\(_2\)] (M) | Initial Rate (M/s) |
|:---------:|:--------------:|:------------------:|
| 0.25      | 0.40           | 0.696              |
| 0.25      | 0.80           | 1.97               |
| 0.50      | 0.80           | 3.94               |

### Step 1: Write the General Rate Law

The general rate law for this reaction is:

\[
\text{Rate} = k[\text{CO}]^m[\text{Cl}_2]^n
\]

Where:
- \( m \) is the order with respect to CO,
- \( n \) is the order with respect to Cl\(_2\),
- \( k \) is the rate constant.

### Step 2: Determine the Reaction Orders \( m \) and \( n \)

To determine the exponents \( m \) and \( n \), we use the method of initial rates by comparing different experiments where only one concentration changes at a time.

#### To find \( n \) (order with respect to Cl\(_2\)):

Compare Experiments 1 and 2. Here, [CO] remains constant, and only [Cl\(_2\)] changes.

\[
\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{k[\text{CO}]^{m}[\text{Cl}_2]^{n}_2}{k[\text{CO}]^{m}[\text{Cl}_2]^{n}_1}
\]

Plugging in the values:

\[
\frac{1.97}{0.696} = \frac{[\text{Cl}_2]_2^{n}}{[\text{Cl}_2]_1^{n}} = \frac{(0.80)^n}{(0.40)^n}
\]

\[
2.83 = \left( \frac{0.80}{0.40} \right)^n = (2)^n
\]

Taking the logarithm:

\[
n \log(2) = \log(2.83)
\]

\[
n = \frac{\log(2.83)}{\log(2)} = \frac{0.451}{0.301} \approx 1.5
\]

Thus, \( n = 1.5 \).

#### To find \( m \) (order with respect to CO):

Now, compare Experiments 2 and 3. Here, [Cl\(_2\)] remains constant, and only [CO] changes.

\[
\frac{\text{Rate}_3}{\text{Rate}_2} = \frac{[\text{CO}]_3^{m}}{[\text{CO}]_2^{m}} = \frac{(0.50)^m}{(0.25)^m}
\]

Plugging in the rates:

\[
\frac{3.94}{1.97} = \left( \frac{0.50}{0.25} \right)^m
\]

\[
2 = (2)^m
\]

Therefore, \( m = 1 \).

### Step 3: Write the Rate Law

The rate law is:

\[
\text{Rate} = k[\text{CO}]^1[\text{Cl}_2]^{1.5}
\]

### Step 4: Determine the Rate Constant \( k \)

To find \( k \), use the rate law with data from any experiment. Let's use the data from Experiment 1:

\[
\text{Rate} = k[\text{CO}]^1[\text{Cl}_2]^{1.5}
\]

Substitute the values:

\[
0.696 = k(0.25)^1(0.40)^{1.5}
\]

First, calculate \( (0.40)^{1.5} \):

\[
0.40^{1.5} = 0.252
\]

Now solve for \( k \):

\[
0.696 = k(0.25)(0.252)
\]

\[
k = \frac{0.696}{0.25 \times 0.252} = \frac{0.696}{0.063} \approx 11.05
\]

Thus, \( k \approx 11.05 \, \text{M}^{-1.5} \text{s}^{-1} \).

### Final Answer:

The rate law is:

\[
\text{Rate} = 11.05 [\text{CO}]^1[\text{Cl}_2]^{1.5}
\]

Would you like further details or clarification on any steps?

---

**Five related questions:**
1. How do you determine the units of the rate constant \( k \) based on the reaction order?
2. What is the significance of fractional reaction orders, like \( n = 1.5 \)?
3. How would changing the initial concentration of both CO and Cl\(_2\) affect the rate?
4. Can the reaction order be determined from the balanced chemical equation?
5. How would temperature affect the value of \( k \)?

**Tip:** When comparing experimental data to determine reaction orders, always select experiments where only one reactant concentration changes while others remain constant.

Determine the rate law and the value of k for the following reaction using the data provided.
CO(g) + Cl2(g) → COCl2(g)

[CO]i (M)
[Cl2]i (M)
Initial Rate
0.25
0.40
0.696
0.25
0.80
1.97
0.50
0.80
3.94

Learn how to determine the rate law and rate constant (k) for the reaction CO(g) + Cl2(g) → COCl2(g) using experimental data. This problem covers concepts in chemical kinetics, including reaction orders and rate laws. Suitable for undergraduate chemistry students.

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