The rate of decomposition of a gas at a certain temperature is 5.14 and 7.25 in some units for 20% and 5% decomposition respectively. Calculate the order of the reaction.

Answer

To calculate the order of the reaction based on the given rates of decomposition at different percentages, we can follow these steps:

### Step 1: Understand the given data
We have the following information:
- Rate of decomposition (r1) at 20% decomposition = 5.14
- Rate of decomposition (r2) at 5% decomposition = 7.25

### Step 2: Define the concentrations
Let the initial concentration of the gas be \( [A] \).
- For 20% decomposition, the concentration of A after decomposition is \( [A] - 0.2[A] = 0.8[A] \).
- For 5% decomposition, the concentration of A after decomposition is \( [A] - 0.05[A] = 0.95[A] \).

### Step 3: Write the rate law expression
The rate of a reaction can be expressed as:
\[ r = k [A]^n \]
where \( k \) is the rate constant and \( n \) is the order of the reaction.

### Step 4: Set up the ratio of rates
Using the rates and concentrations at two different times, we can set up the following equation:
\[ \frac{r_1}{r_2} = \frac{k [0.8[A]]^n}{k [0.95[A]]^n} \]
Since \( k \) is constant, it cancels out:
\[ \frac{5.14}{7.25} = \frac{(0.8[A])^n}{(0.95[A])^n} \]
This simplifies to:
\[ \frac{5.14}{7.25} = \left(\frac{0.8}{0.95}\right)^n \]

### Step 5: Calculate the left side
Calculating the left side:
\[ \frac{5.14}{7.25} \approx 0.707 \]

### Step 6: Express the right side
Now we can express the right side:
\[ 0.707 = \left(\frac{0.8}{0.95}\right)^n \]
Calculating \( \frac{0.8}{0.95} \):
\[ \frac{0.8}{0.95} \approx 0.8421 \]

### Step 7: Set up the logarithmic equation
Now we can write:
\[ 0.707 = (0.8421)^n \]
Taking the logarithm of both sides:
\[ \log(0.707) = n \cdot \log(0.8421) \]

### Step 8: Solve for n
Calculating the logarithms:
- \( \log(0.707) \approx -0.151 \)
- \( \log(0.8421) \approx -0.075 \)

Now substituting these values into the equation:
\[ -0.151 = n \cdot (-0.075) \]
Solving for \( n \):
\[ n = \frac{-0.151}{-0.075} \approx 2.01 \]

### Conclusion
Thus, the order of the reaction is approximately **2**.

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