The rate of a reaction increased from 2 units to 54 units due to change in temperature from `27^(@)C` to `57^(@)C`. What is the approximate temperature coefficient of the reaction.

Explanation

To find the approximate temperature coefficient of the reaction, we will follow these steps:

### Step 1: Understand the Temperature Coefficient
The temperature coefficient (μ) is defined as the ratio of the rate of reaction at two different temperatures. It can be expressed mathematically as:
\[ \mu = \frac{R_f}{R_i} \]
where \(R_f\) is the rate of reaction at the final temperature and \(R_i\) is the rate of reaction at the initial temperature.

### Step 2: Identify Given Values
From the question, we have:
- Initial rate of reaction, \(R_i = 2\) units
- Final rate of reaction, \(R_f = 54\) units
- Initial temperature, \(T_i = 27^\circ C\)
- Final temperature, \(T_f = 57^\circ C\)

### Step 3: Calculate the Change in Temperature
Calculate the change in temperature (\(\Delta T\)):
\[ \Delta T = T_f - T_i = 57^\circ C - 27^\circ C = 30^\circ C \]

### Step 4: Calculate the Temperature Coefficient
Using the formula for temperature coefficient:
\[ \mu = \frac{R_f}{R_i} \]
Substituting the values:
\[ \mu = \frac{54}{2} = 27 \]

### Step 5: Determine the Value of \(n\)
The temperature coefficient can also be expressed in terms of the change in temperature and a factor \(n\):
\[ \mu = \mu_0^{\frac{\Delta T}{10}} \]
Where \(n\) is the number of 10-degree intervals in the change of temperature. Since \(\Delta T = 30^\circ C\):
\[ n = \frac{\Delta T}{10} = \frac{30}{10} = 3 \]

### Step 6: Relate the Temperature Coefficient to \(n\)
Now we can express the relationship:
\[ \mu^n = \mu^{3} \]
From our previous calculation:
\[ 27 = \mu^3 \]
Taking the cube root:
\[ \mu = 27^{\frac{1}{3}} = 3 \]

### Final Answer
Thus, the approximate temperature coefficient of the reaction is:
\[ \mu \approx 3 \]

---