The activation energy for most of the reactions is approximately `"50 kJ mol"^(-1)`. The value of temperature coefficient for such reactions is

Options

A. approx.2
B. approx.3
C. `lt 1`
D. `gt 4`

Explanation

To find the temperature coefficient for reactions with an activation energy of approximately 50 kJ/mol, we can follow these steps:

### Step 1: Understand the Temperature Coefficient
The temperature coefficient (often denoted as \( Q_{10} \)) is defined as the ratio of the rate constants of a reaction at two different temperatures, typically 10°C apart. It can be expressed mathematically as:

\[ Q_{10} = \frac{k(T + 10)}{k(T)} \]

### Step 2: Use the Arrhenius Equation
The Arrhenius equation relates the rate constant \( k \) to temperature \( T \) and activation energy \( E_a \):

\[ k = A e^{-\frac{E_a}{RT}} \]

where:
- \( A \) is the pre-exponential factor,
- \( R \) is the universal gas constant (8.314 J/mol·K),
- \( T \) is the temperature in Kelvin,
- \( E_a \) is the activation energy in Joules.

### Step 3: Calculate the Rate Constants at Two Temperatures
Let’s denote the two temperatures as \( T_1 \) and \( T_2 = T_1 + 10 \) (in Celsius). We need to convert these temperatures to Kelvin:

\[ T_1 = T + 273.15 \quad \text{and} \quad T_2 = T + 10 + 273.15 \]

The rate constants at these temperatures can be expressed as:

\[ k_1 = A e^{-\frac{E_a}{RT_1}} \quad \text{and} \quad k_2 = A e^{-\frac{E_a}{RT_2}} \]

### Step 4: Calculate the Ratio of Rate Constants
Now, we can find the ratio \( \frac{k_2}{k_1} \):

\[ \frac{k_2}{k_1} = \frac{A e^{-\frac{E_a}{RT_2}}}{A e^{-\frac{E_a}{RT_1}}} = e^{-\frac{E_a}{RT_2} + \frac{E_a}{RT_1}} = e^{E_a \left( \frac{1}{RT_1} - \frac{1}{RT_2} \right)} \]

### Step 5: Substitute the Values
Given \( E_a = 50 \, \text{kJ/mol} = 50000 \, \text{J/mol} \) and using \( R = 8.314 \, \text{J/mol·K} \), we can substitute these values into the equation.

Assuming \( T_1 \) is around 25°C (298 K), then \( T_2 \) would be 35°C (308 K):

\[ \frac{1}{RT_1} - \frac{1}{RT_2} = \frac{1}{8.314 \times 298} - \frac{1}{8.314 \times 308} \]

Calculating this gives us a small positive number, which indicates that \( \frac{k_2}{k_1} \) will be greater than 1.

### Step 6: Approximate the Temperature Coefficient
From empirical observations, for many reactions, the temperature coefficient \( Q_{10} \) is approximately 2 when the activation energy is around 50 kJ/mol.

Thus, we conclude that:

\[ Q_{10} \approx 2 \]

### Final Answer
The value of the temperature coefficient for such reactions is approximately **2**.

---