The avrage kinetic energy of molecules in a gas at temperature T is `1.5 KT`find the temperature at which the average kinetic energy of the molecules of hydrogen equals the binding energy of its atoms will hydrogen remain in molecles form at this temperature ? Take `k = 8.62xx 10^(-5) eVK^(-1)`

Explanation

To solve the problem, we need to find the temperature at which the average kinetic energy of hydrogen molecules equals the binding energy of hydrogen atoms. Here's a step-by-step solution:

### Step 1: Understand the relationship between kinetic energy and temperature
The average kinetic energy (K) of molecules in a gas at temperature T is given by the formula:
\[ K = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant.

### Step 2: Set the average kinetic energy equal to the binding energy
According to the problem, we want to find the temperature at which the average kinetic energy of hydrogen molecules equals the binding energy of hydrogen atoms. The binding energy of a hydrogen atom is given as \( 13.6 \, \text{eV} \).

Thus, we can write:
\[ K = 13.6 \, \text{eV} \]

### Step 3: Substitute the expression for kinetic energy
From Step 1, we have:
\[ \frac{3}{2} k T = 13.6 \, \text{eV} \]

### Step 4: Solve for temperature T
Substituting the value of \( k = 8.62 \times 10^{-5} \, \text{eV/K} \):
\[ \frac{3}{2} (8.62 \times 10^{-5}) T = 13.6 \]

Now, rearranging for T:
\[ T = \frac{13.6 \times 2}{3 \times 8.62 \times 10^{-5}} \]

### Step 5: Calculate T
Calculating the value:
1. Calculate \( 3 \times 8.62 \times 10^{-5} \):
\[ 3 \times 8.62 \times 10^{-5} = 2.586 \times 10^{-4} \]

2. Calculate \( 13.6 \times 2 \):
\[ 13.6 \times 2 = 27.2 \]

3. Now divide:
\[ T = \frac{27.2}{2.586 \times 10^{-4}} \approx 105,000 \, \text{K} \]

### Step 6: Conclusion about molecular state
Now, we need to determine if hydrogen will remain in molecular form at this temperature. Given that \( 105,000 \, \text{K} \) is extremely high, hydrogen will not remain in molecular form at such a high temperature. At this temperature, hydrogen molecules would dissociate into individual atoms due to the high kinetic energy overcoming the binding energy.

### Final Answer
The temperature at which the average kinetic energy of hydrogen molecules equals the binding energy of its atoms is approximately \( 105,000 \, \text{K} \). At this temperature, hydrogen will not remain in molecular form.

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