If the energy absorbed by each `X_2` molecule of a gas is `3.5xx10^-19` J and the bond energy of `X_2` is `180.6 kJ mol^-1` then the kinetic energy per atom will be

Options

A. `2.5xx10^-19J`
B. `5.0xx10^-20J`
C. `2.5xx10^-20J`
D. `1.5xx10^-19J`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Identify the given values
- Energy absorbed by each \(X_2\) molecule: \(E = 3.5 \times 10^{-19}\) J
- Bond energy of \(X_2\): \(BE = 180.6 \text{ kJ/mol}\)

### Step 2: Convert bond energy from kJ/mol to J/molecule
To find the bond energy per molecule, we need to convert the bond energy from kilojoules per mole to joules per molecule. We use Avogadro's number (\(N_A = 6.022 \times 10^{23}\) molecules/mol) for this conversion.

\[ BE \text{ (in J/molecule)} = \frac{180.6 \text{ kJ/mol} \times 1000 \text{ J/kJ}}{6.022 \times 10^{23} \text{ molecules/mol}} \]

Calculating this gives:

\[ BE = \frac{180600 \text{ J/mol}}{6.022 \times 10^{23}} \approx 3.0 \times 10^{-19} \text{ J/molecule} \]

### Step 3: Calculate the kinetic energy of the \(X_2\) molecule
The total energy absorbed by the \(X_2\) molecule is used to break the bond and provide kinetic energy to the atoms. The kinetic energy (\(KE\)) can be calculated as:

\[ KE = E - BE \]

Substituting the values:

\[ KE = 3.5 \times 10^{-19} \text{ J} - 3.0 \times 10^{-19} \text{ J} = 0.5 \times 10^{-19} \text{ J} \]

### Step 4: Calculate the kinetic energy per atom
Since the \(X_2\) molecule dissociates into 2 atoms of \(X\), the kinetic energy per atom is:

\[ KE \text{ per atom} = \frac{KE}{2} = \frac{0.5 \times 10^{-19} \text{ J}}{2} = 0.25 \times 10^{-19} \text{ J} = 2.5 \times 10^{-20} \text{ J} \]

### Step 5: Conclusion
The kinetic energy per atom of \(X\) is \(2.5 \times 10^{-20} \text{ J}\).

### Final Answer
The correct option is **C: \(2.5 \times 10^{-20} \text{ J}\)**.

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