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 find the kinetic energy per atom from the given information, we can follow these steps:

### Step 1: Convert bond energy from kJ/mol to J/molecule
The bond energy of \( X_2 \) is given as \( 180.6 \, \text{kJ/mol} \). To convert this to Joules per molecule, we can use Avogadro's number (\( 6.022 \times 10^{23} \, \text{molecules/mol} \)).

\[ \text{Bond energy per molecule} = \frac{180.6 \, \text{kJ/mol} \times 1000 \, \text{J/kJ}}{6.022 \times 10^{23} \, \text{molecules/mol}} \]

Calculating this gives:

\[ \text{Bond energy per molecule} = \frac{180600 \, \text{J}}{6.022 \times 10^{23}} \approx 3.0 \times 10^{-19} \, \text{J} \]

### Step 2: Calculate the difference in energy absorbed and bond energy
The energy absorbed by each \( X_2 \) molecule is given as \( 3.5 \times 10^{-19} \, \text{J} \). To find the kinetic energy, we subtract the bond energy from the energy absorbed:

\[ \text{Kinetic energy per molecule} = \text{Energy absorbed} - \text{Bond energy} \]

\[ \text{Kinetic energy per molecule} = 3.5 \times 10^{-19} \, \text{J} - 3.0 \times 10^{-19} \, \text{J} = 0.5 \times 10^{-19} \, \text{J} \]

### Step 3: Calculate the kinetic energy per atom
Since each \( X_2 \) molecule consists of 2 atoms, we need to divide the kinetic energy per molecule by 2 to find the kinetic energy per atom:

\[ \text{Kinetic energy per atom} = \frac{0.5 \times 10^{-19} \, \text{J}}{2} = 0.25 \times 10^{-19} \, \text{J} \]

### Step 4: Express the final answer in standard form
We can express \( 0.25 \times 10^{-19} \, \text{J} \) in standard form:

\[ \text{Kinetic energy per atom} = 2.5 \times 10^{-20} \, \text{J} \]

### Final Answer
The kinetic energy per atom is \( 2.5 \times 10^{-20} \, \text{J} \).

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