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The average translational kinetic energy...

The average translational kinetic energy of nitrogen gas molecule is `0.02eV (1eV - 1.6 xx 10^(-19)J)`. Calculate the temperatuire of the gas. Boltzmann constant `k = 1.38 xx 10^(-23) J//K`.

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To calculate the temperature of nitrogen gas given the average translational kinetic energy, we can follow these steps: ### Step 1: Write down the formula for average translational kinetic energy. The average translational kinetic energy (K.E.) of a gas molecule is given by the formula: \[ K.E. = \frac{3}{2} k T \] where \( k \) is the Boltzmann constant and \( T \) is the temperature in Kelvin. ### Step 2: Convert the given kinetic energy from electron volts to joules. The average translational kinetic energy is given as \( 0.02 \, \text{eV} \). To convert this to joules, we use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \): \[ K.E. = 0.02 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 3.2 \times 10^{-21} \, \text{J} \] ### Step 3: Substitute the values into the kinetic energy formula. Now we can substitute the values of \( K.E. \) and \( k \) into the formula: \[ 3.2 \times 10^{-21} \, \text{J} = \frac{3}{2} (1.38 \times 10^{-23} \, \text{J/K}) T \] ### Step 4: Solve for temperature \( T \). Rearranging the equation to solve for \( T \): \[ T = \frac{2 \times 3.2 \times 10^{-21}}{3 \times 1.38 \times 10^{-23}} \] ### Step 5: Calculate the value of \( T \). Calculating the numerator: \[ 2 \times 3.2 \times 10^{-21} = 6.4 \times 10^{-21} \] Calculating the denominator: \[ 3 \times 1.38 \times 10^{-23} = 4.14 \times 10^{-23} \] Now substituting these values into the equation for \( T \): \[ T = \frac{6.4 \times 10^{-21}}{4.14 \times 10^{-23}} \approx 1548.78 \, \text{K} \] ### Final Answer: The temperature of the nitrogen gas is approximately \( 1548.78 \, \text{K} \). ---
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