Question

If the mean kinetic energy of one mole of helium gas at \( 400\,K \) temperature is \( 5000\,J \), then that for one mole of neon gas at \( 800\,K \) is

• A. \( 5000\,J \)
• B. \( 50000\,J \)
• C. \( 10000\,J \)
• D. \( 2500\,J \)
• E. \( 500\,J \)

Hint:

For one mole of an ideal gas, \[ K=\frac{3}{2}RT \] So kinetic energy depends only on temperature, not on whether the gas is helium, neon, or any other ideal gas.

Answer 1

The Correct Option is C

Step 1: Recall the expression for kinetic energy of one mole of an ideal gas.
For one mole of an ideal gas, the total translational kinetic energy is: \[ K = \frac{3}{2}RT \] This depends only on temperature, not on the nature of the gas.

Step 2: Note the key idea.
Helium and neon are different gases, but for one mole of an ideal gas, mean translational kinetic energy depends only on \( T \).
So if temperature changes, kinetic energy changes in the same proportion.

Step 3: Write the given information.
For helium at \( 400\,K \), \[ K_1 = 5000\,J \] We need the kinetic energy for neon at \( 800\,K \).

Step 4: Use proportionality with temperature.
Since \[ K \propto T \] for one mole of ideal gas, we get \[ \frac{K_2}{K_1}=\frac{T_2}{T_1} \]

Step 5: Substitute the temperatures.
\[ \frac{K_2}{5000}=\frac{800}{400} \] \[ \frac{K_2}{5000}=2 \]

Step 6: Solve for \( K_2 \).
\[ K_2 = 5000 \times 2 = 10000\,J \] Thus, for one mole of neon at \( 800\,K \), \[ K_2 = 10000\,J \]

Step 7: Final conclusion.
Hence, the required kinetic energy is \[ \boxed{10000\,J} \] Therefore, the correct option is \[ \boxed{(3)\ 10000\,J} \]