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: Understanding the Concept:
The mean kinetic energy (or internal energy) of one mole of an ideal gas depends only on its temperature and its atomicity (i.e., the number of atoms in a molecule, which determines its degrees of freedom). It does not depend on the mass or type of the gas atoms. Both helium (He) and neon (Ne) are monatomic gases.
Step 2: Key Formula or Approach:
The mean kinetic energy (\(E\)) of one mole of an ideal gas is given by:
\[ E = \frac{f}{2}RT \] where \(f\) is the number of degrees of freedom, \(R\) is the universal gas constant, and \(T\) is the absolute temperature.
For a monatomic gas (like helium and neon), the degrees of freedom \(f = 3\) (for translational motion in x, y, and z directions).
So, the formula becomes:
\[ E = \frac{3}{2}RT \] From this, we can see that the mean kinetic energy is directly proportional to the absolute temperature, \(E \propto T\).
Step 3: Detailed Explanation:
Let \(E_{He}\) and \(T_{He}\) be the energy and temperature of helium gas.
Let \(E_{Ne}\) and \(T_{Ne}\) be the energy and temperature of neon gas.
We are given:
\(E_{He} = 5000\) J
\(T_{He} = 400\) K
\(T_{Ne} = 800\) K
We need to find \(E_{Ne}\).
Since both gases are monatomic, their mean kinetic energy is directly proportional to temperature. We can set up a ratio:
\[ \frac{E_{Ne}}{E_{He}} = \frac{T_{Ne}}{T_{He}} \] Substitute the given values into the equation:
\[ \frac{E_{Ne}}{5000 \text{ J}} = \frac{800 \text{ K}}{400 \text{ K}} \] \[ \frac{E_{Ne}}{5000} = 2 \] Now, solve for \(E_{Ne}\):
\[ E_{Ne} = 2 \times 5000 \text{ J} = 10000 \text{ J} \] Step 4: Final Answer:
The mean kinetic energy of one mole of neon gas at 800 K is 10000 J. This corresponds to option (C).