Question

If the average kinetic energy of a molecule of a hydrogen gas at 300 K is $E$, the average kinetic energy of a molecule of a nitrogen gas at the same temperature is

• A. $7E$
• B. $\frac{E}{14}$
• C. $14E$
• D. $\frac{E}{7}$
• E. $E$

Hint:

At same temperature, all gases have equal average kinetic energy — only speeds differ.

Answer 1

Answer: (E)

Concept: According to the kinetic theory of gases, the average kinetic energy of a single molecule of an ideal gas is given by: \[ E = \frac{3}{2} kT \] where:
• $k$ = Boltzmann constant
• $T$ = absolute temperature (in Kelvin)

Step 1: Key observation from formula
The expression $\frac{3}{2}kT$ shows that:
• Average kinetic energy depends only on temperature
• It is completely independent of:
• Mass of the molecule
• Type of gas (hydrogen, nitrogen, oxygen, etc.)

Step 2: Given condition
Both hydrogen gas and nitrogen gas are at the same temperature: \[ T = 300 \, \text{K} \]

Step 3: Apply formula for both gases: \[ E_{\text{hydrogen}} = \frac{3}{2} kT \] \[ E_{\text{nitrogen}} = \frac{3}{2} kT \]

Step 4: Comparison: \[ E_{\text{hydrogen}} = E_{\text{nitrogen}} \]

Step 5: Common misconception clarified
Students often think heavier gases have more kinetic energy. This is incorrect:
• Heavier molecules move slower
• Lighter molecules move faster
• But kinetic energy (average) remains same at same temperature

Step 6: Physical interpretation
Temperature represents the average kinetic energy of molecules. Thus:
• Same temperature → same average kinetic energy
• Independent of molecular mass Final Conclusion: Average kinetic energy of nitrogen molecule = $E$