The helium and argon are put in the flask at the same room temperature (300 K). The ratio of average kinetic energies (per molecule) of helium and argon is :
(Give : Molar mass of helium = 4 g/mol, Molar mass of argon = 40 g/ mol )
Show Hint
- 1 : 10
- 10 : 1
- \( 1 : \sqrt{10} \)
- 1 : 1
The Correct Option is D
Approach Solution - 1
To solve this problem, we need to determine the ratio of average kinetic energies per molecule for helium and argon at the same temperature. The formula for the average kinetic energy per molecule of an ideal gas is given by:
K.E. = \frac{3}{2} k Twhere:
- K.E. is the average kinetic energy per molecule,
- k is the Boltzmann constant, and
- T is the absolute temperature in Kelvin.
Notice that the kinetic energy formula does not depend on the mass or the type of gas, only on the temperature. This is true for all ideal gases.
Since both gases are at the same temperature (300 K), the average kinetic energy per molecule will be the same for both helium and argon.
Therefore, the ratio of the average kinetic energies per molecule for helium and argon is:
1 : 1This is because kinetic energy is solely dependent on temperature, which is the same for both gases in this case.
Conclusion: The correct answer is 1 : 1. This indicates that each gas molecule, regardless of its type, has the same average kinetic energy at a given temperature.
Approach Solution -2
The average kinetic energy of a molecule in an ideal gas is directly proportional to the absolute temperature and is given by the equipartition theorem: \[ K_{avg} = \frac{f}{2} k_B T \] where \( f \) is the number of degrees of freedom of the molecule, \( k_B \) is the Boltzmann constant, and \( T \) is the absolute temperature.
Step 2: Determine the degrees of freedom for helium and argon.
Helium (He) and argon (Ar) are both noble gases and are monatomic. Monatomic gases have only three translational degrees of freedom (motion along the x, y, and z axes).
Therefore, for both helium and argon, \( f = 3 \).
Step 3: Calculate the average kinetic energy for helium and argon.
For helium at temperature \( T = 300 \) K: \[ K_{avg, He} = \frac{3}{2} k_B (300) \] For argon at temperature \( T = 300 \) K: \[ K_{avg, Ar} = \frac{3}{2} k_B (300) \]
Step 4: Find the ratio of the average kinetic energies.
The ratio of the average kinetic energies per molecule of helium and argon is: \[ \frac{K_{avg, He}}{K_{avg, Ar}} = \frac{\frac{3}{2} k_B (300)}{\frac{3}{2} k_B (300)} = 1 \]
Thus, the ratio is 1 : 1. The molar masses of helium and argon are irrelevant for determining the average kinetic energy per molecule at the same temperature.
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