Kinetic Molecular Theory Of Gases

Isotones are atoms of different elements that have the same number of neutrons but different numbers of protons. In other words, isotones have the same neutron number ( ) but different atomic numbers ( ).

Now, let's analyze the given options:

• Option A: and
  • These are isotopes of hydrogen, not isotones. They have the same number of protons but different numbers of neutrons.
• Option B: and
  • These are isotones. Both have 118 neutrons ( and ), but different numbers of protons (79 for Au and 80 for Hg).
• Option C: and
  • These are not isotones. They have the same number of nucleons (3), but different numbers of protons and neutrons.
• Option D: and
  • These are isotopes of chlorine, not isotones. They have the same number of protons (17), but different numbers of neutrons.
• Option E: and
  • These are isotopes of uranium, not isotones. They have the same number of protons (92) but different numbers of neutrons.

Therefore, the correct answer is:

For an ideal gas, the root mean square (rms) speed is related to temperature by the equation: where is Boltzmann constant and is the molar mass. The slope of the graph between and is proportional to .

RMS speed is: For oxygen at temperature : For hydrogen at : Given both are equal: Squaring both sides:
Hence, the correct answer is:

Step 1: Understanding the Concept:
The molar specific heat capacity at constant volume, denoted as , of an ideal gas depends fundamentally on its active degrees of freedom .
The Law of Equipartition of Energy states that each active degree of freedom contributes exactly to the internal molar energy of the gas.
Step 2: Key Formula or Approach:
The internal energy for one mole of a gas is given by .
The molar specific heat capacity at constant volume is then defined as the derivative of internal energy with respect to temperature: .
Step 3: Detailed Explanation:
A standard rigid diatomic gas possesses exactly 5 degrees of freedom (3 translational modes and 2 rotational modes).
However, for a non-rigid diatomic gas, typically considered at higher temperatures, an additional vibrational mode becomes active.
A single active vibrational mode inherently introduces two extra degrees of freedom (one for kinetic energy and one for potential energy).
Therefore, the total degrees of freedom for a non-rigid diatomic gas is .
Substituting into the formula gives:
Step 4: Final Answer:
The specific heat capacity at constant volume is .