Because a diatomic gas molecule has an array of rotational and vibrational modes to store kinetic energy, a diatomic gas will have a higher molar heat capacity than an ideal gas, which can only store thermal energy in the form of translational kinetic energy.
Why do some substances possess a higher molar heat capacity than other substances? The molar heat capacity of real and ideal gases depends on the available translational and rotational 'places' available for kinetic energy.
Picture an imaginary merry-go-round on wheels, a vehicle which can hold kinetic energy not only in translational motion but also in its rotation and, as well, the oscillations of the horses up and down. If a certain amount of kinetic energy were imparted on this vehicle, the energy could be distributed throughout all these 'partitions', some would increase its translational motion, but there is also room for energy in the rotations and vibrations. Because there are so many places to put energy, it would take a lot of energy before any of the partitions would become vigorous.
As a simple example, picture a diatomic molecule, such as O2. In the terminology of kinetic theory, a diatomic molecule possesses more degrees of freedom than a single atom. O2 can have vibrational and rotational motions as well as translational motion. An increase in temperature will cause the average translational energy to increase, but it also causes the energy associated with vibrational and rotational motion to increase. The oxygen gas will require a higher energy input to change the temperature by a certain amount. O2 has a higher molar heat capacity than a monatomic gas such as helium.
Large, complex molecules possess many different modes of rotation and vibration, so it takes a large amount of heat flow before the motion becomes vigorous. The energy is divided into so many partitions. The particles of the ideal gas, though, can only move translationally, in the x,y, and z directions, so the molar heat capacity of an ideal gas is the theoretical minimum.
The rule of Dulong and Petit is an example of this principle that particles that can move similarly will have similar molar heat capacity; the molar heat capacity of most solid metals is very nearly the same. The fact that some metals have atoms much larger than others does not matter for molar heat capacity. In a solid metal, the individual metal atoms occupy crystalline structures that allow for kinetic energy to exist in a similar way, so as heat flows in, for example, the amount of temperature change per mole of particles is the same for different metals (about 26 J/mol oC)
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