As I understand, in the process of state transition (fusion or vaporization) the temperature remains constant for the duration the process is happening. The internal energy is a function of only temperature, so does the internal energy not change?
[Physics] Internal energy in phase transition
energyphase-transitiontemperaturethermodynamics
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But for an ideal gas, internal energy is only a function of temperature and so internal energy remains constant here,no change in average kinetic energy of gas particles takes place, so where does the chaos come from to increase entropy of the system.
'Chaos' is not a very well defined term in context of statistical physics. It is not necessary to use it to understand entropy in this theory.
One way to understand this entropy is the Boltzmann-Planck definition: entropy is logarithm of number of distinct possible states that are compatible with macroscopic variables like $V$,$U$ etc. No chaos is needed in this definition.
You seem to assume that in general, increase in internal energy is necessary to increase entropy. That is not the case, for there are systems where entropy can increase just by making sudden change to the system, without supplying any energy. Consider ideal gas in a chamber that is connected to another chamber via small pipe. At first, the pipe is blocked by a closed valve, so the gas stays in the first chamber. Then the valve is opened and the gas rushes to the second chamber. In the end, the volume is twice the original value.
Now that the gas has twice the original volume, the number of accessible states increased by factor of two. But no energy was supplied or lost, so internal energy stayed the same. According to the above definition, entropy increased by a factor $\ln 2$.
Change in internal energy is zero if temperature is constant because, internal energy is a function of temperature only
This true for an ideal gas, but not true for real gases where we get interactions between the gas particles. The obvious example of this is the Joule expansion. In this process no energy is added an now work is done so the internal energy remains constant. For an ideal gas the temperature does remain constant, but for real gases the temperature can increase or decrease depending on how the gas atoms/molecules interact with each other.
Your example of a dissociating molecule is basically an extension of this where the forces between the gas particles are strong enough to bind them into molecules.
In a Joule expansion some of the gas particles' kinetic energy is converted to potential energy or vice versa. The internal energy is the sum of the kinetic and potential energy while the temperature depends only on the kinetic energy. That's why when when the potential energy is significant the internal energy and the temperature are not simply proportional.
In the specific case of your dissociating molecule, we have to put work in to separate the parts of the molecule from the force binding them together, so the potential energy increases. The heat supplied to the system is going into the increase in the potential energy and leaving the kinetic energy unchanged. So the internal energy is increasing while the kinetic energy, and therefore the temperature, is not changed.
Best Answer
During a change of phase the temperature does not change, but the internal energy does. The internal energy is the sum of the kinetic energy of the molecules and the chemical potential energy of the molecules. During a change of phase, the average kinetic energy of the molecules stays the same (with the temperature), but the average potential energy changes.
For more information on this topic, do visit:
http://electron6.phys.utk.edu/101/CH7/phase_transitions.htm&ei=cJxiShax&lc=en-IN&s=1&m=364&ts=1436773479&sig=AKQ9UO84sBHgLYBnocfErpK9LfMYP6vfkQ