I am confused as to what is the relation between entropy and internal energy. Entropy is always presented as a measure of the randomness in a system. So when we supply heat to a well insulated system say ideal gas in a container with fixed boundary, the internal energy and temperature increase, which implies that the motion of gas particles increases and hence the system becomes more chaotic and thus entropy increases. But if we take the same system, and supply heat isothermally and reversibly, the defnition of entropy change ΔS=Q/T , says that the entropy of system would increase(at the cost of equal entropy drop in surroundings). 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.
How are the two related?
Best Answer
'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 $\ln 2$.