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?