The free expansion of an ideal gas is an adiabatic process andso no heat is transferred. No work is done, so the internal energy doesnot change. Thus, Q/T = 0, yet the randomness of the system and thusits entropy have increased after the expansion. Why does Eq. not apply to this situation?

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Entropy

In physics, entropy means the amount of disorder or the degree of randomness associated with a thermodynamic system. The system is not self-organized when the entropy of the system is high. For instance, when a system is at an elevated temperature, its molecules are in random motion, vibrating and colliding with each other. During this scenario, the probability of finding a molecule at a particular place twice, is low. Under this circumstance, the system can be said to be in a high entropy.