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When a system loses it's internal thermal energy by emitting radiation or by convection what is change of the systems entropy mathematically?

I saw this question's answers for radiation only but still unclear because it doesn't clearly answers my question. Can someone clarify it for me?

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Entropy of a system is a function of the state of the system (i.e., a physical property of the system), and is independent of any process path that caused it to arrive at that state. Therefore, whether a system lost energy and arrived at a given state by radiation, convection, or conduction is immaterial. The entropy change is determined only by the starting state and the final state, and nothing in between.

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  • $\begingroup$ But how can I mathematically show that in convection and radiation, entropy always increase? $\endgroup$
    – user310884
    Commented Aug 8, 2021 at 20:04
  • $\begingroup$ If that is the only method by which the temperature increases, then what you are saying is that the final state has a higher temperature than the initial state. So what you are really asking is that, "if the only difference between the initial state and the final state is an increase in temperature, how do I know that the entropy increased?" Is that what you really mean? $\endgroup$
    – Chet Miller
    Commented Aug 8, 2021 at 20:21
  • $\begingroup$ Yes, I want to say if heat is conducted by means of these, how can I mathematically prove that the system's entropy is also increased? $\endgroup$
    – user310884
    Commented Aug 8, 2021 at 20:46
  • $\begingroup$ I don't know how to prove this using statistical thermodynamics. But, using classical thermodynamics, $$\Delta S=\int{\frac{dQ_{rev}}{T}}$$Would that be able to satisfy you? $\endgroup$
    – Chet Miller
    Commented Aug 8, 2021 at 21:02
  • $\begingroup$ Classical mechanics is fine. But do how we prove it using the formula you presented? $\endgroup$
    – user310884
    Commented Aug 8, 2021 at 21:06

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