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Where did the mathematical description of entropy come from? I understand the qualitative idea of disorder and microstates and all of that but why does dS = dQ/T? This equation is derived from the first law of thermodynamics for an ideal gas heated isothermically. Why is this important? The equation at the end of the derivation was dV/V = dQ/nRT = dS/nR where n and R are the number of moles in the gas and the gas law constant. Why should dQ/T be entropy and not dQ/nRT? I can't figure it out! Can anybody else?

Why should dQ/T be entropy and not dQ/nRT?

dQ/T = dS is derived for an isothermal process. What if T changed and pressure and volume were contant? Then dQ = dU and dU is proportional to T. So, dQ/T is not proportional to T at all (T/T = 1). If no work is done then entropy is independent of temperature?In an adiabatic process where there is no heat then does dS = 0? What if a gas expands adiabatically...then there IS entropy because there are more microstates possible but dS = dQ/T = 0.Since this equation for entropy doesn't work for every process it must be specific to an isothermal process. What might an equation be for other processes?

QuoteWhy should dQ/T be entropy and not dQ/nRT?Entropy has to refer to all systems, not just an ideal gas. Perhaps. n and R are rather gas-specific.

Why is dS = dQ/T not specific to isothermal processes? Here is the derivation:dU = dQ - dW = 0dQ = dW = d(PV) = nRT

dQ/nRT = 1 = dS/nRdS = dQ/TI guess T doesn't have to be constant but I'm pretty sure that my book assumed an isothermally expanding ideal gas.

What kind of process is defined by dU = 0?