Explain how calculate entropy in spontaneous mixing of ideal gases at constant temperature and p?

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When gases freely mix with each other at constant temperature and pressure, each of the gases can be thought of as undergoing a free expansion from initial volume to the same final volume V_f, where V_f = V_a+V_b + .... ; degree of randomness as well as entropy of gas molecules increase. The change in entropy per mole of the mixtureof ideal gases formed can be calculated by

ΔS_(mix) = – R Ʃx_ilnx_i

Where xi is the mole fraction of the ith species in the mixture. Since xi <1, ΔS_(mix) is positive, i.e. entropy increases during mixing.

This is a unique case where entropy (often defined as dq/T) increases though heat transfer (dq) is zero, and yet the behaviour of spontaneity of the process of mixing could be correctly predicted on the basis of entropy alone.

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