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## Main Question or Discussion Point

I understand why

dE=TdS-PdV ...[1]

at constant temperature and pressure

Also, I see that F=E-TS

and therefore

dF=dE-SdT-TdS ...[2]

and how combining [2] and [3] gives

dF=-SdT -PdV

QUESTION: shouldn't [1] be expressed as dE=TdS+SdT-PdV

i.e if you're going to substitute into equation [2] where T isn't kept constant, don't you need to start off with T not being kept constant in your expression for thermal energy? Why are we letting temperature change in equation 2 but not in equation 1?

This seemingly obvious question has me stumped.

dE=TdS-PdV ...[1]

at constant temperature and pressure

Also, I see that F=E-TS

and therefore

dF=dE-SdT-TdS ...[2]

and how combining [2] and [3] gives

dF=-SdT -PdV

QUESTION: shouldn't [1] be expressed as dE=TdS+SdT-PdV

i.e if you're going to substitute into equation [2] where T isn't kept constant, don't you need to start off with T not being kept constant in your expression for thermal energy? Why are we letting temperature change in equation 2 but not in equation 1?

This seemingly obvious question has me stumped.