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The negative sign is there because the internal energy of the system decreases when the system expands, since the system is doing work against the outside pressure and is losing energy. As such, work is negative for expansion (work being done BY the system), and positive for compression (work is being done ON the system).
the minus sign makes it so that the equation represents energy in relation to the system. When work is done on the system (i.e. system is compressed), then ∆V is negative, and (-P∆V) becomes positive. This means that when work is done on the system, the system gains energy. Alternately, when the system does work (i.e. system expands) then ∆V is positive, and (-P∆V) is negative. This way, the equation shows the decrease in energy of the system after doing work.
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