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If you look at Boyle's Law, P1V1 = P2V2, then dividing V1 and P2 from both sides, we get V2/V1 = P1/P2. Therefore, if we substitute that in the equation, then deltaS=nRln(V2/V1) will become deltaS = nRln(P1/P2). In essentiality, it is just using Boyle's Law in conjunction with the change in entropy equation.
The change in entropy formula has final over the initial in that property because we are observing a change from an initial state to a final state. Pressure and volume are inversely proportional due to Boyle's Law and thus, pressure is an exception where it would be initial pressure over final pressure.
It helps to think about it using the ideal gas law, PV=nRT. If moles and temperature remain the same, then you know the entire right hand of the equation is a constant. Say, for example, you increase volume. Then you know since nothing else is changing, pressure has to go down in order for the right hand side of the equation to remain the same number. The same thing would happen if you decrease volume; your pressure increases. This is the basis of Boyle's Law, which is why the relationship is inverse in the ΔS equation.
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