1. The problem statement, all variables and given/known data Determine ΔSsys when 3.0 mol of an ideal gas at 25°C and 1 atm is heatedd to 125°C and expanded to 5 atm. Rationalize the sign of ΔSsys. 2. Relevant equations State Function: dS = (dU)/T + (PdV)/T State Function for Entropy of Ideal Gas: dS = (CV,mdT)/T + (nRdV)/V Ideal gas law: PV = nRT Boyle's Law (I believe this assumes constant temperature): P1V1 = P2V2 3. The attempt at a solution My professor actually worked out this problem for us, but I had a few questions about the solution... Entropy problems are always solved reversibly because the formula for this is a state function. Thus, I drew a graph that would hold one of the variables constant while I solved for entropy in regard to changing one variable at a time. Path 1: Constant temperature, pressure goes from 1 atm to 5 atm. dS = (CV,mdT)/T + (nRdV)/V Since temperature is constant, the (CV,mdT)/T term equals zero. dS = (nRdV)/V ΔS = nRln(Vf/Vi) P1V1 = P2V2 (1atm)Vi = (5atm)Vf Vf/Vi = 1/5 ΔS = (3)(8.31)ln(1/5) While Boyle's law in general assumes temperature to be constant and during this part of the path from initial state to final state temperature is indeed constant, but temperature throughout this process isn't constant, and PV = nRT can also be used to solve for the initial and final volumes to plug into this problem, but when I do this, my volume term is then off by a factor of the difference in temperature and this is not made up for later by using CV,m in path 2. Additionally, there is another homework problem I have where temperature is held constant for part of the pathway, but I have to use PV = nRT to find the correct volume, and if I use Boyle's law to find it, my state function answer for entropy will be wrong. How do I know when to use Boyle's law and when to use the ideal gas law? Path 2: Constant pressure, temperature goes from 298K to 398K. dS = (CP,mdT)/T + (nRdV)/V ΔS = CP,mln(Tf/Ti) ΔS = n(5/2)Rln(Tf/Ti) ΔS = (3)(5/2)(8.31)ln(398/298) Why does the (nRdV)/V term disappear? Just because pressure is constant doesn't mean volume is constant, or at least that's what I thought. Why are we allowed to change the specific heat variable just because pressure is constant for a single part of this pathway? Again, in a third homework problem, pressure is constant for a single part of a pathway, but we can't change from CV to CP then because we "have to remember that this term was derived from dU, which is a state function in terms of CV." Why don't we have to remember that for this problem? In the end the entropies from path 1 and path 2 are summed together to get the final change in entropy.