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One is that residual entropy (the entropy a substance has as a result of not being a perfectly ordered crystal at $\pu{0 K}$) is a thermodynamic entropy. … The other is that "residual entropy is only an apparent entropy and not a real thermodynamic entropy" (J. Non-Cryst. Solids 2009, 355 (10-12), 617–623). …

Also, a "measure of randomness" a crude way to characterize entropy. … that in turn determine the entropy. …

When a system changes from state 1 to state 2, the change in entropy of the system is the same for all processes, because entropy is a state function. … Since the entropy change for the system is independent of the process, the difference in entropy production between rev. and irrev. processes is seen in the entropy change of the surroundings. …

Then we need to determine how the entropy of the argon released into the atmosphere increases as a result of this release. … But, when doing an entropy calculation, we do need to account for the fact that the atmosphere already contains 0.93% Ar. …

Here I'm assuming you are interested in comparing the magnitude of $\Delta S^{\circ}_{fus}$ at $T_{fus}$ with that of $\Delta S^{\circ}_{vap}$ at $T_{vap}$, for the elements in their standard states ( …

To do this, I need an estimate of the absolute molar entropy of water at those conditions, enabling me to calculate W using: $$W= e^{S/k} $$ Are there tables of absolute (as opposed to standard) molar … would then be an easy matter to add the difference in entropy between ice at $T_\mathrm{fus}$ and water at $\pu{298 K}$. …

I did find this paper from 2005 in which Zielkiewicz estimates the molar entropy of water at $\pu{298 K}$ [1]. It's behind a paywall, but the subsequent correction from 2006 is not [2]. … : https://en.wikipedia.org/wiki/Standard_molar_entropy Thus one can simply look up the standard molar entropy of water at 298K, which is 75.28 J/(mol K) (https://en.wikipedia.org/wiki/Water_(data_page …

Entropy is indeed a state function, and thus depends only on the state of the system. Hence it doesn't matter how you get from state A to state B, the entropy change will be the same. … But, since $dS= \frac{\text{đ}q_{rev}}{T}$, one can calculate the entropy change by integrating along a path connecting the two states, but that path must be reversible. …

This should address your confusion about constant T: $$G=H-TS\Rightarrow\mathrm dG=\mathrm dH-T\,\mathrm dS-S\,\mathrm dT$$ $$\Rightarrow \text{ (at constant $T$) } \mathrm dG=\mathrm dH-T\,\mathrm dS …

Hence, for the same heat flow from the surroundings (which is given by $-\Delta H^{o}_{rxn}$), the resulting unfavorable entropy change in the surroundings is less. …

The OP is specifically interested in the efficiency of removing $\ce{CO_2}$ from ambient air, rather than from point sources like power plants. The former is obviously less efficient because the $\ce …

It's about the change in the entropy of the universe. … Thus: $$đw_{non-pV,max} = -T dS_{univ}$$ In sum, it's all about the entropy, baby! …

The textbook is referring to the entropy change of the system. While the textbook is correct that absolute zero can never be attained, its statement that the entropy change is infinite is wrong. … The absolute entropy is given by integrating the entropy change from absolute zero to whatever temperature the substance is at: $\text {Absolute entropy at } T' \equiv S(T') =\Delta S_{0 \rightarrow T' …

Also, some general comments on the issue of the entropy of a substance at absolute zero: It is possible the question of whether substances need to be perfect crystals to have zero entropy at $\pu{0 K} … It's still an open question whether a substance needs to be a perfect crystal to have zero entropy at $\pu{0 K}$. Some in the field argue it does, some argue it doesn't. …

Your textbook's derivation is done under the assumption of constant $T$, which means $T_{sys} = T_{surr} =T$. However, this does not mean $dG_{sys}$ is always zero. Let's start with the following: $ …