Entropy Change — Overview & Calculation

Recall that entropy is dependent on the internal energy of a system.
$$dS=\frac{1}{T}dU+\frac{P}{T}dV$$ In turn, the internal energy is dependent on heat and work. $$dU=đQ+đW$$ If we rearrange our first equation, $$dU=TdS-PdV$$ we have two equations for internal energy which must be equivalent. Let's break this down, starting with work.

Work done on an object is given by force times distance, so in differential form, $$\begin{align}đW&=Fd\ell \\ &=Fd\ell \cdot \frac{\ell ^{2}}{\ell ^{2}} \\ &= \frac{F}{\ell ^{2}}d\ell ^{3} \\ &= PdV \end{align}$$ Of course, we are considering how much work our object, or system, does on the surrounding, thus, $$đW=-PdV$$ Substituting this into our equation for internal energy, and comparing, $$dU=đQ-PdV$$ $$dU=TdS-PdV$$ it becomes obvious that heat and entropy are related in the following manner, $$đQ=TdS$$

Applications

Often times chemists want to calculate the entropy change as things heat and cool. Thus, we want to rearrange our equation, $$\begin{align} dS&=\frac{đQ}{T} \\ \Delta S&=\int_{T_{1}}^{T_{2}}\frac{đQ}{T} \end{align}$$ But we run in to a bit of an issue. Our integral is describing two different temperature, but our differential is is heat. Luckily, we can use heat capacity to change our differential to temperature. $$đQ=mC_{V,P}dT$$ $m$ is the mass of our system, and $C_{V,P}$ is either the constant volume heat capacity or the constant pressure heat capacity. The choice of which to use should be determined on a case by case basis. Plugging this in to our equation, $$\Delta S=\int_{T_{1}}^{T_{2}}\frac{mC_{V,P}dT}{T}=mC_{V,P}ln(\frac{T_{1}}{T_{2}})$$ This is how to calculate the entropy change the reversible heating of a single phase system.

When systems undergo a phase change, the entropy is calculated as by dividing the enthalpy change by the temperature. $$\Delta S=\frac{\Delta H}{T}$$
This is because phase changes occur at constant pressure, thus, $$dH=TdS+VdP=TdS+0=Tds$$ $$dH=TdS$$

(Video) Entropy Change For Melting Ice, Heating Water, Mixtures

by The Organic Chemistry Tutor

In this video, The Organic Chemistry Tutor demonstrates entropy change calculations. In this section of the video, he illustrates three examples. Each one deals with a temperature or phase change. After watching this video, you'll understand how to calculate entropy changes!