## Entropy

October 25th, 2016

Let’s construct a heat engine that converts heat into mechanical work. The heat engine consist of the next main elements: the working body (gas under the piston), heater, cooler (fig.1).

Fig. 1. The  main elements of the heat engine.

The ideal heat engine is called the heat engine in which the work carried out by the working body and the difference between the amount of heat, received from the heater and allotted into the cooler  is equal to:

$W= \Delta Q_H - \Delta Q_C$      (1)

The ideal is a heat engine, which is a closed cycle ( Carnot’s cycle ) consisting of two isotherms and two adiabats (fig.2).

Fig.2 . Carnot’s cycle

If the heat engine is working on the Carnot cycle, we have the relation:

$\frac {\mid \Delta Q_H \mid}{T_H}=\frac {\mid \Delta Q_C \mid}{T_C}$   (2)

The quantity $\Delta S= \frac {\Delta Q}{T}$ is called the reduced heat, or the entropy change. The infinitely small the change of the entropy is equal to:

$dS=\frac {dQ}{T}$       (3)

Integrating the last equality we get the formula for the calculation of the entropy change:

$\Delta S= \int_{1}^{2}\frac{dQ}{T}$      (4)

(The information has been taken from e-book «Fundamentals of classical physics” (author V.D. Shvets), which was published in 2007 by “1C-Multimedia Ukraine” publishing company. Copyright is reserved by certificate Number 55955 from 06.08.2014. More information can be found at https://ipood-kiev.academia.edu/ValentynaShvets/Books ).

Task 1. Ice with weight m at the temperature T=273 K melts slowly, turning the water at the same temperature. Define the entropy change.

Task 2. Define the entropy change for isothermal expansion of the oxygen with weight m  from the volume $V_1$ to the volume $V_2$.

Task 3. Define the entropy change of $\nu$ moles of gas, which is heated under isobaric conditions from the temperature $T_1$ to the temperature $T_2$.

Task 4. Define the entropy change of hydrogen with mass m, heated under isochoric conditions from the temperature  $T_1$ to the temperature $T_2$ .

Task 5. Define the entropy change of water with weight of m, which is cooled from $T_1$$T_2$ .