Chapter 7 Thermodynamics and Entropy PDF

**Ch. 7** **Thermodynamics and Entropy** We summarized thermodynamic laws in Chapter 3 (**3.4**). Although the second law of thermodynamics may not seem obvious at first sight, it is one of the fundamental and far-reaching principles of universe. In this short chapter, we take a closer look upon it...

**Ch. 7** **Thermodynamics and Entropy** We summarized thermodynamic laws in Chapter 3 (**3.4**). Although the second law of thermodynamics may not seem obvious at first sight, it is one of the fundamental and far-reaching principles of universe. In this short chapter, we take a closer look upon it. **7.1 Review of Thermodynamic Laws (3.4)** Followings are the laws of thermodynamics. The [zeroth law of thermodynamics](https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics); *If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.* The [first law of thermodynamics](https://en.wikipedia.org/wiki/First_law_of_thermodynamics); *In a process without transfer of matter, the change in [internal energy](https://en.wikipedia.org/wiki/Internal_energy)* [*ΔU*]{.math.inline} *of a [thermodynamic system](https://en.wikipedia.org/wiki/Thermodynamic_system) is equal to the energy gained as heat*[*Q*]{.math.inline} *less the thermodynamic work* [*W*]{.math.inline} *done by the system on its surroundings.* [*ΔU* = *Q* − *W*]{.math.inline} (energy conservation) The second law of thermodynamics; *Total entropy change associated with thermal contact between systems is always positive*. Alternatively, one may state; *Heat does not spontaneously flow from a colder body to a hotter body.* (increase of entropy, entropy = degree of disorder) The [third law of thermodynamic](https://en.wikipedia.org/wiki/Third_law_of_thermodynamics); *As the temperature of a system approaches absolute zero, all processes cease and the entropy of the system approaches a minimum value.* **7.2 Entropy Change in Thermodynamic Process** The second law of thermodynamics is associated with entropy. And it states that **the entropy of any isolated system (eventually that of the whole universe) is always increasing**. Entropy relates to spontaneity i.e.; the more spontaneity exists in a process, the higher is its entropy. There are other alternative expressions about the fact that entropy cannot decrease. Entropy is a thermodynamic state quantity, i.e., function of temperature, pressure, and volume. In a chemical reaction, the change in entropy occurs as a result of the rearrangement of atoms and molecules of the system. The Entropy change is represented as ΔS, which can be calculated by the following formula; \ [\$\$\\mathbf{\\mathrm{\\Delta}S = \\ }\\frac{\\mathbf{\\text{dQ}}}{\\mathbf{T}}\$\$]{.math.display}\ where, *dQ* is heat transferred in the thermodynamic system and *T* is its absolute temperature. The SI unit of entropy is \[J/K\] (or often \[J/mol/K\]). Clausius discovered the concept of entropy from his study of the steam engine. This change in entropy formula provides an idea about the spontaneity of a process or a chemical reaction. For a spontaneous process, there is an increase in entropy leading to total amount of ΔS being greater than zero. Let us take an example. Suppose water 100 gram of [*T* = 10 *C*]{.math.inline} is mixed with water 100 gram of [*T* = 90 *C*]{.math.inline}, and they soon become 200 gram of [*T* = 50 *C*]{.math.inline}. What is the entropy change during this process? (hint: the heat capacity of water = 1 \[cal/gram/K\] = 4.184 \[J/gram/K\], T(K)=T(C)+273) Gain of entropy for cool water (take its average temperature as [*T* = 30 *C*]{.math.inline}) is approximately; [\$\\mathrm{\\Delta}S = \\ \\frac{\\text{dQ}}{T} = \\frac{100 \\times 40 \\times 4.184}{273 + 30} = 55.234\\ \$]{.math.inline}\[J/K\] Loss of entropy for hot water (take its average temperature as [*T* = 70 *C*]{.math.inline}) is approximately; [\$\\mathrm{\\Delta}S = \\ \\frac{\\text{dQ}}{T} = \\frac{- 100 \\times 40 \\times 4.184}{273 + 70} = - 48.793\\ \$]{.math.inline}\[J/ K\] Then the total entropy change during the mixing process is [*ΔS* = 55.234 − 48.793 = 6.44]{.math.inline} \[J/K\] (exact calculation using [\$\\int\_{}\^{}{\\frac{\\text{dT}}{T} = \\ln T}\$]{.math.inline} yields [*ΔS* = 55.315 − 48.848 = 6.47]{.math.inline} \[J/K\]) Now that you understand the idea, can you imagine any process which would result in entropy decrease? **7.3 Definition of Entropy from Statistical Mechanics** Entropy is a measure of disorder or randomness, and entropy change can be defined as the change in the state of disorder of a system. Simply a system with a high degree of disorder has more entropy. Boltzmann's formula for entropy is given as; \ [*S* = *k*ln *W*]{.math.display}\ where *W* is the number of ways the [atoms](https://en.wikipedia.org/wiki/Atom) or [molecules](https://en.wikipedia.org/wiki/Molecule) of a certain kind of [thermodynamic system](https://en.wikipedia.org/wiki/Thermodynamic_system) can be arranged (called degeneracy function) and *k* is the Boltzmann constant; [*k* = 1.38065 × 10^ − 23^ ]{.math.inline}\[J/K\]. The calculation of entropy by using this formula is often formidable except certain ideal situations. Let us take an example to assess the entropy and its change based on the Boltzmann formula. Consider a tiny system composed of three particles only and each particle's energy can be represented by up/down state. [↑]{.math.inline} : up state with energy [ + *ΔE*]{.math.inline} [↓  ]{.math.inline}: down state with energy [ − *ΔE*]{.math.inline} If the system has total energy of [*ΔE*]{.math.inline}, then we have *W*=3 for this system, because there are three possible states as; [↑ ↑ ↓ ]{.math.inline}, [↑ ↓ ↑]{.math.inline} , and [↓ ↑ ↑ ]{.math.inline}. Therefore, the entropy of this system would be [*S* = *k*ln 3]{.math.inline}. Now consider another tiny system (isolated from the first system) composed of four same particles and its total energy is zero. Then we find *W*=6 for this second system, because there are six possible states as; [↑ ↑ ↓ ↓ ]{.math.inline}, [↑ ↓ ↑ ↓]{.math.inline} , [↑ ↓ ↓ ↑ ]{.math.inline} , [↓ ↑ ↓ ↑]{.math.inline} , [↓ ↓ ↑ ↑]{.math.inline} , [ ↓ ↑ ↑ ↓]{.math.inline}. Therefore, the entropy of this second system would be [*S* = *k*ln 6]{.math.inline}. The total entropy of both systems should be [*S* = *k*ln 3 + *k*ln 6 = *k* *ln*18]{.math.inline}. ([ln 3 = 1.0986]{.math.inline} , [ln 6 = 1.7918]{.math.inline} , [ln 18 = 2.8904]{.math.inline}) Now, we take the two system in thermal contact, so that their total energy is [*ΔE* + 0 = *ΔE*]{.math.inline} but the degeneracy function changes. One may see clearly that the degeneracy function of the combined system should be *W*=[ C~4~^7^ = 35]{.math.inline}. Followings are two of the 35 possible states. \ [↑ ↑ ↓ ↑ ↓ ↑ ↓]{.math.display}\ \ [↑ ↑ ↑ ↓ ↓ ↑ ↓]{.math.display}\ Then the entropy of the combined system is; [*S* = *k*ln 35]{.math.inline} ([ln 35 = 3.5553]{.math.inline} \> [ln 18 = 2.8904)]{.math.inline}. Therefore, we find the entropy has increased through the combining process. **7.4** [*Δ***S**~universe~ **\>** **0**]{.math.inline} *In the shorthand of mathematics, Clausius chose* [*S*~universe~]{.math.inline} *to stand for the total entropy of the universe; the Capital Greek letter,* [*Δ*]{.math.inline} *to stand for 'the net change in... ;' and the symbol \> to stand for 'is always greater than... ' Therefore, his startling result boiled down to this equation:* \ [*ΔS*~universe~ \> 0]{.math.display}\ *In plain English: 'The net change in the total entropy of the universe is always greater than zero.' That is, at any given moment, the Strurm and Drang of existence always left the universe with more entropy than it had the moment before, the positive entropy changes always exceed the negative ones. - Michael Guillen in Five Equations that Changed the World* **7.5 Temperature Scales** Most common temperature scales are Celsius, Fahrenheit, and Kelvin scales. In Celsius scale, zero degree was defined as the freezing temperature of water, while 100 degree was defined as boiling temperature of water (both under the common atmospheric pressure; 1 atm, which is [1.013 × 10^5^Pa]{.math.inline}=1013hPa). Conversion between temperature scales; \ [\$\$T\\left( F \\right) = \\frac{9}{5}\\text{\\ T}\\left( C \\right) + 32\$\$]{.math.display}\ \ [*T*(*K*) = *T*(*C*) + 273.15]{.math.display}\ As an example, 0 degree in Celsius corresponds to 32 degrees in Fahrenheit and 273 degrees in Kelvin. At zero degree in Kelvin (or simply 0 K), all the movements of molecules and atoms stop -- i.e. zero kinetic energy! **7.6 Ideal Gas Law** The volume of ideal gas molecule is negligible to zero. And usual air of the atmosphere can be approximately treated as ideal gas, for the volumes of nitrogen and oxygen are quite small. The ideal gas law is given as; \ [*PV* = *NkT* = *nRT*]{.math.display}\ where P, V, and T are the pressure, volume, and absolute temperature of the gas. N is the number of gas molecule. k is the Boltzmann constant; [1.38065 × 10^ − 23^]{.math.inline}J/K. R is the ideal gas constant; 8.314 J/mol/K. 1 mol=[6.02214 × 10^23^]{.math.inline}, which is called Avogadro number and is the number of hydrogen atom of 1 gram. Consider a small volume consisted of N ideal gas molecule (mass of each molecule = m, temperature T(K)) (fig). The momentum of a gas molecule moving with [*v*~*x*~]{.math.inline}, is simply [*mv*~*x*~]{.math.inline} and each time it collide the right-side wall, it will deliver a momentum of [2*mv*~*x*~]{.math.inline} (elastic collision). And the number of this collision in one second would be [\$\\frac{v\_{x}}{2L}\$]{.math.inline}. Therefore, the average impulse(momentum) exerted by a molecule to the side would be [\$\\frac{mv\_{x}\^{2}}{L}\$]{.math.inline}. And the average pressure exerted is [\$\\frac{\\frac{m{\< v}\_{x}\^{2} \>}{L}}{L\^{2}} = \\frac{m \< v\_{x}\^{2} \>}{V}\$]{.math.inline}. For N molecules, the exerting average pressure is simply; [\$N\\frac{mv\_{x}\^{2}}{V}\$]{.math.inline}. About the gas molecules movement, it can be written as; [*v*^2^ = *v*~*x*~^2^ + *v*~*y*~^2^ + *v*~*z*~^2^]{.math.inline}. Assuming no preferred direction we have [ \  =  \  =  \ and so [ \ = 3 \ .]{.math.inline} \ [\$\$PV = \\frac{N}{3}m \< v\^{2} \> \\ \$\$]{.math.display}\ Compare this with the ideal gas law, we have \ [\$\$kT = \\frac{1}{3}\\ m \< v\^{2} \> \\ \$\$]{.math.display}\ Similarly, we have \ [\$\$v\_{\\text{rms}} = \\sqrt{\\frac{3kT}{m}}\$\$]{.math.display}\ Here, we check a few amounts; \ [\$\$R = N\_{\\text{avo}}\\ k = 6.02214 \\times 1.38065 = \> 8.3145\\ \\lbrack\\frac{J}{\\frac{\\text{mol}}{K}}\\rbrack\$\$]{.math.display}\ Average speed of oxygen molecule in a room(T=20C); \ [\$\$v\_{\\text{rms}} = \\sqrt{\\frac{3 \\times 1.38065 \\times (273 + 20)}{2 \\times 16/6.02214} \\times 1000} = 478\\ \\lbrack m/s\\rbrack\$\$]{.math.display}\ \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-- **References** College Physics, Serway, Cengage Learning Thermal Physics, Kittel, Freeman Five Equations that Changed the World, Michael Guillen, Hyperion NY 17 Equations that Changed the World, Ian Stewart, Basic Books NY Thermodynamics; [[https://en.wikipedia.org/wiki/Thermodynamics]](https://en.wikipedia.org/wiki/Thermodynamics) \* The content of this chapter is abridged from the books and other sources as well as Wikipedia. **\[Homework\]** \[1\] Check the entropy change in **7.2** again for yourself (describe in detail).