User:Retired Pchem Prof/Sandbox01

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Starting work here on some entropy related issues.

Issues to be addressed

In the article on Entropy under "Entropy change formulas for simple processes" there are formulas given with no derivation or references.

Proposed actions:

(1) Insert references

(2) Insert derivations for the cases where the basic fomula can be directly applied (heating/cooling, phase transitions)

(3) Add links to articles with derivations for more complex cases that don't belong here (isothermal expansion).

(4) Add a section to the Isothermal process article on how to calculate entropy change for the reversible isothermal expansion of an ideal gas. There is already a version of this in the Joule expansion article that should be moved, with a link and short explanation left in the Joule expansion article.

This has now been done.

In the article on Entropy production under "Examples" it says: "The expression for the rate of entropy production in the first two cases will be derived in separate sections." Those being "heat flow through a thermal resistance" and "fluid flow through a flow resistance such as in the Joule expansion or the Joule-Thomson effect". There are no links. Results for the first and the Joule expansion are given, without derivation or references, under "Expressions for the entropy production".

There is no explanation of how to calculate $\Delta S_{\mathrm {univ} }$. Overall, the article needs a lot of work. Very little on talk page, (backwater?), but it is a "requested article". There is a more qualitative article on Irreversible process.

Proposed actions:

(5) For heat flow, include a derivation (and references) using the appropriate results from the entropy article and explaining how the reversible result can be applied to an irreversible process.

(6) For Joule expansion add references, explain how the result from the isothermal expansion article can be used, clarify that this is only for ideal gases, explain relation to Joule-Thomson expansion.

Entropy changes

This section has been inserted in the article on Isothermal process.

Isothermal processes are especially convenient for calculating changes in entropy since, in this case, the formula for the entropy change, $\Delta S$, is simply

$\Delta S={\frac {Q_{\text{rev}}}{T}}$

where $Q_{\text{rev}}$ is the heat transferred reversibly to the system and $T$ is absolute temperature. This formula is valid only for a hypothetical reversible process; that is, a process in which equilibrium is maintained at all times.

A simple example is an equilibrium phase transition (such as melting or evaporation) taking place at constant temperature and pressure. For a phase transition at constant pressure, the heat transferred to the system is equal to the enthalpy of transformation, $\Delta H_{\text{tr}}$, thus $Q=\Delta H_{\text{tr}}$. At any given pressure, there will be a transition temperature, $T_{\text{tr}}$, for which the two phases are in equilibrium (for example, the normal boiling point for vaporization of a liquid at one atmosphere pressure). If the transition takes place under such equilibrium conditions, the formula above may be used to directly calculate the entropy change

$\Delta S_{\text{tr}}={\frac {\Delta H_{\text{tr}}}{T_{\text{tr}}}}$.

Another example is the reversible isothermal expansion (or compression) of an ideal gas from an initial volume $V_{\text{A}}$ and pressure $P_{\text{A}}$ to a final volume $V_{\text{B}}$ and pressure $P_{\text{B}}$. As shown in Calculation of work, the heat transferred to the gas is

$Q=-W=nRT\ln {\frac {V_{\text{B}}}{V_{\text{A}}}}$.

This result is for a reversible process, so it may be substituted in the formula for the entropy change to obtain

$\Delta S=nR\ln {\frac {V_{\text{B}}}{V_{\text{A}}}}$.

Since an ideal gas obey's Boyle's Law, this can be rewritten, if desired, as

$\Delta S=nR\ln {\frac {P_{\text{A}}}{P_{\text{B}}}}$.

Once obtained, these formulas can be applied to an irreversible process, such as the free expansion of an ideal gas. Such an expansion is also isothermal and may have the same initial and final states as in the reversible expansion. Since entropy is a state function, the change in entropy of the system is the same as in the reversible process and is given by the formulas above. Note that the result $Q=0$ for the free expansion can not be used in the formula for the entropy change since the process is not reversible.

The difference between the reversible and free expansions is found in the entropy of the surroundings. In both cases, the surroundings are at a constant temperature, $T$, so that $\Delta S_{\text{sur}}=-Q/T$; the minus sign is used since the heat transferred to the surroundings is equal in magnitude and opposite in sign to the heat, $Q$, transferred to the system. In the reversible case, the change in entropy of the surroundings is equal and opposite to the change in the system, so the change in entropy of the universe is zero. In the free expansion, $Q=0$, so the entropy of the surroundings does not change and the change in entropy of the universe is equal to $\Delta S$ for the system.

Entropy change formulas for simple processes

From the article on Entropy.

For certain simple transformations in systems of constant composition, the entropy changes are given by simple formulas.

Isothermal expansion or compression of an ideal gas

For the expansion (or compression) of an ideal gas from an initial volume $V_{0}$ and pressure $P_{0}$ to a final volume $V$ and pressure $P$ at any constant temperature, the change in entropy is given by:

$\Delta S=nR\ln {\frac {V}{V_{0}}}=-nR\ln {\frac {P}{P_{0}}}.$

Here $n$ is the number of moles of gas and $R$ is the ideal gas constant. These equations also apply for expansion into a finite vacuum or a throttling process, where the temperature, internal energy and enthalpy for an ideal gas remain constant.

Cooling and heating

For heating or cooling of any system (gas, liquid or solid) at constant pressure from an initial temperature $T_{0}$ to a final temperature $T$, the entropy change is

$\Delta S=nC_{P}\ln {\frac {T}{T_{0}}}$.

provided that the constant-pressure molar heat capacity (or specific heat) C_P is constant and that no phase transition occurs in this temperature interval.

Similarly at constant volume, the entropy change is

$\Delta S=nC_{v}\ln {\frac {T}{T_{0}}}$,

where the constant-volume heat capacity C_v is constant and there is no phase change.

At low temperatures near absolute zero, heat capacities of solids quickly drop off to near zero, so the assumption of constant heat capacity does not apply.

Since entropy is a state function, the entropy change of any process in which temperature and volume both vary is the same as for a path divided into two steps - heating at constant volume and expansion at constant temperature. For an ideal gas, the total entropy change is

$\Delta S=nC_{v}\ln {\frac {T}{T_{0}}}+nR\ln {\frac {V}{V_{0}}}$.

Similarly if the temperature and pressure of an ideal gas both vary,

$\Delta S=nC_{P}\ln {\frac {T}{T_{0}}}-nR\ln {\frac {P}{P_{0}}}$.

Phase transitions

Reversible phase transitions occur at constant temperature and pressure. The reversible heat is the enthalpy change for the transition, and the entropy change is the enthalpy change divided by the thermodynamic temperature. For fusion (melting) of a solid to a liquid at the melting point T_m, the entropy of fusion is

$\Delta S_{\text{fus}}={\frac {\Delta H_{\text{fus}}}{T_{\text{m}}}}.$

Similarly, for vaporization of a liquid to a gas at the boiling point T_b, the entropy of vaporization is

$\Delta S_{\text{vap}}={\frac {\Delta H_{\text{vap}}}{T_{\text{b}}}}.$

Expressions for the entropy production

From the article on Entropy production

Suitable links needed from Entropy production#Examples of irreversible processes.

Heat flow

In case of a heat flow ${\dot {Q}}$ from T₁ to T₂ the rate of entropy production is given by

${\dot {S}}_{i}={\dot {Q}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right).$

If the heat flow is in a bar with length L, cross-sectional area A, and thermal conductivity κ, and the temperature difference is small

${\dot {Q}}=\kappa {\frac {A}{L}}(T_{1}-T_{2})$

the entropy production rate is

${\dot {S}}_{i}=\kappa {\frac {A}{L}}{\frac {(T_{1}-T_{2})^{2}}{T_{1}T_{2}}}.$

Flow of matter

In case of a volume flow ${\dot {V}}$ from a pressure p₁ to p₂

${\dot {S}}_{i}=-\int _{1}^{2}{\frac {\dot {V}}{T}}\mathrm {d} p.$

For small pressure drops and defining the flow conductance C by ${\dot {V}}=C(p_{1}-p_{2})$ we get

${\dot {S}}_{i}=C{\frac {(p_{1}-p_{2})^{2}}{T}}.$

The dependences of ${\dot {S}}_{i}$ on (T₁-T₂) and on (p₁-p₂) are quadratic. This is typical for expressions of the entropy production rates in general. They guarantee that the entropy production is positive.

Entropy of mixing

In this Section we will calculate the entropy of mixing when two ideal gases diffuse into each other. Consider a volume V_t divided in two volumes V_a and V_b so that V_t = V_a+V_b. The volume V_a contains n_a moles of an ideal gas a and V_b contains n_b moles of gas b. The total amount is n_t = n_a+n_b. The temperature and pressure in the two volumes is the same. The entropy at the start is given by

$S_{t1}=S_{a1}+S_{b1}.$

When the division between the two gases is removed the two gases expand, comparable to a Joule-Thomson expansion. In the final state the temperature is the same as initially but the two gases now both take the volume V_t. The relation of the entropy of n moles an ideal gas is

$S=nC_{V}\ln {\frac {T}{T_{0}}}+nR\ln {\frac {V}{V_{0}}}$

with C_V the molar heat capacity at constant volume and R the molar ideal gas constant. The system is an adiabatic closed system, so the entropy increase during the mixing of the two gases is equal to the entropy production. It is given by

$S_{i}=S_{t2}-S_{t1}.$

As the initial and final temperature are the same the temperature terms plays no role, so we can focus on the volume terms. The result is

$S_{i}=n_{a}R\ln {\frac {V_{t}}{V_{a}}}+n_{b}R\ln {\frac {V_{t}}{V_{b}}}.$

Introducing the concentration x = n_a/n_t = V_a/V_t we arrive at the well known expression

$S_{i}=-n_{t}R[x\ln x+(1-x)\ln(1-x)].$

Joule expansion

The Joule expansion is similar to the mixing described above. It takes place in an adiabatic system consisting of a gas and two rigid vessels (a and b) of equal volume, connected by a valve. Initially the valve is closed. Vessel (a) contains the gas under high pressure while the other vessel (b) is empty. When the valve is opened the gas flows from vessel (a) into (b) until the pressures in the two vessels are equal. The volume, taken by the gas, is doubled while the internal energy of the system is constant (adiabatic and no work done). Assuming that the gas is ideal the molar internal energy is given by U_m = C_VT. As C_V is constant, constant U means constant T. The molar entropy of an ideal gas, as function of the molar volume V_m and T, is given by

$S_{m}=C_{V}\ln {\frac {T}{T_{0}}}+R\ln {\frac {V_{m}}{V_{0}}}.$

The system, of the two vessels and the gas, is closed and adiabatic, so the entropy production during the process is equal to the increase of the entropy of the gas. So, doubling the volume with T constant, gives that the entropy production per mole gas is

$S_{mi}=R\ln 2.$

Entropy production

This is from Joule expansion. It would be more appropriate in Isothermal process where there are already calculation of W and Q.

It is awkward to calculate the entropy production in this process directly, because between the time the partition is opened and the time equilibrium is reached, the system passes through states that are far from thermal equilibrium. However, entropy is a function of state, and therefore the entropy change can be computed directly from the knowledge of the final and initial equilibrium states. For an ideal monatomic gas, the entropy as a function of the internal energy U, volume V, and number of moles n is given by the Sackur–Tetrode equation:

$S=nR\ln \left[\left({\frac {V}{N}}\right)\left({\frac {4\pi m}{3h^{2}}}{\frac {U}{N}}\right)^{\frac {3}{2}}\right]+{\frac {5}{2}}nR.$

In this expression m the particle mass and h Planck's constant. For a monatomic ideal gas U = (3/2)nRT = nC_VT, with C_V the molar heat capacity at constant volume. In terms of classical thermodynamics the entropy of an ideal gas is given by

$S(V,T)=S_{0}+nR\ln \left({\frac {V}{V_{0}}}\right)+nC_{V}\ln \left({\frac {T}{T_{0}}}\right)$

where S₀ is the, arbitrary chosen, value of the entropy at volume V₀ and temperature T₀. It is seen that a doubling of the volume at constant U or T leads to an entropy increase of ΔS = nR ln(2). This result is also valid if the gas is not monatomic, as the volume dependence of the entropy is the same for all ideal gases.

A second way to evaluate the entropy change is to choose a route from the initial state to the final state where all the intermediate states are in equilibrium. Such a route can only be realized in the limit where the changes happen infinitely slowly. Such routes are also referred to as quasistatic routes. In some books one demands that a quasistatic route has to be reversible, here we don't add this extra condition. The net entropy change from the initial state to the final state is independent of the particular choice of the quasistatic route, as the entropy is a function of state.

Here is how we can effect the quasistatic route. Instead of letting the gas undergo a free expansion in which the volume is doubled, a free expansion is allowed in which the volume expands by a very small amount δV. After thermal equilibrium is reached, we then let the gas undergo another free expansion by δV and wait until thermal equilibrium is reached. We repeat this until the volume has been doubled. In the limit δV to zero, this becomes an ideal quasistatic process, albeit an irreversible one. Now, according to the fundamental thermodynamic relation, we have:

$\mathrm {d} U=T\mathrm {d} S-P\mathrm {d} V.$

As this equation relates changes in thermodynamic state variables, it is valid for any quasistatic change, regardless of whether it is irreversible or reversible. For the above defined path we have that dU = 0 and thus TdS=PdV, and hence the increase in entropy for the Joule expansion is

$\Delta S=\int _{i}^{f}\mathrm {d} S=\int _{V_{0}}^{2V_{0}}{\frac {P\,\mathrm {d} V}{T}}=\int _{V_{0}}^{2V_{0}}{\frac {nR\,\mathrm {d} V}{V}}=nR\ln 2.$

A third way to compute the entropy change involves a route consisting of reversible adiabatic expansion followed by heating. We first let the system undergo a reversible adiabatic expansion in which the volume is doubled. During the expansion, the system performs work and the gas temperature goes down, so we have to supply heat to the system equal to the work performed to bring it to the same final state as in case of Joule expansion.

During the reversible adiabatic expansion, we have dS = 0. From the classical expression for the entropy it can be derived that the temperature after the doubling of the volume at constant entropy is given as:

$T=T_{i}2^{-R/C_{V}}=T_{i}2^{-2/3}$

for the monatomic ideal gas. Heating the gas up to the initial temperature T_i increases the entropy by the amount

$\Delta S=n\int _{T}^{T_{i}}C_{V}{\frac {\mathrm {d} T'}{T'}}=nR\ln 2.$

We might ask what the work would be if, once the Joule expansion has occurred, the gas is put back into the left-hand side by compressing it. The best method (i.e. the method involving the least work) is that of a reversible isothermal compression, which would take work W given by

$W=-\int _{2V_{0}}^{V_{0}}P\,\mathrm {d} V=-\int _{2V_{0}}^{V_{0}}{\frac {nRT}{V}}\mathrm {d} V=nRT\ln 2=T\Delta S_{gas}.$

During the Joule expansion the surroundings do not change, so the entropy of the surroundings is constant. So the entropy change of the so-called "universe" is equal to the entropy change of the gas which is nR ln 2.