Preface
有些东西【只是数学关系】。
有些东西比起【数学形式】更加需要思考【物理意义】.
工程原则：简化测量，简化运算，自圆其说。
每一个“非自然变量”(除 V, T, N等等可以被自然获得和感知)，都是我们对每一件事背后真相的探索的里程碑。
中间一大段中文懒得翻译了反正不会有人看，就扔在这里吧。
Basics Concepts
Steady state = No accumulation
Equilibrium = No change with time
State Function = Path Independent
Intensive/Extensive Properties = Whether properties change with mass/amount
Simple System assumption is necessary for almost all conclusion we make.
State postulate，Intensive state
PV图上都是可逆过程，表示不出不可逆过程，不可逆过程最好看TS图。
Ideal Gas Law
For the famous formula $PV=RT$ , $V$ is molar volume whose unit is $L/mol$.
Though mathematically it is extra step (since the exercises always give $\underline{V}$ and $N$ instead of $V$ directly. But the physical meaning is that in collaborate two Extensive Properties into a single Intensive property. For condition of well-mixed open system, we don’t have to consider the mole change if we use $V$.
First Law
Remember that U is state function and W,Q are process variable
Reversibility, Dissipation, Quasi-static are better to be discussed in Second Law.
For MATHEMITICAL CONVENIENCE，we then introduce $C_V$, $C_P$
Calculation of internal energy
$U=U(T,V)$,($T$ is the macroscopic value measuring average molecular kinetic energy，$V$ is the macroscopic value measuring average molecular kinetic energy)
Here we still don’t know the expression for internal energy, but we can use the approach of 【Derivative and Integration】, if we are not able to measure the whole value directly, then somehow we measure the changes.
$dU=(\frac{\partial U}{\partial T})_V+(\frac{\partial U}{\partial V})_T$ . Then for 【Ideal Gas】$U=U(T)$，$(\frac{\partial U}{\partial V})_T=0$
Therefore $(\frac{\partial U}{\partial T})_V \equiv C_V$
Though it is called constant-volume heat capacity，$dU=C_V dT$ not only applicable to isochoric condition. It is a UNIVERSAL expression. It is called CONSTANT-VOLUME only because when we calculate partial derivative, we need to set other variable to be constant.
To understand its physical meaning, in real life （and in most exercise） processes are irreversible. For two state of ideal gas $U_1, U_2$，the core of state function is PATH INDEPENDENT. Therefore we can imagine an isobaric reversible process and calculate the change of internal energy. It is same as the irreversible process, since they have same initial and final state.
Calculation of Work
Remember that W is path independent. So $𝛿𝑊 = −𝑃𝑑\underline{V}$ only applicable for Quasi-static。If it is sudden expansion, $P$ is environment pressure.
Calculation of Heat
The trick of First Law lies in that you cannot calculate heat directly but only through First Law.
Enthalpy
高中化学选修四对于所有的反应热都用的是$\Delta H$，但是我记得的是一定有恒压这个条件，反应焓变才等于反应热。上了大学之后才开始真正思考焓变到底是什么。
最后发现这个问题压根没有意义。因为最好理解焓的方式就是把它当成一个为了【数学计算方便】而产生的量。
具体来说，就是我们经常需要计算Q，所以我们就经常要算U+PV，嫌麻烦所以有了H，同样地，为了计算H的方便，所以才有了$C_P$，有了$dH=C_PdT$
刚好在open system的first law也可以写成H的形式，形式简单，而且可以用$C_P$计算，何乐而不为。
Tricks
For transient system, remember we can convert $d\underline{U}$ to $dU$ and $dN$ and then integrate.
Second Law
Second Law mainly discussed about the direction of process, and how we define, describe and measure the process.
Heat engine and Carnot
Heat engine = convert HEAT to WORK
Carnot engine set up the maximum of efficiency for heat engine.
卡诺热机的公式以及重要的$ \frac{|Q_H|}{|Q_C|} = \frac{T_H}{T_C} $。其实仔细想想这个公式还挺神奇的，Q是过程量，T是状态量，却能够在数量关系划等号。中间有一个重要的【数学】步骤就是四个状态的体积的比例关系。
这个体积关系在PV图上的几何意义是，不同等温线上的两点连线却是平行的。所以说一开始学卡诺的时候觉得很奇怪，因为以前至知道在PV图上横平竖直地走，比如迪赛尔，卡诺开始却要斜着跨过等温线，没有地方是直线的，而现在看来，“直线”其实是隐藏在其中的。
另外关于卡诺热机为什么是天花板，不用反证法的话，这个解答的解释比较好——不能从PV图（因为PV图只能反映可逆过程），而是要从TS图中找到答案。
Entropy
做定性分析，熵衡量混乱程度，越混乱则熵越大。
顺带一提，熵是一个状态量，这也很好理解，熵就是一个状态，该有多乱有多乱，无论怎么变来变去，有这么乱就是这么乱。
这样很好理解，但是不定量分析的话，我们就无法衡量一个系统的混乱度有没有变化。
微观的熵的表达式，$\underline{S}=k_B ln \Omega$ 其中$\Omega$ 是微观状态的数目。对这个公式定性分析的话很好理解，可能情况越多那就意味着越混乱，关于怎么和宏观联系起来我还在思考。
宏观的乍一看不知道怎么做，那就先假设有这样找一个熵不变的过程，找它的起始和终点，然后看看这个过程中有什么量不变，那它就是熵了。
但是再我们量化熵之前，我们根本没法比较两个状态的熵是不是一样了，那我们只能跟原状态比较——既然熵是状态量，和自己比那肯定一样了吧。
那这样一个起点是自己、终点也是自己的过程，我们叫做一个cyclic process。于是，我们要找一个cyclic process中一个不变的状态量。
很巧的是，卡诺循环是个cyclic process，卡诺定理我们发现 $\delta Q/T$ 是不变的，所以我们定义为熵。
（这个自圆其说看上去强词夺理甚至有点无厘头，但是是我到此为止的一个理解了）
虽然有了定义式，但是熵的计算也要讲究，一定要分成系统和reservoir两种情况，原因且看下文。
Calculation of Second law
回到最初的问题，我们要看系统的混乱度也就是熵有没有变化，那既然有了个等式就可以计算了。
前人总结出了second law的文字表述，再把这个意思变成公式，一句话概括second law的公式就是【孤立系统熵增】。用公式表达就是$d\underline{S}>=0$
但是，我们生活中哪里有那么多孤立系统，那没办法，对于open system，只好把inflow和outflow带来的熵变、还有所研究的系统以外的整个宇宙的熵变也考虑进去，这样我们才能算出来系统的熵变。
关于system本身的熵变，可以假设system进行的是reversible(in this case, quasi-static)的过程，这样 就可以用W=PV计算功，再假设ideal gas，U也可以算出来，就可以算出Q，就可以算出S。
这个时候就要考虑到系统以外的整个宇宙——也就是所谓的环境，对于系统来说也就是reservoir——的熵变。直接算是不可能的，因为公式里的$\delta Q$ 得要用 $d\underline{U}$ 算吧，$d\underline{U}=NC_VdT$，对那么大的环境（宇宙）来说，$N\rightarrow \infty, dT\rightarrow 0$，没有办法直接算。
所以看系统给环境多少Q。
但是系统算 $Q$ 是我们假设reversible了，但是环境我们怎么知道是不是reversible呢。于是这里在note里定义了环境是mechanical reversible，这样就可以保证系统的 $Q$ 可以代进去算了。
至于inflow和outflow，就分别算$S{in}n{in}$, $S{out}n{out}$就好了（题目会给）（装死.jpg）
Exergy
让我们先回到卡诺热机，为什么会有这个天花板在？有人就提出了，能量不仅有多少（量）也是有优劣（质）的。
能量“质”的指标是根据它的做功能力来判断的，因此，可以根据能量转换能力将能量分为三种来行。
- 可以完全转换的能量，如机械能、电能等。理论上可以百分之百地转换成其他形式地能量。这种能量的“量”和“质”完全统一，它的转换能力不受约束。
- 可部分转换的能量，如热量、内能等，这种能量的“量”和“质”不完全统一，它的转换能力受热力学第二定律约束。
- 不能转换的能量，如环境内能，这种能量只有“量”没有“质”。
（摘抄自《工程热力学》）
于是除了按照能量的产生和存在分为机械能、电能、内能等以外，我们又有了新的区分能量的办法。（但是要注意这里还是人为的定义，是能源学科为了区分能源质量做出的定义。）
能量=㶲+㷻
Energy=Exergy+Anergy
问题来了，我们怎么知道有用能——㶲有多少呢。
㶲其实就是卡诺热机做了多少功。
因为卡诺热机实际不存在，首先的首先我们不可能直接测出来。
卡诺热机是个可逆热机，是把内能转化成机械能等其他能量的。气体膨胀的PV公式是不能用的（为什么），没办法直接算，所以我们要用First Law算。但是我们没办法知道Q有多少，但是这个时候我们有Second Law， 我们可以得出一个Steady state, Reversible的系统有$\frac{\dot{Q}{rev}}{T}+(S{in}-S{out})\dot{n}=0$ ，这样我们知道了$S{in}$和$S_{out}$就可以算$Q$了。这样代回First Law我们就可以算卡诺热机的功了。
解释这个式子，就是inflow其实有点能量，但是outflow有更多㶲，过程可逆（没有能量耗散），所以这个差值乘 $\dot{n}$ 就是卡诺热机的有用功。也就是所谓的ideal shaft work。
同时，我们把个表达式定义为㶲 $X$ ，在【数值上】等于$H-TS$。
（我目前还没有get到这个式子的物理含义，因为我就没有get到H的物理意义，也没有get到S=δQ/T的物理意义，就姑且认为只是数学公式）
Second efficiency
如果说之前的efficiency我们在讨论总能量里的效率的话，这里讨论的就是相对于㶲的效率
Non ideal gases
勒让德变换——where do H, A, G, F come from
自由能，它限制了系统对外可做的总功。
吉布斯自由能，它限制了系统对外可做的非体积功。
反过来说，无需外界做功驱动的热力学过程一定有dF<=0; 无需外界做非体积功驱动的热力学过程，一定有dG<=0。
后者包括各类在恒定气压、恒定温度的环境中发生的化学反应，因为化学反应中物质改变体积，体积功是我们所不可控、也不用操心考虑的。故而吉布斯自由能可以作为化学反应能否在热力学意义上发生的判据。
我们永远没可能达到卡诺热机的效率，但是既然我们知道了天花板在哪里，我们就能不断逼近。这就是人类追求卓越的历程。
Phases
In the first half semester we have already discussed about purity(mainly gases) and mostly about intensive properties. Then we turn to liquid, and mixtures and how their properties change if we change their P, T, V etc.
First thing to talk about is the deviation of liquid from ideal gas. We still hope to use ideal gas law (it is so convenient), therefore:
(1) we introduce $\beta$ and $\kappa$ mainly to discuss the incompressibility of liquid.
(2) since we usually regard liquid can not be compressed arbitrarily, we realize that the molar volume cannot be determined through ideal gas law. Empirical formula help us measure molar volume for liquid.
(We seldom discuss solid here)Since we already have two phases, we now discuss Phase Behavior. We want to determine the properties(P,T,V) of each phase
(1)First of all Gibbs’ Phase Rule 𝐹 = 2 + 𝑛 − 𝜋 − 𝑟 tell us how many intensive properties we need to determine multiple phases(and even multiple components)
(2)We need to know the T or V or P where the two phases interchange. That gives us $P^{sat}$ (saturated)vapor pressure (can be determined through empirical formula called Antoine Equation), $T^{sat}$, or so-called boiling point.
(3) We have known the boundary, then when the substance goes across the boundary of phases, it need to absorb/exert heat, which is called latent heat. It can be determined through Clapeyron equation and its empirical correlation.
(4) We learn from Chemistry before that everything process is actually incomplete. Somehow equilibrium exists. It is so for chemical reaction and same for phases interchange. Equilibrium means that even S does not change. By expressing $d\underline{S}=0$, we get $(\frac{\underline{G}^L}{T^L}-\frac{\underline{G}^V}{T^V})dN^L=0$
(这里有个因果关系，因为括号里等于零(能量差)（这里不是真正意义上的能量，但是单位是能量的单位），所以才有平衡，才有dN=0)
（还有一个就是我们之前定义的三个equilibrium在这里终于派上用场！其实我们想要的就是N平衡也就是diffusional equilibrium）
By the way we learn to read those complicated thermodynamics diagram. Mainly, P-T and P-V diagrams.
Mixture
For mixture, we (1) define $x_i, y_i$ for liquid and gas components respectively, and (2)develop another idealized model called ideal gas mixture. (3)Since the properties are not same each part of the mixture, we finally need to turn to extensive properties. We compare the fundamental equations we derive for extensive properties and those for intensive ones, we found that they are different by $(U+PV-TS)dN$. We call molar $U+PV-TS$ by Chemical potential(we will explain what does it physically mean later) Now we can express extensive properties with chemical potential, which is more convenient.
Then, how to find chemical potential for different states.
We know that we can express it in U, G, A etc, holding different variable constant. In real situation, it is easiest for us to hold T and P constant (so we choose G), correspondingly, we can express chemical potential of ideal gas as $ \mu^{ig}{i} = G{i}^{ig} + RTln(y_i) $ and similar form for ideal solution.
For gases, we have residual properties to compensate the difference from ideal gas. Similarly, for liquid, we have excess properties to compensate the difference from ideal solution.
Finally, we are able to determine the properties of components of mixtures.
Before ending, we will discuss about the physical meaning of chemical potential. In lecture notes, it write like this
• It is related to (but not exactly) the “concentration” of a substance. Matter moves from region of high chemical potential to low chemical potential.
• At diffusional equilibrium, the chemical potential is exactly balanced.
• Note the parallelism
(𝑇 : thermal equilibrium : heat)
(𝑃 : mechanical equilibrium : work)
(𝜇 : diffusional equilibrium : flow of matter).
Similar to voltage (or better understanding by Chinese terminology ‘electrical pressure’), current flow from high voltage to low voltage. Same as chemical potential. matter flows from high chemical potential to low chemical potential. Remind the change can be both chemical and physical (fundamentally, not much difference from the aspect of energy changes, but people like to differentiate them)
Fugacity
First of all, fugacity is a surrogate function for the chemical potential because the drawbacks of chemical potential as followings:
• It is negatively infinite when pressure approaches zero (i.e. at ideal gas limit).
• It is negatively infinite when the mole fraction approaches zero (i.e. at the infinite dilution limit).
• It is on an arbitrary scale like internal energy, and only the difference in chemical potential (not the absolute value) is meaningful.
We seek for more practical appliance of what we have defined before, unfortunately, chemical potential is hard to be measured directly, but it is so important to obtain its value for calculating. Therefore, engineers define another property called fugacity to replace it. (1) Fugacity is attributed with the unit of pressure, make it fairly easy to be measured. (2) Fugacity is defined as the deviation of chemical potential (RT and exponential can be simply treated as conversion of unit) from $\mu{ref}$ , where $\mu{ref}=\mu^{ig}_i(T,P=1bar,y_i=1)=G^{ig}_i(T,P=1bar,y_i=1)$ Again, we get back to G, we can be measured with the help of properties relation. Then we only need to measure fugacity to everything we want.
The measurement of fugacity is divided to many cases. (1) For ideal gas, it simply equals to pressure at the same state
(2) for non-ideal gas, a revise parameter or fugacity coefficient $\phi$ is multiplied to pressure
(3) For liquid, f is equal to vapor pressure at this temperature.
(4) For ideal mixtures, just multiplied by component. (remark: there is an idea called ‘gaseous ideal solutions’)
(5) For gases in non ideal mixtures, expression is more or less the same, while for liquids $\hat{f_i}=a_if_i$ , where $a_i$ is the activity of i. It can also be expressed in $a_i = x_i\gamma_i$ , where $\gamma_i$ the activity coefficient of 𝑖.
Eventually, our approach to get the macroscopic properties of mixtures work, now we are going to analyze through, diagram. We look into P-x-y diagram and T-x-y diagram.
In real life, sometimes we cannot determine component of mixture directly. Raoult’s Law, which illustrate the relationship of $y_i, x_i, P, P^{sat}$ (where pressure are easily obtained)give a convenient way to obtain component of Binary mixture. Basically, we have the Raoult’s Law for two of the components. Two unknown variables and two equations. We can easily solve for unknowns. Remark that Raoult’s Law has a bunch of restrictions so take care when applying.
If we have more than two component, we can conduct what is called Flash Calculation, in which process we introduced variable $ℒ, 𝒱$ , and $z_i=ℒx_i+𝒱y_i$ . I personally regards it as mathematical tricks. So I will skips the formula here.
We mentioned the strict restrictions for Raoult’s Law. For air and many other gases they actually violates the law. Therefore we introduce Henry’s Law, which is in similar form to Raoult’s Law, but $P^{sat}$ replaced by $ℋ_i$, Henry’s Constants, which we could find through literature.
Realizing the strict condition of Raoult’s Law, we modified it for liquid a bit with $\gamma_i$.
By the way for Azeotropes we have $x_i=y_i$
Chemical reaction and equilibrium
Finally, we chemical engineers build up the bridge between mechanical engineering and chemistry. We are to analyze how energy affect reaction. Actually this part should be the focus of another course called “reaction engineering” and now we are pathing our way to it.
First, combining stoichiometry (actually, mass balance) and 1st law (energy balance) and require $d\underline{S}=0$, we realize that Gibbs Free Energy and chemical potential imply the direction of reaction. (we may have already known this in high school, but we only learned about the conclusion but don’t know why then). Then we define equilibrium constant K, which can be obtained by standard Gibbs free energy of reaction $\Delta_rG^0$, which can be determined through Gibbs free energy of formation and combustion
Le Châtelier’s Principle(very simple so simply skip it this topic)