Thermodynamics
[TOC]
Primer
Thermodynamics: Thermal macroscopic theory
The first law of thermodynamics 10.1 & work heat energy
First law of thermodynamics
- That part of the energy system related to thermal phenomena: the internal energy
- First law of thermodynamics:
\ [Q = \ Delta E + W \\ \ bar dQ = dE + \ bar dW \\ Q: heat from \ quad W absorbed outside: the external work done \]
- The first perpetual motion machine: do not need the outside world to provide energy to the system, but they can continue to do work outside
- The first law of thermodynamics another expression: the first perpetual motion is not possible
Internal capacity
Adiabatic system state function, regardless of the route
Gong
- In limited quasi-static process, the system volume $ V_1 \ rightarrow V_2 $, the system to the outside world to do the work are:
\[W=\int \bar dW=\int ^{V_2}_{V_1}pdV \]
- FIG pV: total work done by the system is equal to the entire area under the curve
- It is a reactive process variables, path-dependent
Heat and heat capacity
definition
Heat Q: thermal interaction between systems (or due to a temperature difference) of the energy transfer
Heat capacity C:
\[ C=\lim_{\Delta T\rightarrow 0}\frac{\Delta Q}{\Delta T}=\frac{\bar dQ}{dT} \]
- Specific heat capacity c:
\[ c=\frac{C}{m} \]
Molar heat capacity: heat capacity material 1mol
Molar heat capacity for $ C_ {p, m} $:
\[ C_{p,m}=\lim_{\Delta T\rightarrow 0}\left( \frac{\Delta Q}{\Delta T}\right)_p =\left( \frac{\bar d Q}{dT}\right)_p \]
- Molar constant volume heat capacity $ C_ {V, m} $:
\[ C_{p,m}=\lim_{\Delta T\rightarrow 0}\left( \frac{\Delta Q}{\Delta T}\right)_V =\left( \frac{\bar d Q}{dT}\right)_V \]Relationship between heat capacity at constant pressure and constant volume heat capacity
- Meyer formula:
\ [C_ {p, m} = C_ {V, m} + R \\ [generally used with equipartition of energy principle: C_ {V, m} = \ frac {i} {2} R] \]
- Specific heat ratio:
\[ \gamma=\frac{C_{p,m}}{C_{V,m}} \]
10.2 & the first law of thermodynamics applied to 3
Process of change Summary
| process | Feature | Process equation | Q | W | \ (\ Delta E \) |
|---|---|---|---|---|---|
| Peers | \(V=c\) | \(\frac{P}{T}=c\) | \(\frac{m}{M}C_v(T_2-T_1)\) | 0 | \(\frac{m}{M}C_v(T_2-T_1)\) |
| Isobaric | \(p=c\) | \(\frac{V}{T}=c\) | \(\frac{m}{M}C_p(T_2-T_1)\) | \(P(V_2-V_1)\\\frac{m}{M}R(T_2-T_1)\) | \(\frac{m}{M}C_v(T_2-T_1)\) |
| isothermal | \(T=c\) | \(PV=c\) | \(\frac{m}{M}RTln\frac{V_2}{V_1}\\\frac{m}{M}RTln\frac{P_1}{P_2}\) | \(W_r=Q_r\) | 0 |
| Heat insulation | \(Q=0\) | \ (PV ^ \ gamma = C_1 \ quad [Poisson Equation] \\ TV ^ {\ gamma-1} = C_2 \\\ frac {p ^ {\ gamma-1}} {T ^ \ gamma} = C_3 \ ) | 0 | \(\frac{1}{\gamma-1}(p_1V_1-p_2V_2)\) | \(-W\) |
Multi-party process
Process equation
\ [PV ^ n = Constant \]Multi-index: n
- When n = 0, represents isobaric process
- When n = 1, represents isothermal process
- n = $ \ when infty $, isometric process represents
- n = $ \ when gamma $, represents adiabatic process
The formula of the multi-function process
\(W=\frac{1}{n-1}(p_1V_1-p_2V_2)\)
Polytropic process molar heat capacity formula
\(C_m=\frac{\gamma-n}{1-n}C_{V.m}\)
- If $ n \ in (1, \ gamma) $, the work done is larger than the gas outside the heat it absorbs, to reduce the
- If $ n \ notin (1, \ gamma) $, the work done is smaller than the external gas it absorbs heat, can increase the
10.4 & Carnot cycle and cycle
Heat engine concepts
Thermodynamic cycle
- Thermodynamic cycle: a system starting from a certain state, through a series of process back to the original state of the process
- FIG circulating pV:
- The area surrounded by the closed loop of the loop process made equal in value to the net work
- Clockwise: positive cycle, thermal cycling
- Counterclockwise: reverse cycle, cooling cycle
hot
- Thermal machine: capable of continuously heat converted into work machine
- Substances are used to absorb heat in the heat engine and the external work: the working substance
- Cycle: positive cycle
- Cycle characteristics:
- \ (\ Delta E = 0 \)
- \(Q_1-Q_2=W\)
- Heat engine efficiency: total calories of work to do net work of foreign substances and it absorbs heat from high-temperature ratio
- \(\eta =\frac{W}{Q_1}=\frac{Q_1-Q_2}{Q_1}=1-\frac{Q_2}{Q_1}\)
Refrigerator
- Cycle: reverse cycle
- Cooling factor:
- Definition: The refrigerator low temperature heat source to draw from the work of heat with the outside world than in a loop
- \(\varepsilon =\frac{Q_2}{W}=\frac{Q_2}{Q_1-Q_2}\)
Carnot cycle
Carnot heat engine
- Conditions: temperature only exchange heat with two heat sources, there are no heat, leakage and other factors
- Carnot cycle: the cycle of the substance of their work
composition
- Two isothermal
- Isothermal expansion process
- \(Q_1=\frac{m}{M}RT_1ln\frac{V_2}{V_1}\)
- Isothermal compression process
- \(Q_2=\frac{m}{M}RT_2ln\frac{V_3}{V_4}\)
- Two adiabatic process
- Adiabatic expansion
- Adiabatic compression process
Carnot cycle efficiency
- Carnot cycle efficiency formula: \ (\ ETA = l- \ FRAC T_2 {} {} T_l \)
- Derivation Principle:
- \(W=Q_1-Q_2\)
- \(T_1V_2^{\gamma-1}=T_2V_3^{\gamma-1}\quad \rightarrow \quad \frac{V_2}{V_1}=\frac{V_3}{V_4}\)
- in conclusion:
- Carnot cycle efficiency over the quasi-static process gas is determined by only two source temperature
- Improve the efficiency of the process: improving high-temperature heat source temperature (low-temperature heat source temperature can also be reduced, but generally very difficult and uneconomical)
Reverse Carnot cycle efficiency (coefficient of performance):
\(\varepsilon =\frac{T_2}{T_1-T_2}\)
Carnot heat engine working diagram:
graph LR style a fill:#FAB,stroke:#000000,stroke-width:2px style b fill:#FFA,stroke:#000000,stroke-width:2px style c fill:#AFF,stroke:#000000,stroke-width:2px style d fill:#AFB,stroke:#000000,stroke-width:2px a["高温热源 T1"]-->|Q1|b(("工作物质")) b-->|Q2|c["低温热源 T2"] b-->d["W"]
- Carnot refrigerator work diagram:
graph LR style a fill:#FAB,stroke:#000000,stroke-width:2px style b fill:#FFA,stroke:#000000,stroke-width:2px style c fill:#AFF,stroke:#000000,stroke-width:2px style d fill:#AFB,stroke:#000000,stroke-width:2px a["高温热源 T1"]-->|Q1|b(("工作物质")) b-->|Q2|c["低温热源 T2"] d["W"]-->b
Otto cycle (internal combustion engine)
stage
Is a constant circulation is heated
- Adiabatic compression process
- Peer endothermic process (explosion)
- Adiabatic expansion (work processes)
- Peer exothermic process
Cycle efficiency
- 公式：\(\eta = 1-\frac{1}{(\frac{V}{V_0})^{\gamma-1}}=1-\frac{1}{(r)^{\gamma-1}}\)
- Compression ratio: \ (R & lt = \ FRAC V_0} {V} {\)
- Improve the efficiency of the process: continuous improvement of compression ratio
10.5 & second law of thermodynamics and irreversible processes
Directional natural processes
Classic process:
- Heat Transfer Process
- Work to heat phenomenon
- Free expansion
- Diffusion process
Common feature: the process actually occurs spontaneously in nature are directional
Described in the Second Law of Thermodynamics
Kelvin statements
"Is impossible to prepare a heat engine cycle operation, only a single heat source and absorbs heat from the useful work becomes completely without any impact"
"The second category is made impossible perpetual motion"
The second category perpetual motion: 100% efficiency of the heat engine
Clausius statements
- "Impossible to heat to high temperature and low temperature objects from the object without causing any other changes."
- "Heat can not automatically transferred from the hot object to the low temperature object"
Reversible and irreversible processes
Reversible process
- Definition: a system from state A, go through a process $ A \ rightarrow B $ to reach another state B, if the system reply from the state B to the state A, the outside world restitution. He said the $ A \ rightarrow B $ process is reversible process
Irreversible process
- Definition: a system from state A, go through a process $ A \ rightarrow B $ to reach another state B, if the system reply from the state B to the state A, by any means impossible to make full restitution system and the outside world. He said the $ A \ rightarrow B $ process is irreversible process
Carnot's theorem
content:
- Operating in the same high-temperature heat source (T_l temperature $ $) between the same low temperature heat source (T_2 temperature $ $) and all reversible heat engine efficiency are the same, regardless of the working substance
- In the same high temperature heat source (T_l $ $ temperature) the same low temperature heat source (T_2 temperature $ $) between all the operating irreversible heat engine efficiency can not be greater than the efficiency of a reversible machine
Heat engine efficiency
\[\eta\leq 1-\frac{T_2}{T_1} \]REFRIGERATOR coefficient
\[\varepsilon\leq \frac{T_2}{T_1-T_2} \]
10.6 & concept of the second law of thermodynamics and entropy of statistical significance
Statistical significance of the second law of thermodynamics
Irreversible process spontaneously isolated system is small probability to the probability of a large macrostate macrostate performed, i.e. comprising a small number of micro-states comprising the macrostate Microstate be more macrostate
Boltzmann equation Entropy entropy principle of entropy increase
entropy
- Definition: a measure of disorder (randomness) of the microscopic particles of the components of the system, is a function of reflecting the state of the system
- Symbols: S
Boltzmann's entropy formula
\(S=k\ln\Omega\)
- k: Boltzmann constant
- \ (\ Omega \) : thermodynamic probability, i.e., the number of micro-states corresponding to a macrostate
Entropy change
\(\Delta S=S_2-S_1=k\ln\Omega_2-k\ln\Omega_1=k\ln\frac{\Omega_2}{\Omega_1}\geq0\)
- Equal sign applies only reversible process
Principle of entropy increase
\(\Delta S\geq0\)
Clausius entropy
\ (\ Delta S = \ ^ 2_1dS you \ geq \ ^ 2_1 you \ frac {\ dQ bar} {T} \)
- For isothermal expansion has: \ (\ S of Delta = \ FRAC Q {} {} T = \ FRAC {m}} M {R & lt \ LN \ FRAC V_2} {} {V_1 \)
- For reversible isobaric heating: \ (\ S of Delta = \ ^ int the 2_1-th \ FRAC {\ bar dQ} {} T = cm & lt \ LN \ FRAC T_2 {} {} T_l \)