질문
열역학 일반관계식, 맥스웰 관계식 증명
kin.naver.com/qna/detail.nhn?dirId=111301&docId=372401470
풀이
\[\left ( \frac{\partial s}{\partial v} \right )_{P}=\frac{1}{T}\left( \frac{\partial h-v\partial P}{\partial v} \right )_{P}\]
\[=\frac{1}{T}\left( \frac {\partial h}{\partial v}\right )_{P}=\frac{1}{T}\left( \frac {\partial h}{\partial T}\right )_{P}\left( \frac {\partial T}{\partial v}\right )_{P}\]
\[=\frac{c_{P}}{T}\frac{v}{v}\left( \frac {\partial T}{\partial v}\right )_{P}=\frac{c_{P}}{Tv\beta }\]
\[\left ( \beta =\frac{1}{v}\left ( \frac{\partial v}{\partial T} \right )_{P} =\alpha \right )\]
\[\left ( \frac{\partial s}{\partial P} \right )_{v}=\frac{1}{T}\left( \frac{\partial u+P\partial v}{\partial P} \right )_{v}\]
\[=\frac{1}{T}\left( \frac {\partial u}{\partial P}\right )_{v}=\frac{1}{T}\left( \frac {\partial u}{\partial T}\right )_{v}\left( \frac {\partial T}{\partial P}\right )_{v}\]
\[=\frac{c_{v}}{T}\left[ -\left( \frac {\partial T}{\partial v}\right )_{P}\left( \frac {\partial v}{\partial P}\right )_{T}\right ]\]
\[=\frac{c_{v}}{T}\frac{v}{1}\left( \frac {\partial T}{\partial v}\right )_{P}\frac{-1}{v}\left( \frac {\partial v}{\partial P}\right )_{T}\]
\[=\frac{c_{v} \kappa}{T\beta}\]
\[\left ( \kappa =\frac{-1}{v}\left ( \frac{\partial v}{\partial P} \right )_{T} =\beta_{T}\right )\]
\[c_P-c_v=Tv\frac{\beta ^2}{\kappa }\]
[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - Mayer 관계식 Mayer's relation
Mayer 관계식 Mayer's relation
Mayer 관계식 Mayer's relation 0. ds 일반관계식 ds general relation 2018/10/26 - [열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - ds 일반관계식 ds general relation 2018/10/25 - [열..
syssurr.tistory.com
\[\left ( \frac{\partial s}{\partial P} \right )_{v}=\frac{\left( c_P-Tv\frac{\beta ^2}{\kappa } \right) \kappa}{T\beta}\]
\[\left ( \frac{\partial s}{\partial P} \right )_{v}=\frac{ c_P \kappa}{T\beta}-v\beta\]
\[\binom{ \beta =\frac{1}{v}\left ( \frac{\partial v}{\partial T} \right )_{P} =\alpha \ }{\kappa =\frac{-1}{v}\left ( \frac{\partial v}{\partial P} \right )_{T} =\beta_{T}}\]
\[\left(\frac{\partial a}{\partial P} \right )_{v}=\left(\frac{\partial u-T\partial s-s\partial T}{\partial P} \right )_{v}\]
\[=\left(\frac{\partial u-\left( \partial u+P\partial v\right )-s\partial T}{\partial P} \right )_{v}=\left(\frac{-P\partial v-s\partial T}{\partial P} \right )_{v}\]
\[=-s\left(\frac{\partial T}{\partial P} \right )_{v}=s\left(\frac{\partial T}{\partial v} \right )_{P}\left(\frac{\partial v}{\partial P} \right )_{T}\]
\[=-s\frac{v}{1}\left(\frac{\partial T}{\partial v} \right )_{P}\frac{-1}{v}\left(\frac{\partial v}{\partial P} \right )_{T}=\frac{-s\kappa}{\beta}\]
\[\left(\frac{\partial a}{\partial v} \right )_{P}=\left(\frac{-P\partial v-s\partial T}{\partial v} \right )_{P}\]
\[=-P-s\left ( \frac{\partial T}{\partial v} \right )_{P}=-P-s\frac{v}{v}\left ( \frac{\partial T}{\partial v} \right )_{P}=-P-\frac{s}{v\beta}\]
\[\left ( \frac{\partial h}{\partial s} \right )_{v}=\left ( \frac{T\partial s+v\partial P}{\partial s} \right )_{v}=T+v\left ( \frac{\partial P}{\partial s} \right )_{v}\]
\[=T-v\left ( \frac{\partial P}{\partial u+P\partial v} \right )_{v}=T-v\left ( \frac{\partial P}{\partial u} \right )_{v}\]
\[=T-vT\left ( \frac{\partial P}{\partial u+P\partial v} \right )_{v}=T\left (1 -v\left ( \frac{\partial P}{\partial u} \right )_{v} \right )\]
\[=T\left (1 +v\left ( \frac{\partial P}{\partial T} \right )_{v}\left ( \frac{\partial T}{\partial u} \right )_{v} \right )=T\left (1 +\frac{v}{c_{v}}\left ( \frac{\partial P}{\partial T} \right )_{v}\right )\]
\[=T\left (1 -\frac{v}{c_{v}}\left ( \frac{\partial P}{\partial v} \right )_{T}\left ( \frac{\partial v}{\partial T} \right )_{P}\right )=T\left (1 +\frac{v\beta}{c_{v}\kappa}\right )\]
[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - 편미분 관계식 partial differential relations
편미분 관계식 partial differential relations
편미분 관계식 partial differential relations 열역학에서 z의 전미분이 exact이면 상태량(property) 이고 변화량은 으로 표기한다 inexact이면 경로함수(path function) 이고 변화량은으로 표기한다. [1]..
syssurr.tistory.com
[열역학 Thermodynamics/12. 일반관계식 Thermodynamic Relations ] - Maxwell 관계식 Maxwell relations
Maxwell 관계식 Maxwell relations
Maxwell 관계식 Maxwell relations 1. 열역학 퍼텐셜 thermodynamic potentials 정의 definition U : 내부에너지 internal energy H : 엔탈피 enthalpy A : 헬름홀츠 자유에너지 Helmholtz function G : 깁스..
syssurr.tistory.com
[열역학 Thermodynamics/7. 엔트로피 Entropy] - T ds 관계식 Tds equation
T ds 관계식 Tds equation
T ds 관계식 T ds equation 가정 Assumption ΔKE, ΔPE 등 무시한 단순 압축성 물질 neglecting changes in kinetic and potential energies, and assuming a simple compressible substance 준평형 과정 a quas..
syssurr.tistory.com
압축계수 팽창계수 compressibility & expansion coefficient
압축계수 팽창계수 compressibility & expansion coefficient 체팽창계수 thermal expansion coefficient 등온압축계수 isothermal compressibility 등엔트로피압축계수 Isentropic compressibility 압..
syssurr.tistory.com
'열역학 Thermodynamics > 문제와 풀이 Problems and Solutions' 카테고리의 다른 글
| 압력밥솥 문제 - 포화액 포화증기 혼합물 (0) | 2021.05.23 |
|---|---|
| 개방계 시스템 터빈/압축기 일과 엔탈피 관계 (0) | 2021.05.20 |
| 정압비열, 정적비열 (cp-cv) 일반 관계식 문제(Mayer's Relation) (0) | 2020.10.29 |
| 정압과정에서 내부에너지 감소? (0) | 2020.10.21 |
| 이상기체의 정적비열과 정압비열 관계 (0) | 2020.10.20 |
