# 位溫

## 位溫嘅公式

${\displaystyle dQ=dU+dW}$

${\displaystyle dQ=c_{v}dT+pd\left({\frac {1}{\rho }}\right)}$

${\displaystyle c_{v}dT+pd\left({\frac {1}{\rho }}\right)=0}$

${\displaystyle d\left({\frac {p}{\rho }}\right)=pd\left({\frac {1}{\rho }}\right)+{\frac {1}{\rho }}dp}$

${\displaystyle pd\left({\frac {1}{\rho }}\right)=d\left({\frac {p}{\rho }}\right)-{\frac {1}{\rho }}dp}$

${\displaystyle c_{v}dT+d\left({\frac {p}{\rho }}\right)-{\frac {1}{\rho }}dp=0}$

${\displaystyle p=\rho RT}$（呢度都係用緊氣象學單位）

${\displaystyle {\frac {p}{\rho }}=RT}$

${\displaystyle d\left({\frac {p}{\rho }}\right)=RdT}$ 同埋 ${\displaystyle {\frac {1}{\rho }}={\frac {RT}{p}}}$

${\displaystyle c_{v}dT+RdT-{\frac {1}{\rho }}dp=0}$

${\displaystyle (c_{v}+R)dT-{\frac {RT}{p}}dp=0}$

${\displaystyle c_{p}dT-{\frac {RT}{p}}dp=0}$

${\displaystyle c_{p}dT={\frac {RT}{p}}dp}$

${\displaystyle {\frac {dT}{T}}={\frac {R}{c_{p}}}{\frac {dp}{p}}}$

${\displaystyle \int _{T_{1}}^{T_{2}}{\frac {dT}{T}}={\frac {R}{c_{p}}}\int _{p_{1}}^{p_{2}}{\frac {dp}{p}}}$

${\displaystyle [\ln |T|]_{T_{1}}^{T_{2}}={\frac {R}{c_{p}}}[\ln {p}]_{T_{1}}^{T_{2}}}$

${\displaystyle \ln {\frac {T_{2}}{T_{1}}}={\frac {R}{c_{p}}}\ln {\frac {p_{2}}{p_{1}}}}$

${\displaystyle {\frac {T_{2}}{T_{1}}}=\left({\frac {p_{2}}{p_{1}}}\right)^{\frac {R}{c_{p}}}}$

${\displaystyle {\frac {\theta }{T}}=\left({\frac {P'}{P}}\right)^{\frac {R}{c_{p}}}}$

${\displaystyle \theta =T\left({\frac {P'}{P}}\right)^{\frac {R}{c_{p}}}}$

${\displaystyle P'}$ 係海平面氣壓，即係 ${\displaystyle P'=P_{s}}$，條式就變成：

${\displaystyle \theta =T\left({\frac {P_{s}}{P}}\right)^{\frac {R}{c_{p}}}}$

## 位溫嘅特性

### 同熵嘅關係

${\displaystyle \theta =T\left({\frac {P_{s}}{P}}\right)^{\frac {R}{c_{p}}}}$

${\displaystyle \ln \theta =\ln T+{\frac {R}{c_{p}}}\ln {\frac {P_{s}}{P}}}$

${\displaystyle \ln \theta =\ln T+{\frac {R}{c_{p}}}(\ln P_{s}-\ln P)}$

${\displaystyle {\frac {d\theta }{\theta }}={\frac {dT}{T}}-{\frac {R}{c_{p}}}{\frac {dP}{P}}}$ （留意當 ${\displaystyle d\theta =0}$，即係 ${\displaystyle \theta }$ 唔變嘅時候，右面條式就係代表氣塊嘅絕熱質過程）

${\displaystyle c_{p}{\frac {d\theta }{\theta }}=c_{p}{\frac {dT}{T}}-R{\frac {dP}{P}}}$

${\displaystyle dQ=c_{v}dT+pd\left({\frac {1}{\rho }}\right)}$

${\displaystyle dQ=c_{p}dT-{\frac {RT}{p}}dp}$

${\displaystyle {\frac {dQ}{T}}=c_{p}{\frac {dT}{T}}-R{\frac {dp}{p}}}$

${\displaystyle dS={\frac {dQ}{T}}}$

${\displaystyle dS=c_{p}{\frac {dT}{T}}-R{\frac {dp}{p}}}$

${\displaystyle dS=c_{p}{\frac {d\theta }{\theta }}}$