정압비열, 정적비열 (cp-cv) 일반 관계식 문제(Mayer's Relation)

 

 

 

 

 

 

질문

열역학 정압비열 정적비열 관계식에 대해서 질문드립니다. 

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열역학 정압비열 정적비열 관계식에 대해서 질문드립니다.

 

답변

 

$$ds={\left(\frac{\partial s}{\partial T}\right)}_{v}dT+{\left(\frac{\partial s}{\partial v}\right)}_{T}dv$$

$$=\frac{1}{T}{\left(\frac{\partial u+P\partial v}{\partial T}\right)}_{v}dT+{\left(\frac{\partial s}{\partial v}\right)}_{T}dv$$

$$=\frac{1}{T}{\left(\frac{\partial u}{\partial T}\right)}_{v}dT+{\left(\frac{\partial s}{\partial v}\right)}_{T}dv$$

$$=\frac{c_v}{T}dT+{\left(\frac{\partial s}{\partial v}\right)}_{T}dv$$

 

 

$$ds={\left(\frac{\partial s}{\partial T}\right)}_{P}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

$$=\frac{1}{T}{\left(\frac{\partial h-v\partial P}{\partial T}\right)}_{P}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

$$=\frac{1}{T}{\left(\frac{\partial h}{\partial T}\right)}_{P}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

$$=\frac{c_P}{T}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

 

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$$ds=\frac{c_v}{T}dT+{\left(\frac{\partial s}{\partial v}\right)}_{T}dv$$

$$=\frac{c_P}{T}dT+{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

$$\frac{1}{T}(c_P-c_v)dT={\left(\frac{\partial s}{\partial v}\right)}_{T}dv-{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

$$dT=\frac{T}{(c_P-c_v)}{\left(\frac{\partial s}{\partial v}\right)}_{T}dv-\frac{T}{(c_P-c_v)}{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

 

$$dT={\left(\frac{\partial T}{\partial v}\right)}_{P}dv+{\left(\frac{\partial T}{\partial P}\right)}_{v}dP$$

 

$${\left(\frac{\partial T}{\partial v}\right)}_{P}dv=\frac{T}{(c_P-c_v)}{\left(\frac{\partial s}{\partial v}\right)}_{T}dv$$

$$c_P-c_v=T{\left(\frac{\partial s}{\partial v}\right)}_T{\left(\frac{\partial v}{\partial T}\right)}_P$$

$$={\left(\frac{T\partial s}{\partial v}\right)}_T{\left(\frac{\partial v}{\partial T}\right)}_P$$

$$={\left(\frac{\partial u+P\partial v}{\partial v}\right)}_T{\left(\frac{\partial v}{\partial T}\right)}_P$$

$$=[{\left(\frac{\partial u}{\partial v}\right)}_T+P]{\left(\frac{\partial v}{\partial T}\right)}_P$$

 

$${\left(\frac{\partial T}{\partial P}\right)}_{v}dP=-\frac{T}{(c_P-c_v)}{\left(\frac{\partial s}{\partial P}\right)}_{T}dP$$

$$c_P-c_v=-T{\left(\frac{\partial s}{\partial P}\right)}_T{\left(\frac{\partial P}{\partial T}\right)}_v$$

$$=-{\left(\frac{T\partial s}{\partial P}\right)}_T{\left(\frac{\partial P}{\partial T}\right)}_v$$

$$=-{\left(\frac{\partial h-v\partial P}{\partial P}\right)}_T{\left(\frac{\partial P}{\partial T}\right)}_v$$

$$=-[{\left(\frac{\partial h}{\partial P}\right)}_T-v]{\left(\frac{\partial P}{\partial T}\right)}_v$$

 

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