열역학 맥스웰 관계식 문제 증명 (편미분 방정식)

 

 

 

 

질문

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열역학 관계식 증명 질문있습니다

 

풀이

 

 

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

 

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

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

$$ds=[{\left(\frac{\partial s}{\partial P}\right)}_v+{\left(\frac{\partial s}{\partial v}\right)}_P{\left(\frac{\partial v}{\partial P}\right)}_T]dP+{\left(\frac{\partial s}{\partial v}\right)}_P{\left(\frac{\partial v}{\partial T}\right)}_{P}dT$$

 

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

 

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

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

$$1={\left(\frac{\partial s}{\partial P}\right)}_v{\left(\frac{\partial P}{\partial s}\right)}_T+{\left(\frac{\partial s}{\partial v}\right)}_P{\left(\frac{\partial v}{\partial s}\right)}_T$$

$$1={\left(\frac{\partial s}{\partial P}\right)}_v[-{\left(\frac{\partial T}{\partial v}\right)}_P]+{\left(\frac{\partial s}{\partial v}\right)}_P{\left(\frac{\partial T}{\partial P}\right)}_v$$

$${\left(\frac{\partial T}{\partial P}\right)}_v{\left(\frac{\partial s}{\partial v}\right)}_P-{\left(\frac{\partial T}{\partial v}\right)}_P{\left(\frac{\partial s}{\partial P}\right)}_v=1$$

 

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