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We know that Joule-Thomson coefficient is defined as $$\mu_{jt}=\left(\frac{\partial T}{\partial P}\right)_H$$

Using the cyclic rule for partial derivatives it can be written as $$\mu_{jt}=-\frac{\left(\frac{\partial H}{\partial P}\right)_T}{\left(\frac{\partial H}{\partial T}\right)_P}$$

Which is equivalent to

$$\mu_{jt}=-\frac{\left(\frac{\partial H}{\partial P}\right)_T}{C_p}$$

My question is can we write simmilarly

$$\mu_{jt}=-\frac{\left(\frac{\partial U}{\partial P}\right)_T}{C_v}$$

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