For a simple compressible system v, s, p and T are specific volume, specific entropy, pressure and temperature, respectively. As per Maxwell’s relations, (∂v /∂s)_p is equal to

The correct option is: ${\left(\frac{\partial T}{\partial p}\right)}_s$.

Analysis of the Image:
As shown in the attached image, we can derive the thermodynamic Maxwell relations by utilizing the mathematical property of an exact differential.
For a state function $x$ defined by two independent variables $y$ and $z$, the total differential can be written as:

$$dx=Mdy+Ndz$$

Since $dx$ is an exact differential, the reciprocity relation states that:

$${\left(\frac{\partial M}{\partial z}\right)}_y={\left(\frac{\partial N}{\partial y}\right)}_z$$

Step-by-Step Derivation:
1. For a simple compressible system, we choose specific enthalpy ($h$) as a function of specific entropy ($s$) and pressure ($p$). The fundamental thermodynamic relation for specific enthalpy is given by:

$$dh=Tds+vdp$$

where:
• $T$ is the absolute temperature
• $s$ is the specific entropy
• $v$ is the specific volume (denoted as capital $V$ in the image)
• $p$ is the pressure

2. Comparing this to the standard mathematical differential form $dx=Mdy+Ndz$, we identify:
• $x=h$
• $M=T$, $y=s$
• $N=v$, $z=p$

3. Applying the exact differential reciprocity condition:

$${\left(\frac{\partial T}{\partial p}\right)}_s={\left(\frac{\partial v}{\partial s}\right)}_p$$

Thus, the partial derivative ${\left(\frac{\partial v}{\partial s}\right)}_p$ is equal to ${\left(\frac{\partial T}{\partial p}\right)}_s$.