# Specific volumes of a saturated substance due to pressure

1. Apr 24, 2017

### mech-eng

I cannot understand why specific volume of a saturated liquid rises when pressure rises, specific volume of saturated vapor reduces when pressure rises. This made me remember buckingham-pi theorem. Is there any equational approach or formulation that show this?

Thank you.

2. Apr 26, 2017

### Staff: Mentor

The specific volume of a liquid is typically expressed according to $$dV=V(-\kappa dP+\alpha dT)$$where $\alpha$ is the coefficient of thermal expansion and $\kappa$ is the bulk compressibility. If the specific volume of a saturated liquid increases with saturation vapor pressure, it means that $$\frac{dP_{sat}(T)}{dT}<\frac{\alpha}{\kappa}$$So the thermal expansion wins out over the bulk compression.

The specific volume of a vapor that can be approximated by the ideal gas law is given by:
$$V=\frac{RT}{P}$$
So, $$dV=\frac{V}{T}dT-\frac{V}{P}dP=V\left(-\frac{dP}{P}+\frac{dT}{T}\right)$$ So the coefficient of thermal expansion of an ideal gas is 1/T and the bulk compressibility is 1/P. If the specific volume of the saturated vapor increases with the saturation vapor pressure, it means that $$\frac{dP_{sat}(T)}{dT}>\frac{P_{sat}}{T}$$From the Clausius-Clapeyron equation, all that this requires is for the heat of vaporization to be positive (which it is).

3. Apr 28, 2017

### mech-eng

are bulk compressibility and bulk module the same things?
What is the name of above equation? I cannot remember if I ever saw it in undergraduate thermodynamics books?

Thank you.

4. Apr 28, 2017

### Staff: Mentor

In freshman physics, you learned about thermal expansion and bulk compressibility of solids and liquids. This equation is just the combination of the two.

5. Apr 29, 2017

### mech-eng

Above you have done a very instructive proof but for that proof we have already assumed that "specific volume of a saturated liquid increases with saturatino vapor pressure." Is this an experimental knowledge, being similar to natural laws being known by only experience with no counter situations against observed such as four laws of thermodynamics and Newton's three laws of motion?

Thank you.