# Relative humidity, definition and its dependence on state variables

Relative humidity ($$RH$$) is defined by the following equation:

$$RH = \frac{p_{vap}}{p_{sat}}$$

Where $$p_{vap}$$ is the vapour pressure and $$p_{sat}$$ is the saturation vapour pressure.

VAPOUR PRESSURE

Vapour pressure $$p_{vap}$$ is the partial pressure of water vapour in the air.

SATURATION VAPOUR PRESSURE

Saturation vapour pressure $$p_{sat}$$ is the partial pressure of water vapour at which vapour and liquid/solid states coexist. This pressure can be directly obtained from isothermes in $$P(V)$$ plots where changes in volumes occurs at constant pressure. This is an intrinsic property of water. The saturation vapour pressure depends on temperature only and empirical laws exist to describe its exponential behaviour (see Arden Buck equations for examples).

EXAMPLE: This image shows the saturation vapour pressure at $$T_{1}$$.

QUESTIONS

• What physical process determines the vapour pressure $$p_{vap}$$ in a given state $$(p,V,T)$$ in a closed container? What balance of forces (pressures) fixes the value of $$p_{vap}$$?
• What does vapour pressure $$p_{vap}$$ depends on? Is it a property of water only and its state or depends on environmental conditions like, say, air pressure as well?

CONTEXT

• These questions and others arose while trying to answer the following. Given a closed room with air at a given relative humidity. Imagine to take two glasses and place them upside-down on a table. One covering a certain amount of water (just poured on the table) and the other being empty. How does relative humidity change in the two glasses?

My difficulty in answering this question is in estimating how $$p_{vap}$$ will change in the two situations, I don't understand how mother natures constrains the value of $$p_{vap}$$ :).

• Let's focus on your specific problem. What are your thoughts on it so far? The term you are using for $p_{vap}$ is more properly called just the partial pressure of water vapor in the gas phase. I would not call it the vapor pressure, since that is usually also reserved for the equilibrium vapor pressure. – Chet Miller Dec 24 '18 at 22:49
• I'll add my reasoning soon, but I miss the information about p_vap. That's why I am rather interested by the "questions" and not the "context". I agree on the term "partial pressure", I called it vapour pressure because it was called that way on my old notes and some other source I found on the web. – Worldsheep Dec 25 '18 at 8:56
• Are you comfortable with the concept of partial pressure with respect to, say, oxygen and nitrogen in air. So, for oxygen, the partial pressure is about 0 21 bars, and for nitrogen, it is about 0.79 bars. Do you feel that, conceptually, there is a fundamental difference between these and the partial pressure of water vapor? – Chet Miller Dec 25 '18 at 13:07

Imagine you take two glasses and place them upside-down on a table. One covering a certain amount of water (just poured on the table) and the other being empty. How does relative humidity change in the two glasses?

If the temperature of the room remains constant, the relative humidity of the gas mixture in the glass not covering water will remain the same for the following reason.

As you have indicated, the relative humidity (RH) is defined as

$$RH=\frac{p_v}{p_{sat}}$$

Since the amount of water molecules under the glass is unchanged, for the RH to change the saturation pressure ($$p_{sat}$$) needs to change. The latter depends on temperature (it decreases when temperature decrease and vice versa).

For the glass covering some water, assuming the water is at room temperature, the relative humidity will increase. This is because the water will evaporate increasing the partial pressure of the gaseous $$H_{2}O$$, up to a maximum of 100 %, depending on the amount of water covered and the volume of the glass.

Now, regarding the lead in questions.

What physical process determines the vapor pressure $$p_{vap}$$ in a given state (p,V,T) in a closed container? What balance of forces (pressures) fixes the value of $$p_{vap}$$?

The physical process involves evaporation. The amount of water that can evaporate is limited to the amount of water available to evaporate, the space available for it to fill (volume), and the temperature of the air in the space (which determines the saturation pressure of the air). The greater each of these is the more water in the gaseous phase that can be accommodated. As the water vaporizes the vapor pressure above the surface of the water increases slowing the rate of evaporation. Evaporation balances condensation at the surface when the relative humidity of the container reaches 100 %.

What does vapour pressure $$p_{vap}$$ depends on? Is it a property of water only and its state or depends on environmental conditions like, say, air pressure as well?

The partial pressure of the gaseous $$H_{2}O$$ is the pressure that it alone would exert if all the other gases in the container were removed. It is a property of the water (saturation temperature and pressure of water). It depends on the environment since the temperature of the environment determines the saturation pressure. In the atmosphere at sea level the partial pressure of the water vapor is very small. For example, for a temperature of 20 C and relative humidity of 50% the partial pressure of the water vapor is about only about 1% of the total atmospheric pressure.

Overall your questions concern the subject of Psychrometrics, the study of water vapor in atmospheric air. A convenient reference showing the relationship between thermodynamic variables pertaining to the partial pressure of gaseous water vapor is a Psychrometric Chart. The chart assumes one is dealing with standard atmospheric pressure. The chart includes the following properties:

• Relative humidity

• Humidity ratio

• Dry bulb temperature

• Wet bulb temperature

• Saturation temperature (dew point) (saturation pressure)

• Specific enthalpy

• Specific volume

From the chart, given values of any two of the above properties any of the remaining properties can be determined, or at least estimated. While the partial pressure of the gaseous water vapor $$P_{vap}$$ is not on the chart, it can be calculated from the relative humidity and dry bulb temperature properties on the chart.

Hope this helps.

• Thanks for your reply! I need to take some time to read it through in detail, but I don't understand your explanation at the very beginning. You said all the water will evaporate in the glass. There is no equilibrium between evaporation and condensation. Is that equilibrium reached only at pv=psat. Are you telling that up to RH 100% the water will evaporate and after that no more water will evaporate due to equilibrium (condensation/evaporation). Why not all air is at 100% RH as there is water to be evaporated on our planet? I'll come back as soon as I get the time to read the rest in detail. – Worldsheep Dec 27 '18 at 20:09
• If I look at the P(V) plot I showed, I have the impression that the liquid/gas coexistence happens in the blue region only. What happens if in the glass with water I am in a (P,V) point in the blue region? And for a (P,V) outside the blue region? (both for a given T, say T1). – Worldsheep Dec 27 '18 at 20:19
• @WorldsheepI don’t believe I said all the liquid under the glass will necessarily evaporate. In any case, it will all evaporate if the partial pressure of the gas remains less than the saturation pressure for the gas. All the air on our planet is not at 100% RH because waters are constantly evaporating as part of a cycle of evaporation, condensation (formation of clouds, fog and mist) and precipitation (when the atmosphere becomes saturated with water vapor). – Bob D Dec 29 '18 at 13:00
• @WorldsheepThe blue region on the PV diagram corresponds to the boiling point of water where liquid water and water vapor coexist at various total pressures. For example, at 1 atmosphere pressure the boiling temperature is 100 C. It does not apply to the evaporation of water at 1 atmosphere and room temperature (20 C). Evaporation, on the other hand, involves the transition between the liquid and gas phases at temperatures below the boiling temperature. Microscopically, this is what happens: – Bob D Dec 29 '18 at 13:01
• @Worldsheep The temperature of the water is a measure of the average translational kinetic energy of the molecules of the water. However the kinetic energies of individual molecules are distributed above and below the average. The more energetic molecules at the surface of the water escape and become $H_{2}O$ gas. Some molecules return and condense back into liquid. – Bob D Dec 29 '18 at 13:02

What physical process determines the vapour pressure pvap in a given state (p,V,T) in a closed container? What balance of forces (pressures) fixes the value of pvap?

What does vapour pressure pvap depends on? Is it a property of water only and its state or depends on environmental conditions like, say, air pressure as well?

For a given pure component, vapor pressure is only a function of temperature, as given by the Antoine equation: https://en.wikipedia.org/wiki/Antoine_equation