Posts Tagged ‘formula’

afr01

If the man who almost single-handedly invented chemical thermodynamics – the American mathematical physicist Josiah Willard Gibbs – had owned an automobile, he would have had no trouble figuring out the action of antifreeze.

“The problem reduces to consideration of a binary solution in equilibrium with solid solvent,” I can hear old Josiah saying. “Such a thermodynamic system has two degrees of freedom, so at constant pressure there must be a relation between temperature and composition.”

And indeed there is. The relation corresponds to the observed depression of the freezing point of a solvent by a solute. What’s more, its exact form confirms how antifreeze really works.

– – – –

Computing chemical potential

We have Josiah Willard Gibbs to thank for introducing the concept of chemical potential (μ) as a sort of generalized force driving the flow of chemical components between coexistent phases.

When the phases are in equilibrium at constant temperature and pressure, the chemical potential of any component has the same value in each phase

pe04

The key point to note here is that μi is the chemical potential of component i in an arbitrary state, i.e. in a mixture of components. In order to compute this potential we need to know two things: the chemical potential of the pure substance μi0 at a pressure p (such as that of the atmosphere), and the mole fraction (xi) of the component in the mixture. Assuming an ideal solution, use can then be made of the textbook formula

dce12 …(1)

With pressure and temperature fixed, this equation has a single variable (xi), from which we can draw the conclusion that the variation in chemical potential of a component in an ideal solution is determined solely by its own mole fraction.

The significance of this fact can be appreciated by considering the following diagrams

afr03

Here is water in equilibrium with ice at 273K. The chemical potentials of the solid and liquid phases are equal; there is no net driving force in either direction. Now consider the effect of adding an antifreeze agent to the liquid phase

afr04

Assuming the temperature held constant at 273K, the addition of antifreeze reduces the mole fraction of water, lowering its chemical potential in accordance with equation 1. The coexistent solid phase now has a higher potential, providing the driving force to transform ice into water. Since the temperature is held constant, this equates to the lowering of the freezing point of water in the mixture.

– – – –

Deducing a formula for freezing-point depression

To obtain a formula for the freezing point of water in a solution containing antifreeze, we start with the equilibrium relation

afr06

where the zero superscript indicates a standard potential, i.e. that the solid phase consists of pure ice whose mole fraction x is unity. Substituting the left hand side with

afr07

we obtain

afr08

which after differentiation with respect to temperature at constant pressure and subsequent integration yields the formula for the freezing point of water in a solution containing antifreeze at 1 atmosphere pressure:

afr09

The terms on the right are the molar enthalpy of fusion of water (ΔHf0), the freezing point of pure water (Tf0), the gas constant R and the mole fraction of water (xH2O) in the solution containing antifreeze.

The latter is the only variable, confirming that the freezing point of water in a solution containing antifreeze is determined solely by the mole fraction of water in the mixture – in other words the extent to which the water is diluted by the antifreeze agent.

This is how antifreeze works. There is nothing active about its action. It exerts its effect passively by being miscible and thereby reducing the mole fraction of water in the liquid mixture. There’s really nothing more to it than that.

– – – –

Using the formula

afr09

Values for constants

Enthalpy of fusion of water ΔHf0 = 6.02 kJmol-1
Freezing point of pure water Tf0 = 273.15 K
Gas constant R = 0.008314 kJmol-1K-1

Example

651 grams of the antifreeze agent ethylene glycol (molecular weight 62.07) are added to 1.5 kg of water (molecular weight 18.02). What is the freezing point of water in this solution?

Strategy

1. Calculate the mole fraction of water in the solution

afr10

Number of moles of water = 1500/18.02 = 83.2
Number of moles of ethylene glycol = 651/62.07 = 10.5
Mole fraction of water = 83.2/(83.2 + 10.5) = 0.89

2. Calculate the freezing point of water in the solution

afr11

The solution will give antifreeze protection down to 261.65K or –11.5°C

– – – –

ventus001

I have a digital weather station with a wireless outdoor sensor. In the photo, the top right quadrant of the display shows temperature and relative humidity for outdoors (6.2°C/94%) and indoors (21.6°C/55%).

I find this indoor-outdoor thing fascinating for some reason and revel in looking at the numbers. But when I do, I always end up asking myself if the air outside has more or less water vapor in it than the air inside. Simple question, which is more than can be said for the answer. Using the ideal gas law, the calculation of absolute humidity from temperature and relative humidity requires an added algorithm that generates saturated vapor pressure as a function of temperature, which complicates things a bit.

Formula for calculating absolute humidity

In the formula below, temperature (T) is expressed in degrees Celsius, relative humidity (rh) is expressed in %, and e is the base of natural logarithms 2.71828 [raised to the power of the contents of the square brackets]:

Absolute Humidity (grams/m3) = 6.112 x e^[(17.67 x T)/(T+243.5)] x rh x 18.02
                                                                            (273.15+T) x 100 x 0.08314

which simplifies to

Absolute Humidity (grams/m3) = 6.112 x e^[(17.67 x T)/(T+243.5)] x rh x 2.1674
                                                                                        (273.15+T)

This formula is accurate to within 0.1% over the temperature range –30°C to +35°C

– – – –

gif format (decimal separator = .)

ah3

gif format (decimal separator = ,)

ah3a

jpg format (decimal separator = .)

ah1

jpg format (decimal separator = ,)

ah1a

– – – –

Additional notes for students

Strategy for computing absolute humidity, defined as density in g/m^3 of water vapor, from temperature (T) and relative humidity (rh):

1. Water vapor is a gas whose behavior approximates that of an ideal gas at normally encountered atmospheric temperatures.

2. We can apply the ideal gas equation PV = nRT. The gas constant R and the variables T and V are known in this case (T is measured, V = 1 m3), but we need to calculate P before we can solve for n.

3. To obtain a value for P, we can use the following variant[REF, eq.10] of the Magnus-Tetens formula which generates saturated vapor pressure Psat (hectopascals) as a function of temperature T (Celsius):

Psat = 6.112 x e^[(17.67 x T)/(T+243.5)]

4. Psat is the pressure when the relative humidity is 100%. To compute the pressure P for any value of relative humidity expressed in %, we multiply the expression for Psat by the factor (rh/100):

P = 6.112 x e^[(17.67 x T)/(T+243.5)] x (rh/100)

5. We now know P, V, R, T and can solve for n, which is the amount of water vapor in moles. This value is then multiplied by 18.02 – the molecular weight of water ­– to give the answer in grams.

6. Summary:
The formula for absolute humidity is derived from the ideal gas equation. It gives a statement of n solely in terms of the variables temperature (T)  and relative humidity (rh). Pressure is computed as a function of both these variables; the volume is specified (1 m3) and the gas constant R is known.

– – – –

UPDATES

– – – –

Formula recommended for use in monitoring comfort levels for exotic pets

March 2017: A post has appeared on Reddit concerning an Arduino Uno with T&RH sensor and LCD screen, which the poster is using to improve temperature and humidity monitoring of a pet’s habitat – in this particular case a Bearded Dragon (not the one illustrated).

The post has attracted much interested discussion and comment, including a recommendation from one participant to use AH rather than RH, citing my conversion formula. The rationale for the change is so neatly expressed that I would like to quote it:

“May I recommend absolute humidity instead of relative? Relative humidity only tells you how “full” the air is of moisture, and it’s entirely dependent on temperature; the same amount of moisture will read lower relative humidity at higher temperatures, and vice versa. Whereas absolute humidity is measured in grams of water per cubic meter of air. You can implement this simple conversion formula in your code: (URL for this blogpost)
0-2 is extremely dry, 6-12 is your average indoors, and 30 is like an Amazon rainforest.”

See the Reddit post here:
https://www.reddit.com/r/arduino/comments/5ysmo5/i_noticed_my_bearded_dragons_habitat_could_use_a/

See the Arduino project here:
https://create.arduino.cc/projecthub/ThothLoki/portable-arduino-temp-humidity-sensor-with-lcd-a750f4

– – – –

Igor uses my formula to keep his cellar dry

igor01

October 2016: I am impressed by this basement humidity control system developed by Igor and reported on Amperka.ru forum.

Inside the short pipe is a fan equipped with a 3D-printed circumferential seal. The fan replaces basement air with outdoor air, and is activated when absolute humidity in the cellar is 0.5 g/m^3 higher than in the street, subject to the condition that the temperature of the outdoor air is lower. This ensures that water in the cellar walls is drawn into the vapor phase and pumped out; the reverse process cannot occur. на русском здесь.

igor02

– – – –

Formula powers AH measurements from high-precision RH&T sensor

sht75

The SHT75 RH&T sensor from SENSIRION

April 2016: Prof. Antonietta Frani has made a miniature device for measuring absolute humidity, using my formula to power an Arduino Uno microcontroller board equipped with an SHT75 RH&T sensor which connects to a computer via a USB cable. Systems Integrator Roberto Valgolio has developed an interface to transfer the data to Excel spreadsheets with their associated graphical display functions.

– – – –

Formula powers online RH←→AH calculator

reckoner

March 2016: German website rechneronline.de is using my formula to power an online RH/AH conversion calculator.

– – – –

Formula cited in academic research paper

ahcitat

January 2016: A research article in Landscape Ecology (October 2015) exploring microclimatic patterns in urban environments across the United States has used my formula to compute absolute humidity from temperature and relative humidity data.

– – – –

Formula finds use in humidity control unit

August 2015: Open source software/hardware project Arduino is using my absolute humidity formula in a microcontroller designed to control humidity in basements:

arduino

“The whole idea is to measure the temperature and relative humidity in the basement and on the street, on the basis of temperature and relative humidity to calculate the absolute humidity and make a decision on the inclusion of the exhaust fan in the basement. The theory for the calculation is set forth here – carnotcycle.wordpress.com/2012/08/04/how-to-convert-relative-humidity-to-absolute-humidity.” на русском здесь.

More photos on this link (text in Russian):http://arduino.ru/forum/proekty/kontrol-vlazhnosti-podvala-arduino-pro-mini

– – – –

AH computation procedure applied in calibration of NASA weather satellite

June 2015: My general procedure for computing AH from RH and T has been applied in the absolute calibration of NASA’s Cyclone Global Navigation Satellite System (CYGNSS), specifically in relation to the RH data provided by Climate Forecast System Reanalysis (CFSR). The only change to my formula is that Psat is calculated using the August-Roche-Magnus expression rather than the Bolton expression.

The CYGNSS system, comprising a network of eight satellites, is designed to improve hurricane intensity forecasts and was launched on 15 December 2016.

Reference: http://ddchen.net/publications (Technical report “An Antenna Temperature Model for CYGNSS” June 2015)

– – – –

Formula cited in draft paper on air quality monitoring

May 2015: Metal oxide (MO) sensors are used for the measurement of air pollutants including nitrogen dioxide, carbon monoxide and ozone. A draft paper concerning the Air Quality Egg (AQE) which cites my formula in relation to MO sensors can be seen on this link:

MONITORING AIR QUALITY IN THE GRAND VALLEY: ASSESSING THE USEFULNESS OF THE AIR QUALITY EGG

– – – –

Formula used by US Department of Energy in Radiological Risk Assessment

June 2014: In its report on disused uranium mines, Legacy Management at DoE used my formula for computing absolute humidity as one of the meteorological parameters involved in modeling radiological risk assessment.

– – – –