Tricia W.

asked • 06/12/22

An apartment has the dimensions 17 m by 9 m by 6 m. The temperature is 30°C, and the relative humidity is 55 percent. What is the total mass (in kg) of water vapor in the air in the apartment?

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Grigoriy S. answered • 06/12/22

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Knowing that at 30°C the saturated vapor pressure of water is 31.8 mm Hg, it is easy to find that the vapor pressure in the room is P = 31.8·0.55 = 17.5 mm Hg.

The volume of the room is V = 17m x 9m x 6m = 918 m3.

Now I will use the fact that with the accuracy of 5 percent the numerical value of pressure of water, measured in mm Hg, is the same as the numerical value of density of water vapor, measured in g /m3.

Because in 1 m3 we have 17.5 g of water vapor, multiplying this number by 918,

we will have 16 065 g or 16 kg of water vapor.

More precise answer will be a little bit less. For big volumes of rooms this 5 % difference does not play a crucial role.



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Tricia W.

incorrect answer according to the test
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06/12/22

Grigoriy S.

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I am 100% certain in my solution. The answer could be 15.3 kg, if we consider 5 % accuracy. Could you please then tell me the test answer?
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06/12/22

Grigoriy S.

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You need to understand the method of solution. Your instructor must give you the precision of the answer or amount of significant digits. You see in front of you 2 different approaches to the same problem. And you are saying that our answers are wrong. Sorry, cannot agree. Jacques D. method also a right one. I teach my student first of all to look at physics of the problem. It is not a multiple choice question. So try to understand solution. If you have questions, we will be happy to answer them.
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06/12/22

Grigoriy S.

tutor
I put the numbers in Jacques D. solution. The answer is still 15.3 kg. If 2 solutions are giving the same numeric value with different approaches, it means they could be trusted. I am repeating: for big volumes of rooms, it is better to give an answer with the error +- 0.5kg.
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06/12/22

JACQUES D. answered • 06/12/22

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The relative humidity is the PH2O/ PVP,H2O where you have to look up the vapor pressure of water at 30°.(31.8 mmHg).


PH2O = .55 * 31.8 mmHg/ (760 mmHg/atm) = .023013 atm H2O


Now just use Ideal gas law for water vapor in order to solve for moles. Then convert to grams.


ni = Pi V/(RT) V = 17x9 x6 m3 * 1000 liters/m3 Use R = .0821 l-atm/mole-K and T in Kelvin



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Tricia W.

wrong answer
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06/12/22

JACQUES D.

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You need to convert moles to grams, and then kg of water.
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06/12/22

Grigoriy S.

tutor
You cannot convert moles to grams. We use moles to count huge amount of particles. We use grams to measure mass.
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06/16/22

JACQUES D.

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Gregory, this is the second time you've commented inappropriately on my responses. Either show how to finish up the problem or keep your educational insights to yourself. You can express the quantity of material given in moles to kg using the multipliers g/mole * kg/g Or multiply by 18.02 g/mole and 1kg/1000g.
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06/16/22

Grigoriy S.

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Ok. Let me show another solution using equation for ideal gas law PV = nRT. Here P - pressure in Pa, V – volume in m3, n = m / M - number of moles in mol ( m – mass in kg, M – molar mass in kg/mol), R = 8.31 J / (mol ·K) – universal gas constant, T – absolute temperature in K. For our case 273 + 30 = 303 K. From this formula m = PVM / RT (*) Knowing that saturated vapor pressure of water at 30 degrees Celsius is 4.24x103 Pa, we can find that the pressure of water vapor in the room. It is P = .55 · 4.24 x103 Pa = 2.33 x103 Pa. The volume of the room is 918 m3. Molar mass of water is 18 x 10 – 3 kg/mol. Putting all values to (*), we will get M = 2.33 x103 Pa · 918 m3 · 18 x 10 – 3 kg/mol / (8.31 J / (mol ·K) · 303 K = 15.3 kg Reminder: mole of matter is the amount of substance containing as many particles as there atoms in 0.012 kg of carbon-12. Mass is a measure of inertia of a body measured in kg. Molar mass of a substance is the mass of 1 mole of a substance. And measured in kg / mol. I recommend to use only SI units for physics. Any questions?
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06/17/22

JACQUES D.

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This problem is more typical of chemistry than physics. Also, the natural "mole" used is the gram mole. There's a built-in conversion when you give molar masses as kg/mole given the values on the periodic table where moles of different elements are given in grams. Any constant connection between extensive variables can be utilized as pseudo-conversion factors and extends the usefulness of factor labeling. Maybe the answer is supposed to be to two decimal places (not correct by sig figs) . Nice explanation, Grigoriy.
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06/17/22

Grigoriy S.

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Let me remind you that before all gas laws and humidity were covered in a regular high school physics curriculum. About 20 years ago those topics were moved to chemistry course. I am a professional and know the majority of pitfalls of the students. When I take new class, I ask my students who are taken chemistry now or took it before. I warn them that in chemistry a broad variety of units are used and many of them are not SI units like cubic centimeters, liters, grams, grams per mol and so on. In physics in order to come to the right answer fast, it is a good habit right away to express all units in SI. One big misunderstanding is about the amount of matter. Many people believe that mass is the measure of the amount of mater in an object, which is wrong. What is bad that, if you search internet, you will find this definition over there. In reality unit of the amount of matter, mol, was added in 1971 to 6 unit international of SI. So that now we have 7 of them. The amount of substance is defined as the QUANTITY OR NUMBER of discrete particles in it divided by the Avogadro constant. I can use moles not only to count atomic-scale particles but also stars. On the contrary, mass is a measure of inertial, gravitational and energetic properties of matter. The more we now study the nature of mass - the more we are confused. I would say that today we do not completely understand what is mass. We can build all physics using only the concept of energy, linear and angular momentums. But mass and the amount of matter are connected to each other by simple formula m = M n, where n - number of moles and M - molar mass. There is another formula for mass of the substance m = m0 N, where m0 - mass of a molecule and N - number of them. This particular problem is interesting because if I use not SI units for pressure and density, I can come to the answer much faster without the use of universal gas law (see my first solution).
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06/17/22

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