The relative humidity of air equals the ratio of the par- tial pressure of water in the air to the equilibrium vapor pressure of water at the same temperature times 100%. If the relative humidity of the air is 58% and its temperature is 68 °F, how many molecules of water are present in a room measuring12ft * 10ft * 8ft?

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Hey everyone today, we're being asked to calculate the number of water molecules present in a room with a volume of five x 5 x five m. And the relative humidity of the room is 26% at 25°C. So right off the bat, since we need to use pressure and volume and temperature in order to find the mass or the number of water molecules present in the room, we can determine that we're going to need to utilize our ideal gas law formula or PV. Is equal to N. R. T. And we can rearrange this all for the end term, which is the number of molds because that will help us find out how many molecules we have at the end. So let's just rewrite that and is equal to P. V. By R. T. And we'll keep that to the side for now and come back to it in a little bit. Now in the problem, there is one very important thing that has stated that the relative humidity is the ratio of partial pressure of water in the air to the vapor pressure of water at the given temperature. So let's write that out. The relative humidity, which I will denote with our H. Is equal to the partial pressure of water divided by the vapor pressure of water at that temperature, water at T. Now, the vapor pressure is a value that can be calculated experimentally and using that experimental data, we know that at 25°C The Vapor pressure of H 20. Is equal to 0. A. T. M's 0.30313 Atmospheres. So we need to find the pressure since we're already given the relative humidity and the vapor pressure. So let's go ahead and do that. So we can rearrange and say that the partial pressure of water is equal to the relative Humidity times the vapor pressure. So that will be 26% or 0. Times 0.0313 Atmospheres. Which will give us a final value of 8. times 10 to the negative third A. T. M. So this is our first value, the value for P that is the partial pressure or the pressure of water that we're going to need to use to find the volume V. It's also quite easy as to give us the parameters that we need. We're told that the room is five x 5 x five m. And so let's try that out. We have five Cubedm essentially, Which is 125 m cubed. However, we still need to convert this to Leaders because in the ideal gas law volume is calculated in Leaders. So we can recall that for one m cubed. There are 1000 leaders in a one By one x 1 m Cube Square or Cube. We will have 1000 L of volume. So using that conversion factor, We can arrive at the answer that the volume present here that we need to use is 1.25 times 10 to the 5th leaders going back to our ideal gas law and specifically this one right here, the one that we isolated and rearranged for N. We can now use the values given to solve so N is equal to P V by R T. Which in this case is 8.138 times 10 to the negative 3rd 80 M Times vol, which is 1.25 times 10 to the 5th. Leaders divided by The gas constant r which is 0.08 206 80 M. leaders buy more kelvin And the temperature will be in Kelvin. So we need to do 25 plus 273.15 to convert or 298.15 Kelvin. And that gives us a final value of forties one .578 moles. Well we're not done here. Remember according to avocados law we have Uh 6.02, 2 Times 10 to the 23rd molecules molecule per mole. So if we want to find out how many water molecules there are but we have to multiply this mole value by avocados number. So 41.578 moles times 6.02, 2 times 10 to the 23rd molecules over one more Will equal a final value of 2. times 10 to the 25 molecules. So the number of water molecules present in the room at this, at this temperature will be 2.5 times 10 to the 25 molecules. I hope this helps, and I look forward to seeing you all in the next one.

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Key Concepts

Relative Humidity

Partial Pressure

Ideal Gas Law