$4.77^\circ\textrm{C}$

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This is College Physics Answers with Shaun Dychko. We're going to rearrange this formula for relative humidity, vapor density divided by maximum possible vapor density and solve it for vapor density. And we're going to multiply both sides by saturation vapor density and that solves for vapor density. And so we're told that the relative humidity at 20 degrees Celsius is 39 percent, so I've written that as a decimal here 0.39. And then we multiply by the saturation vapor density at 20 degrees Celsius for water which is 17.2 grams per cubic meter. And that means our vapor density is 6.71 grams per cubic meter. Now, to find the temperature at which the dew point would occur or the point which water would start to come out of the air and form droplets, we need to figure out at what temperature is this, the saturation vapor density. Now, this number is not appearing in this Table 13.5. And so we have to interpolate between two values that are given and figure out where the temperature would be for this number of 6.71. So from Table 13.5, we have a couple of values. For the temperature column, we have zero degrees Celsius. The saturation vapor density is 4.84 and at five degrees Celsius, the saturation vapor density is 6.80. And so, the temperature is somewhere between zero and five, much closer to five because 6.71 is pretty close to 6.8. So, we could just guesstimate and say might be about 4.8 or so, but we can do a little bit better than that. And we're going to do so by interpolating between these two values. And we're going to assume that there's a linear change in the saturation vapor density with respect to temperature in this range from zero to five degrees. So, we're going to say that this can be modeled with a linear equation, <i>y</i> equals <i>mx</i> plus <i>b</i> where the <i>y</i> value is the dependent variable saturation vapor density and <i>x</i> is the independent variable being temperature and <i>m</i> is the slope and <i>b</i> is the y-intercept. So, we can say <i>y</i> equals the slope times <i>x</i> plus the y-intercept which is the <i>y</i> value when <i>x</i> is zero. And <i>x</i> is zero right here, so that means the <i>y</i> value is 4.84 when <i>x</i> is zero, that's the y-intercept. And then the slope is the change in <i>y</i>, 6.8 minus 4.84 divided by the change in <i>x</i> five degrees Celsius minus zero degrees Celsius. And this fraction works out to 0.392. And so here's our equation. And we can use this equation to figure out <i>x</i>. We can subtract 4.84 from both sides and then afterwards divide by this factor in front of <i>x</i>, that's slope in other words. And we get this equation to get <i>x</i>, the temperature. So the temperature is going to be the saturation vapor density minus 4.84 divided by 0.392. Now the saturation vapor density is going to be 6.71, and so we plug that in and this all works out to 4.77 degrees Celsius, which is close to our guesstimation of 4.8 but this is a number precise to the hundredths place. So this is the temperature at which there will be a dew point given a vapor density of 6.71 gram per cubic meter which is what we have when there's 39 percent relative humidity at 20 degrees Celsius.