The magnitude of the atmospheric pressure at different heights above sea level is shown in the diagram. The atmospheric pressure is shown at sea level and at a height ℎ at which the atmospheric pressure is 0.5 times the atmospheric pressure at sea level. One atmosphere of pressure is equivalent to 101 kilopascals of pressure. How does the atmospheric pressure change as the height above sea level increases?
To answer this question, let’s look at our graph, which shows us the height above sea level plotted against atmospheric pressure. We can see that our vertical axis begins at sea level and then moves up to the very top of Earth’s atmosphere. On the horizontal axis, we see that the atmospheric pressure values range from zero pressure up to a pressure of one atmosphere. For this part of our question, we want to see how atmospheric pressure changes as we move up through Earth’s atmosphere.
We’re going to give a general answer to this question. That is, we’ll describe the general trend of this curve that we see plotted. If we begin at an elevation of sea level, we see that the corresponding atmospheric pressure there is one atmosphere. However, as we move up through the atmosphere all the way to the top, we see we reach an atmospheric pressure of zero pressure. As our height above sea level has increased then, the atmospheric pressure has decreased. That then will be our answer to this part of the question. As the height above sea level increases, the atmospheric pressure decreases.
Let’s look now at part two of our question.
How does the rate of change of the atmospheric pressure change as the height above sea level increases?
This may seem similar to the question we just answered. But in fact, now we’re talking about the rate of change of atmospheric pressure and how that changes. That is, we can see that as we ascend through the atmosphere, atmospheric pressure correspondingly decreases. What we now want to figure out is, for example, if we move through some set height through the lower portion of the atmosphere and atmospheric pressure changes as a result, how does that change in atmospheric pressure compared to the change if we move through the same set distance but through the upper atmosphere. Which one of these equal changes in elevation result in a greater change in atmospheric pressure? If we can answer that, then we’ll have an understanding of how the rate of change of atmospheric pressure changes as height above sea level increases.
Here’s one way we can think about this question. Notice that atmospheric pressure is marked out at one atmosphere, at 0.5 atmospheres, and at zero pressure or zero atmospheres. So the change in atmosphere from one to 0.5 atmospheres is the same as from 0.5 to zero. Earlier, we saw that at sea level, the atmospheric pressure is one atmosphere. Looking now at our vertical axis, we notice that if we move upward a height ℎ from sea level, then we’re then at an altitude with a corresponding pressure of 0.5 atmospheres. In other words, by moving up from sea level a height ℎ, we’ve cut the atmospheric pressure in half.
Now notice how the height ℎ here compares to the remaining height from that height all the way up to the top of the atmosphere. The remaining distance is much larger than ℎ. And yet over this remaining distance, as we rise through that portion of the atmosphere, the atmospheric pressure, we see, only changes by 0.5 atmospheres from 0.5 atmospheres down to zero pressure. Therefore, this relatively small change in altitude through the lower atmosphere resulted in as larger change in atmospheric pressure as this much larger change in elevation here. This confirms to us that atmospheric pressure changes much more rapidly in the lower portions of the atmosphere than in the upper portions.
Therefore, as height above sea level increases as we move up from that elevation, the rate of change of atmospheric pressure decreases. As we move from sea level up through the lower atmosphere, atmospheric pressure changes relatively rapidly. But the closer we get to the top of Earth’s atmosphere, the less atmospheric pressure changes for a resulting increase in height.
Let’s look now at part three of our question.
What is the atmospheric pressure at sea level to the nearest kilopascal?
We’ve seen that at sea level the atmospheric pressure in units of atmospheres is one atmosphere. But in this part of our question, we want to report that pressure in units of kilopascals. In our problem statement, we’re told to treat one atmosphere as effectively equal to 101 kilopascals. Since the atmospheric pressure at sea level is one atmosphere, to the nearest kilopascal, that’s 101 kilopascals.
The next part of our question asks this, what is the atmospheric pressure at the top of Earth’s atmosphere, to the nearest kilopascal?
Way up here at the top of Earth’s atmosphere, we’ve seen that the pressure is zero. That tells us our answer because regardless of the units we use, zero pressure in whatever units will be zero. At the top of Earth’s atmosphere to the nearest kilopascal, the pressure is zero kilopascals.
Let’s look now at the last part of our question.
Is the height ℎ at which the atmospheric pressure is 0.5 times the atmospheric pressure at sea level more than, less than, or exactly halfway from sea level to the top of the atmosphere?
We see this height ℎ on our graph just above sea level. Earlier, we saw that the distance from height ℎ all the way up to the top of the atmosphere is much greater than from sea level up to height ℎ. In this part of our question, we’re comparing these two distances in blue and in orange. We see that the length of the orange line is much less than the length of the blue one. This tells us that the height ℎ is less than halfway between the sea level and the top of Earth’s atmosphere.