# Determining pressure at various altitudes using integration

• joseph_kijewski
In summary, the problem involves finding the normal pressure at different elevations assuming air density is proportional to respective pressure at each height. Using the ideal gas law, the equation can be simplified to dp=-dy*density*g. By substituting in the density as 1.29*P/P(sea level), the equation can be integrated to find the pressure at different elevations. However, errors in sign and forgetting the constant of integration may lead to incorrect results.
joseph_kijewski
1. The problem statement

Solve the following problems assuming air density is proportional to respective pressure at each height: What is the normal pressure at the atmosphere at the summit of a. Mt. McKinley, 6168m above sea level and b. Mt. Everest, 8850m above sea level c. At what elevation is the air pressure equal to one fourth of the pressure at sea level (assume the same air temperature).

## The Attempt at a Solution

I started by considering a block of air with the bottom at sea level and the top at the top of each of the mountains, such that (P+dp)A+dy*A*density*g-PA=0, which simplifies to dp=-dy*density*g. I can't figure out how to go on from here though, as the density is variable.

joseph_kijewski said:
the density is variable.
Yes, and it tells you how to handle that:
joseph_kijewski said:
assuming air density is proportional to respective pressure at each height:

I just don't really understand how to do that. I've been trying to formulate an equation but I assume it involves integration which I can do in some cases but don't conceptually understand how to apply to this problem.

joseph_kijewski said:
I just don't really understand how to do that. I've been trying to formulate an equation but I assume it involves integration which I can do in some cases but don't conceptually understand how to apply to this problem.
$$\rho=kp$$ where k can be determined from the ideal gas law using the pressure and temperature at the Earth's surface.

I might (?) be making progress here. Assuming temperature is constant, nRT should be constant as well, thus P is inversely proportional to V, and hence directly proportional to density. Thus I can substitute in 1.29*P/P(sea level) for density, yielding the equation dp=g*(1.29P/101,325)*dy. From there I get dp/P=1.247668*10^-4. I then integrate from 0 to 6168 meters and get lnP=1.247668*10^-4*6168. Unfortunately, the answer I get is way off. Where am I going wrong here?

joseph_kijewski said:
Where am I going wrong here?
Two errors.
You need to pay attention to signs. Does the pressure go up or down as y increases?
You forgot the constant of integration.

## 1. How is pressure at various altitudes determined using integration?

The pressure at different altitudes can be determined by using the equation P = ρgh, where P is the pressure, ρ is the density, g is the acceleration due to gravity, and h is the altitude. By integrating this equation, we can calculate the pressure at various altitudes.

## 2. What is the significance of determining pressure at various altitudes?

Determining pressure at various altitudes is important in understanding how air pressure changes with altitude and how it affects things like weather, aviation, and human physiology. It also helps in predicting and understanding atmospheric phenomena.

## 3. How does air pressure change with increasing altitude?

In general, air pressure decreases with increasing altitude. This is because the higher you go, the less air there is above you, resulting in less weight and therefore less pressure. This decrease is not linear and varies depending on factors such as temperature, humidity, and weather conditions.

## 4. Are there any limitations to using integration to determine pressure at various altitudes?

Integrating the pressure equation assumes that the change in pressure with altitude is constant, which is not always the case. Other factors such as temperature, humidity, and wind can also affect pressure at different altitudes. Additionally, the accuracy of the results may be affected by the precision of the measurements used in the integration process.

## 5. How is the pressure at sea level related to pressure at higher altitudes?

The pressure at sea level is considered to be the standard atmospheric pressure. As altitude increases, the pressure decreases. For every 100 meters increase in altitude, the pressure decreases by about 1 kPa. This relationship between altitude and pressure can be described using the pressure-altitude relationship or the barometric formula.

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