How Does Temperature Constancy Affect Atmospheric Pressure with Altitude?

In summary, the conversation discusses a problem related to the Ideal Gas Law and how to derive a differential equation for pressure using this law. The goal is to show that if the temperature in the atmosphere is independent of altitude, then the pressure can be represented by the function p=(p initial)e^(-mgt/kT), where m is the average mass per molecule of air. The conversation also explores different approaches to solving the problem, including using h instead of y and finding the rate of change of pressure with respect to h. However, the individual is still seeking guidance on how to get started.
  • #1
calculusisrad
20
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Ideal Gas Law Problem (Please help!)

Show that if the temperature in the atmosphere is independent of altitude, then the pressure as a function of altitude y is

p=(p initial)e^(-mgt/kT)

Where m is the average mass per molecule of air.

I really have no idea where to start. Please help! Thank you so much!

I thought maybe you use the ideal gas law and manipulate the variables to get it to match. I tried, but I couldn't get anywhere. Thanks
 
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  • #2


Try deriving a differential equation for P by using the ideal gas law and calculating the pressure from above and below for a slither of air

Also in the function you gave p is constant as a function of y (fix this and try using h instead of y this might trigger something)
 
  • #3


Thank you for your response, but I still can't figure out what to do. Find the rate of change of pressure with respect to h, right? doesn't that wind up as a constant if I'm using p=p+gy? That probably makes no sense lol, but I thought I'd show that I attempted to figure it out. I would really appreciate some more help and/or directly what I need to do to get started please!
 

1. What is the Ideal Gas Law and how is it used in problem-solving?

The Ideal Gas Law is a mathematical equation that describes the relationship between pressure, volume, temperature, and the number of moles of a gas in a closed system. It can be written as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature. This law is used in problem-solving to calculate any of the variables when the others are known, assuming the gas behaves ideally.

2. What are the units used in the Ideal Gas Law?

The units used in the Ideal Gas Law depend on the values of the variables. Pressure can be measured in atmospheres (atm), kilopascals (kPa), or millimeters of mercury (mmHg). Volume can be measured in liters (L) or cubic meters (m^3). Temperature can be measured in Kelvin (K) or Celsius (°C). The gas constant, R, has a value of 0.0821 L·atm/mol·K, which can be used to convert between different units.

3. How do I determine if a gas behaves ideally?

The Ideal Gas Law assumes that the gas molecules have no volume and do not interact with each other. In reality, most gases deviate from this behavior at high pressures or low temperatures. To determine if a gas behaves ideally, you can compare its behavior to that of an ideal gas under similar conditions. If the gas follows the Ideal Gas Law closely, it can be considered an ideal gas.

4. Can the Ideal Gas Law be used for all gases?

The Ideal Gas Law is an approximation and is not accurate for all gases. As mentioned before, most gases deviate from ideal behavior at high pressures or low temperatures. Additionally, gases with larger molecules or stronger intermolecular forces may not follow the Ideal Gas Law. In these cases, more complex equations, such as the van der Waals equation, may be used.

5. How can I use the Ideal Gas Law to solve real-world problems?

The Ideal Gas Law can be used to solve a variety of practical problems, such as determining the volume of a gas needed for a specific pressure and temperature, or finding the pressure inside a container given its volume and the number of moles of gas present. It is commonly used in industries such as chemistry, physics, and engineering to calculate gas properties and make predictions. However, it is important to consider the limitations and assumptions of the Ideal Gas Law when applying it to real-world situations.

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