Equation p(h): Explained and Derived From Boltzmann's Law

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In summary, the equation p(h) = (p0e)-(mgh/kT) is known as the Law of Atmospheres and is derived from the hydrostatic equilibrium equation. It predicts the pressure at a given height, h, in terms of the surface pressure, p0, temperature, T, and other constants including Boltzmann's constant, k, and the mean molecular mass of air, m. The value of e in the equation represents Euler's number. It should also be noted that g, the acceleration due to gravity, is not the same as G, the gravitational proportionality constant.
  • #1
yeveat
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p(h) = (p0e)-(mgh/kT)

where p0 is the atmosphere at sea level, m is the mass of an object at height h, g is the gravitational proportionality constant...

is there a specific name for this equation? is this derived from the Boltzmann’s distribution law? Also, I'm really confused about what e is... Thank you!
 
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  • #2
e is Euler's number, about 2.7. Search wiki for it.

It's the pressure at a height, h at a temperature T. k is Boltzmann's constant.
 
  • #3
Also g is not the same thing as G. G is the gravitational proportionality constant, g is the acceleration due to gravity.
 
  • #4
Also,the p_o should not be inside the parentheses. Only the e is being raised to that power. It is not a Boltzmann factor, that would be statistical physics. This is a solution to a force equation, the hydrostatic equilibrium that looks like dp/dh = -rho * g, where rho = mp/kT from the ideal gas law and p is gas pressure.
 
  • #6
And m is the mean molecular mass of air.
 

1. What is the significance of the equation p(h)?

The equation p(h) is a mathematical representation of the probability of finding a particle at a certain height (h) in a gas system. It is derived from Boltzmann's Law, which describes the relationship between the energy of a particle and its probability of being in a particular state.

2. How is the equation p(h) derived from Boltzmann's Law?

The equation p(h) is derived by considering a system of particles in a gas at different heights, with each particle having a specific energy level. Boltzmann's Law is then used to calculate the probability of each particle being in a particular energy state at a given height. These probabilities are then summed up to obtain the overall probability of finding a particle at a certain height.

3. What is the role of Boltzmann's Law in p(h)?

Boltzmann's Law is a fundamental equation in statistical mechanics that relates the energy of a particle to its probability of being in a particular state. In the context of p(h), Boltzmann's Law is used to calculate the probabilities of particles at different heights, which are then combined to obtain the overall probability distribution.

4. What are the applications of the equation p(h)?

The equation p(h) has various applications in fields such as thermodynamics, statistical mechanics, and gas dynamics. It can be used to analyze the behavior of particles in a gas system, predict the distribution of particles at different heights, and study the properties of gases at different temperatures and pressures.

5. How can the equation p(h) be experimentally verified?

The equation p(h) can be experimentally verified by conducting experiments on gas systems and measuring the probabilities of particles at different heights. These probabilities can then be compared to the values predicted by the equation p(h) to validate its accuracy and applicability.

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