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ideal_gas_analysis.tex
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\documentclass{article}
\usepackage[a4paper,left=2cm,top=2cm]{geometry}
\usepackage{parskip}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{hyperref}
\begin{document}
\title{Analysis of Superpressure Balloon using the ideal gas law}
\author{Richard Meadows 2016}
Analysis of Superpressure Balloon using the ideal gas law.
\[
P_{super} = P_{gas} - P_{air} \ \ \ \ (1)
\]
We can write the ideal gas equation for the gas inside the balloon:
\[
P_{gas}V = {m_{gas}\over{M_{gas}}} R T_{gas} \ \ \ \ (2)
\]
Since the system is floating, we know the mass of the air displaced is
$m_{system}$. So we can also write the idea gas equation for the air
displaced by the balloon.
\[
P_{air}V = {m_{system}\over{M_{air}}} R T_{air} \ \ \ \ (3)
\]
We assume the volumes are equal, so we can subsitiute one into the other.
\[
P_{gas} = P_{air} \bigg[ {{{m_{gas}\over{M_{gas}}} R T_{gas}}\over{{m_{system}\over{M_{air}}} R T_{air}}} \bigg] \ \ \ \ (4)
\]
Re-arrange and cancel $R$:
\[
P_{gas} = { P_{air} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } \bigg]} \ \ \ \ (5)
\]
Now we can use the definition of superpressure (1):
\[
P_{super} = P_{gas} - P_{air} \ \ \ \ (1)
\]
\[
P_{super} = { P_{air} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } - 1\bigg]} \ \ \ \ (6)
\]
Substituting in our expression for $P_{air}$:
\[
P_{air} = {{m_{system}R T_{air}}\over{M_{air}V}} \ \ \ \ (3)
\]
\[
P_{super} = { {{m_{system}R T_{air}}\over{M_{air}V}} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } - 1\bigg]} \ \ \ \ (7)
\]
\[
P_{super} = { {R\over{V}} \bigg[ { {m_{gas}}\over{M_{gas}} } T_{gas} - { {m_{system}}\over{M_{system}} } T_{air}\bigg]} \ \ \ \ (8)
\]
We define supertemperature in the same way as superpressure:
\[
T_{super} = T_{gas} - T_{air} \ \ \ \ (9)
\]
\[
P_{super} = { {R\over{V}} \bigg[ \Big( {m_{gas}\over{M_{gas}}} - {m_{system}\over{M_{air}}} \Big)T_{air} + {{m_{gas}}\over{M_{gas}}}T_{super} \bigg]} \ \ \ \ \ \ (10)
\]
We can reasonably say the superpressure due to the temperature dominates, so
\[
{P_{super}\over{T_{super}}} \approx {{m_{gas}}\over{M_{gas}}}{R\over{V}} \ \ \ \ \ (11)
\]
\end{document}