Climate Physics Chapter 2: Thermodynamics

How can we use thermodynamics to understand the structure of our atmosphere?

Temperature decreases with height until a critical height, called the tropopause, above which it increases with height. The atmosphere below the tropopause is called the troposphere, and the portion immediately above is the stratosphere. This is a typical pattern that we observe in many planetary atmospheres.

In thermodynamics, we are concerned with three main variables:

Temperature (\(T\)): Proportional to the average kinetic energy per molecule. We say two objects have the same temperature when they are in thermodynamic equilibrium, where heat flows from hotter areas to cooler areas no longer occur.

Pressure(\(p\)): Force per unit area exerted on a surface in a direction perpendicular to the surface.

Density(\(\rho\)): mass divided by volume.

The ideal gas law relates these three variables:

In the first equality, \(k_{B}\) is Boltzmann’s constant, with a value of \(1.3806\textrm{ kg}\left(\frac{\textrm{m}}{\textrm{s}}\right)^{2}\textrm{K}^{-1}\), and \(n\) is the number of molecules per unit volume. In the second equality, we instead write the equation in terms of mass density \(\rho\) and \(R\), the gas constant, defined as \(R=\frac{R^{*}}{M}\), where the universal gas constant \(R^{*}=8314.5\left(\frac{\textrm{m}}{\textrm{s}}\right)^{2}\textrm{K}^{-1}\) and the molecular weight \(M\) is an integer telling the number of protons and neutrons in a molecule.

A bicycle tire with mass \(0.1\) kg when empty and a volume of \(3\) liters is pumped up with Earth air to a pressure of 4 bars. Its temperature remains at the ambient air temperature of \(290\) K. What is the mass of the tire after it has been pumped up?

We have \(R_{dryair}=287\frac{\textrm{m/s}^{2}}{\textrm{K}}\).