Skip navigation      The “baby kinetic theory” required here is
a simple model of atoms bouncing elastically about inside a
container. Their average energy is given to be
[tex]\frac{1}{2}mv^2_{\rm RMS} = \frac{3}{2}k_B T,[/tex]
implying that the root-mean-square speed scales as the one-half
power of the temperature: [tex]v_{\rm RMS} \propto
T^{1/2}.[/tex] The other necessary ingredients are Clausius’s
definition of the entropy, [tex]dQ = TdS,[/tex] where for
business majors we say that the “d” just means “a small amount”:
add a small amount of heat dQ to a system at temperature T, and
you increase the entropy by the small amount dS. We also use the
conservation of energy in its thermodynamic clothing, [tex] dU =
dQ + dW = dQ – PdV,[/tex] and the Ideal Gas Law [tex] PV = Nk_B
T = \alpha U.[/tex] The coefficient α is 3/2 for a monatomic
ideal gas (in three dimensions).  Consider an isothermal
process, i.e., one the gas stays at constant temperature. The
Ideal Gas Law tells us that the energy U must also remain