In Part V, when studying fluid dynamics, we shall encounter an adiabatic index

[Eq. (13.2)] that describes how the pressure P of a fluid changes when it is compressed adiabatically (i.e., compressed at fixed entropy, with no heat being added or removed).

Derive an expression for Г for an ideal gas that may have internal degrees of freedom (e.g., Earth’s atmosphere). More specifically, do the following.

(a) Consider a fluid element (a small sample of the fluid) that contains N molecules. These molecules can be of various species; all species contribute equally to the ideal gas’s pressure P =(N/V )k_BT and contribute additively to its energy. Define the fluid element’s specific heat at fixed volume to be the amount of heat TdS that must be inserted to raise its temperature by an amount dT while the volume V is held fixed:

Deduce the second equality from the first law of thermodynamics. Show that in an adiabatic expansion the temperature T drops at a rate given by C_VdT = −PdV.

(b) Combine the temperature change dT = (−P/C_V)dV for an adiabatic expansion with the equation of state PV = Nk_BT to obtain Г = (C_V + Nk_B)/C_V .

(c) To interpret the numerator C_V + Nk_B, imagine adding heat to a fluid element while holding its pressure fixed (which requires a simultaneous volume change). Show that in this case the ratio of heat added to temperature change is

Combining with part (b), conclude that the adiabatic index for an ideal gas is given by

a standard result in elementary thermodynamics.

Equation 13.2.

Transcribed Image Text:

r=- - P (ainv), V S (5.42)