Question

.4 Recall that the adiabatic index is the ratio of the heat capacities at constant pressure and at constant volume. Show that, for an ideal classical gas, the critical temperature gradient for the onset of convection (3.23) can be written as dx con where Cp is the thermal capacity per unit mass at constant pressure and g is the acceleration due to gravity. (Note that if the thermal capacity is high because of the absorption of heat by the excitation and/or the dissociation of the consti- tuent particles, then the temperature gradient needed for convection is less steep.)

Answer 1

For adiabatic process pv^2 = a const: p (R T/p)^2 = a const: p^1-r T^r = a const: T^2 (1 - r) p^-r dp + r T^r - 1 p^1 - r dT = 0 (1 - r) dp + r p/T dt = 0 or, r p/T dT = (r - 1) dp dT/dx = T/p (r - 1)/r dp/dx we have r = c_p /w implies r - 1/r = 1 - 1/r = 1 - c_v/c_p = c_p - c_v/c_p = R/c_p therefore dT/dx = R T/p c_p (dp/dx) = v/c_p dp/dx, where p v = +R T we have, dp/dx = -row g rho = n/v dT/dx = -v/c_p rho g = -Mg/c_p dT/dn = -g/c_p, for unit mass