Question

Which of the following is/are the correct explanation(s) for an increase in the adiabatic temperature gradient from the lower mantle (LM) to the outer core (OC)?

• A. OC has lower specific heat capacity and higher temperature than LM
• B. OC has higher specific heat capacity and lower temperature than LM
• C. OC has lower specific heat capacity and lower temperature than LM
• D. OC has higher specific heat capacity and higher temperature than LM

Hint:

The adiabatic gradient dT/dr is proportional to T/Cp; check how temperature and specific heat capacity each change going from silicate lower mantle into the metallic liquid outer core.

Answer 1

The Correct Option is A

The steepness of the adiabatic temperature gradient inside the Earth is fixed by the thermodynamic relation

\[ \left(\frac{dT}{dr}\right)_{ad} = -\frac{g\,\alpha\,T}{C_p} \]

where \(g\) is the local gravity, \(\alpha\) is the volumetric thermal expansion coefficient, \(T\) is the absolute temperature, and \(C_p\) is the specific heat capacity at constant pressure. This tells us that the magnitude of the adiabatic gradient scales directly with \(T\) and inversely with \(C_p\): a hotter material, or a material that stores heat less efficiently per unit temperature rise (lower \(C_p\)), has a steeper adiabat.

Two independent changes happen simultaneously as we cross the core-mantle boundary (CMB) from the base of the lower mantle into the liquid outer core:

• Temperature: A strong thermal boundary layer (the D" region) sits atop the CMB, across which temperature jumps by several hundred to about a thousand kelvin. So the outer-core side of the boundary is hotter than the lower-mantle side, \(T_{OC} > T_{LM}\).
• Specific heat capacity: The outer core is a dense Fe-Ni-rich liquid alloy, while the lower mantle is silicate (bridgmanite/post-perovskite + ferropericlase). Metallic liquids have a distinctly lower specific heat capacity per unit mass than silicate minerals (roughly 700-800 J/kg/K for the outer core versus about 1200 J/kg/K for lower-mantle silicates), so \(C_p^{OC} < C_p^{LM}\).

Because the adiabatic gradient is proportional to \(T/C_p\), an increase in \(T\) together with a decrease in \(C_p\) both act in the same direction and reinforce each other, producing a markedly steeper (larger) adiabatic gradient on entering the outer core. This is exactly the combination stated in option (A).

\(\boxed{\text{OC has lower specific heat capacity and higher temperature than LM}}\)