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## Homework Statement

Derive an expression for the change of temperature of a solid material that is compressed adiabatically and reversible in terms of physical quantities.

(The second part of this problem is: The pressure on a block of iron is increased by 1000 atm adiabatically and reversible. What is the temperature change? The initial temperature of the iron is 298K. You are also given α,β, and cp for Iron.)

## Homework Equations

$$ dS=\frac{c_P}{T} dT - Vα dP$$

## The Attempt at a Solution

My initial thought was to use some combination of $$PV=nRT$$ or $$dU=c_V dT$$ but those only apply to ideal gases, and it is given that this is a solid.

I then decided to do a Legendre transform of the equation for dS, so I could get S(T,V). The logic here was that I know ΔS=0 since this is reversible, and so I could get an equation with dT (which I am trying to solve for) and dV (since this is compression, which is a volumetric change). The transform gave me

$$dS=(\frac{c_P}{T} - \frac{Vα^2}{β}) dT + \frac{α}{β} dV$$

From here, after I set dS=0, I solved for dT.

$$dT=\frac{-αT}{c_P β+Vα^2 T} dV$$

This is not a separable differential equation and I don't see a way to really solve it.

I'm pretty confident that the math is all correct, but I'm not sure if my approach is.