Isothermal and Adiabatic Compression of a Solid

[Spiral1183]

Homework Statement

A 200g cylinder of metallic copper is compressed isothermally and quasi-statically at 290K in a high-pressure cell.
A) Find the change in internal energy of the copper when the pressure is increased from 0 to 12kbar.
B) How much heat is exchanged with the surrounding fluid?
C) If the process is instead carried out adiabatically, find the temperature increase of the copper.
For copper,
C_P=16J(mol*K)^-1, β=32x10^-6K^-1, κ=0.73x10^-6atm^-1, and v=7cm³mol^-1

Homework Equations

For the isothermal compression, we found Q=-Tvβ(P_f-P_i)

For the adiabatic change in temperature, ΔT=T(βv/C_P)(P_f-P_i)

The Attempt at a Solution

For part A, it is asking me to find the internal energy, which is the heat minus the work done correct? If there is no change in temperature, wouldn't this be zero?

For part B in the isothermal process, is it as simple as plugging in the known variables to find the heat exhange? I'm thinking that I should be taking into account the mass of the copper which should change the volume, correct?

Part C I am looking at the same way, but not sure if I need to factor in the mass of the copper.

 

[Chestermiller]

The internal energy is a function of temperature only for an ideal gas. It is also a good approximation for solids, but in this case, the pressure change is very large, and you are being asked to determine the change in internal energy as a function of pressure at constant temperature.

 

[Spiral1183]

So in that case what equation would I use to determine the change in internal temperature? Everything I am finding seems to deal with ideal gas or changes in temperature with constant pressure, not the other way around.

 

[Chestermiller]

So in that case what equation would I use to determine the change in internal temperature? Everything I am finding seems to deal with ideal gas or changes in temperature with constant pressure, not the other way around.

I don't know what level of thermo you are taking. But even many introductory texts derive expressions for the partial derivatives of the thermodynamic functions with respect to pressure at constant temperature (or the partial derivatives with respect to specific volume at constant temperature). Hopefully your textbook covers this. If not, see Smith and van Ness, or Hougan and Watson.