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.

 

[paranormal]

I would first clarify that the given values for specific heat capacity (CP), coefficient of thermal expansion (β), bulk modulus (κ), and molar volume (v) are all for copper. This information is important for accurately solving the problem.

For part A, you are correct that the change in internal energy is equal to the heat exchanged minus the work done. In this case, since the process is isothermal, there is no change in temperature and therefore no change in internal energy. This can also be seen from the equation for internal energy: ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat exchanged, and W is the work done. Since there is no change in temperature, ΔU and W are both zero, making Q also equal to zero.

For part B, you are correct that you need to take into account the mass of the copper. The heat exchanged in an isothermal process is given by the equation Q = -TΔV. Since the process is quasi-static, the change in volume can be calculated using the equation ΔV = βVΔP, where V is the initial volume of the copper and ΔP is the change in pressure. Plugging in the given values, we get Q = -(290K)(32x10^-6K^-1)(7cm^3mol^-1)(12x10^5Pa) = -268.8J. Note that the negative sign indicates heat is being transferred out of the copper and into the surrounding fluid.

For part C, the change in temperature can be calculated using the equation ΔT = T(βV/CP)(Pf - Pi). Again, we need to take into account the mass of the copper, so we can rewrite this equation as ΔT = T(βρ/CP)(Pf - Pi), where ρ is the density of copper. Plugging in the given values, we get ΔT = (290K)(32x10^-6K^-1)(8.96g/cm^3)/(16J/molK)(12x10^5Pa) = 0.258K. This is the temperature increase of the copper in an adiabatic process.

Overall, it is important to take into account the properties of the material (in this case, copper) when solving thermodynamic problems. It is also important to carefully read and understand the given