Compressing a rod: The equation of state for a rod is:

1. Compressing a rod: The equation of state for a rod is: Again, if the compression is done quasi-statically (e.g., with a vise), then

2. From the eq. of state: So: Isothermal and adiabatic processes: Compressing the rod may generate some internal energy, and the rod may heat up. But if the process is carried out VEEEEEEERYYYYYYYY SLOOOOOOWLYYYYY then the heat will dissipate, and the rod will not change its T. We can then calculate the isothermal work:

3. However, if the process is donevery quickly, the Heat has no time to escape, so Q=0. Then the First Law becomes: 0 = ΔU + W Such a process is called adiabatic. More about quasi-static work: We have already calculated the work done by expanding gas:

4. In this apparatus, described on Page 20, the piston is loaded with sand. We can remove the sand grain after grain and thus slowly expand the gas. At any moment we can reverse the process – i.e., start putting sand grains back on the pile, and we can return to the initial state going through exactly the same intermediate states as when the gas was expanded – or going back along the same path In the p-V space.

5. HEAT CAPACITY It is an extremely important quantity in thermodynamics because it can be relatively easily measured (e.g., measurments the heat capacity of metals like Cu or Al are often done in high-school physics classes). Adding or removing thermal energy from a system (in the form o heat) results in a temperature change. But remember – this is not always true! You can keep adding heat to boiling water, and its temperature will be 212 F and will not change as long as there is water in the cauldron!!! But it is so because in the process of boiling water changes its phase – in other words, it undergoes a phase transition (a.k.a.“phase trans- formation”). So, the above applies only to systems that do not undergo a phase transition.

6. However, for now let’s forget about phase transitions. So, an infinitesimal reversible heat transfer causes a temperature change dT, and: The coefficient Cαis called the heat capacity. Why the subscript α ? Well, heat may be transferred to the system in different ways. We may keep one of the system Parameters constant, and α refers to that parameter – e.g.: for a process in which V = const. for a process in which p = const.

7. Heat capacities can be expressed in terms of derivatives of state parameters. This is a very important part of the heat capacity theory!!! Let’s recall the mathematical formula expressing the First Law: For a gas, expanded or compressed But we can also take advantage of the fact that U is a function of two parameters in this particular process – namely, U = U(T,V). So:

8. Combining the two equations, we get: If the volume is kept constant, dV = 0, so: Meaning that:

9. ENTHALPY: • In thermodynamics, in addition to the “natural” state • parameters, we introduce a number of state functions • that help to organize and to simplify the calculations. • The most important of these are: • The “Helmholtz free energy”, traditionally denoted as F ; • Enthalpy, traditionally denoted as H ; • The “Gibbs function”, traditionally denoted as G. • Mathematically, these functions are Legendre Transfor- • mations of the internal energy U, which can be though of • as a function of entropy S, volume V, and the number of • particles N. But we will not discuss the theory of the • Legendre transformation now. • In literature, the F, H, and G functions are often called • “thermodynamic potentials”.

10. The enthalpy is defined simply as: Let’s take the differential of H: But we also know that (First Law): By combining, we get: Because H is a state parameter, we can write H = H(T,p) and take the differential: Then, we combine the last two equations:

11. We get: At constant pressure, dp = 0, so that: for p = const. From which we immediately obtain:

12. EXAMPLES: A wire is kept at zero tension between two firmly planted vertical posts: Initial temperature is TH and the tension τ = 0. Then the wire is cooled down to temperature TL which causes that the wire gets tensioned because It “wants” to get shorter, but cannot, because the length L does not change!

13. Suppose that we know the Young modulus of the wire material, and its thermal expansion coefficient. We want to find the expression for the tensionin a cooled wire. Let’s take the length L as a function of T and the tension: Then: But dL = 0 , so that:

14. Now, we can use some coefficient that have been introduced in order to characterize the “response” of materials to external influences. The isothermal Young modulusETis a commonly Used “material characteristic” – you can find the values of ETlisted in many books – it is defined as: The coefficient of linear thermal expansion is defined as: (in other words, it’s the elongation of a wire of unit length heated up by a unit of temperature).

15. So, we can rewrite: If the temperature change is small enough that αL and ET do not change significantly, we get: