Reading Question "2"

Read all of chapter 2:

• Before long we'll deal with issues related to the new sections 2.4 and 2.5.
• In 2.6, Carter makes elegant use of the 'cyclical relation' which should hold true for any system with that kind of regular state function we've been discussing.This section will be harder to digest. At this point, just skim this to see what's going on, and why the cyclical relation is useful.

  Questions to respond to:

  1. Looking at that 3-d part of the $P-v-T$ diagram in figure 2.3... Which of $P(v,t)$, $v(P,T)$, and $T(P,v)$ are *regular functions*...
     1. on the vapor-only part of the surface?
     2. on the liquid-only part of the surface?
     3. on the triple line?

     For a function of 1 dimension, you do a "vertical line test" for a function: see if there's any line parallel to the $y$-axis that pierces the graph of the function at more than one point.

     To test if $T(P,v)$ is a regular function, you should do a "vertical" line test--that is, lines parallel to the $T$ axis test--to see if any such line pierces the surface more than once. ...And then consider lines parallel to $P$ to test $P(T,v)$; and lines parallel to $v$ to test $v(T,P)$.

     Hugh found this 3-d manipulatable visualization of the surface in Fig. 2.3 that might also be useful.

  2. There are three paths on the $P-T$ diagram (below) of some substance.
     
     
     Give a short "narrative" for each path saying what the starting phase is, and then subsequent phases the system passes through as the pressure/volume/temperature changes (increases? decreases?) along each path.
  3. Current predictions for average warming due to excess carbon-dioxide in the atmosphere (global warming) are 2-4 C by 2100.

     Even if all the ice (and glaciers) did *not* melt, the oceans would still rise due to the thermal expansion of water.
     
     

     Water has an expansion coefficient at room temperature of $\frac{\Delta V}{\Delta T}\frac{1}{V}=\frac{\Delta V}{V}\frac{1}{\Delta T}=\beta\approx 2\times 10^{-4}$ / degree C, and the oceans have an average depth of 12,000 feet. View the oceans as "water in a glass", and assume that the dimensions of the containing "glass" do not appreciably change with such a small temperature rise. Take the higher value of 4C. By what amount would the sea level rise?[*]

     (Optional) Use the National Oceanic and Atmospheric Agency sea level rise viewer to estimate what parts of the United States would be at greatest risk! Make sure to examine the little piece of Goshen College in Florida: Our Marine Biology Laboratory is on Long Key in the Florida Keys.

  4. Carter says that the coefficient of volume expansion, $\beta$ , and the isothermal compressability, $\kappa$, are qualitatively different for liquids/solids and for ideal gases. In what way?
  5. Muddy questions? Things you wonder about?

  [*] Actually, water with a temperature close to freezing has a negative expansion coefficient. An appreciable fraction of ocean water is close to freezing. The IPCC estimates of ocean level rise are about 1/3 of what you have calculated using the assumption that all of the ocean is at room temperature.

  But still... NOAA: Miami broke an all-time record for high tide floods in 2019..