## Studio Session 11

### Thermometric properties

Any property of a substance which changes with temperature is called a thermometric property.   Some examples of thermometric properties are the length of a metal rod, the length of a column of liquid in a glass thermometer, the pressure of a constant volume of gas, the volume of a fixed amount of gas at constant pressure, etc.

The temperature is a measure of the internal, disordered energy of a substance.  The absolute temperature of any substance is proportional to the average kinetic energy associated with the random motion of the molecules of the substance.

½m<v2> = (3/2)kBT

In SI units the scale of absolute temperature is Kelvin (K).  But we use different temperature scales in everyday situations.  The Kelvin scale is identical to the Celsius (oC) scale, except it is shifted so that 0 oC equals 273.15 K.

If an equation in physics contains the temperature T, this temperature is always the absolute temperature.  In SI units it is always measured in K.  If an equation contains a temperature difference ΔT, this temperature difference can be measured in K or oC in SI units, since both scales give the same ΔT.

In this session you will investigate some thermometric properties of gases.  You will also complete an exercise on disorder and entropy.

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  Address the points highlighted in blue.  Answer all questions.

The ideal gas law states that for a fixed volume of an ideal gas PV/T = nR = constant.  Here P and V are the pressure and volume of the gas at absolute temperature T.  Theoretical derivations of the ideal gas law neglect the forces that the gas molecules exert on each other.  Real gases therefore do not strictly obey the ideal gas law.  However, at sufficiently low densities, intermolecular forces do not play a significant role and the ideal gas law becomes increasingly accurate.  For instance, at 20 atm pressure and room temperature, the volume of 1 mole of oxygen gas is about 2.3% smaller than predicted by the ideal gas law, but at 1 atm pressure the volume is only about 0.13% smaller.

• If the temperature T is constant the ideal gas law yields Boyle's law, PV = constant (at constant T).
• If the pressure is held constant, the ideal gas law yields Charles's law, V/T = constant (at constant P).

Exploration

Use an on-line simulation from the University of Colorado PhET group to verify Boyle's law and Charles' law.

(a)  Explore the interface!  (Click "Help" for hints.)

• You can pump two types of molecules into the chamber.
• You can specify constant volume, pressure, or temperature.
• You can work in an environment with or without gravity.
• You can heat up the gas in the chamber.
• You can use various measuring tools. (Explore what they can be used for.)

(b)  Reset the simulation, choose constant volume and a gravity-free environment and add gas to the chamber at 300 K until the pressure is between 1 and 2 atm.
Switch to constant temperature and verify Boyle's law.  Boyle's law states  that PV = constant for constant T, or P = constant/V.  A plot of P versus 1/V should yield a straight line.  If it does, you have verified Boyle's Law.

(c)  Switch to constant pressure and verify Charles' law.  Charles' law states V = constant * T for constant P, where T is the absolute temperature.  A plot of V versus T should yield a straight line.  If it does, you have verified Charles' Law.

(d)  Reset the simulation, choose constant volume and a gravity-free environment and add two species gas to the chamber at 300 K until the pressure is between 1 and 2 atm.

Compare the average speeds of the two types of particles.  Explain your results.

(e)  In an environment with lots of gravity explore how the pressure varies with height.

Exercise

We can state the second law of thermodynamics in various ways.

• Heat cannot, of itself, flow from a cold to a hot object.
• Heat cannot be taken in at a certain temperature with no other change in the system and converted into work.
• The total entropy of a closed system is always increasing.

### How do we connect heat flow and entropy?

Discuss and answer all questions (blue font color).

(a) Assume 50 students registered for a course.  You need to report the total number of students who actually attend.  You have every student in attendance sign a list.

• Which of the following would constitute a macrostate description of the class attendance and which would constitute a microstate description?
• A list of the names of each student present today.
• The number of students in attendance.

(b)  Consider the following macrostates of the class.

1. All 50 students are present.
2. 25 students are present.
3. 1 student is present.
4. No students are present.
• Which macrostate(s) would have the highest entropy?
• Which macrostate(s) would have the lowest entropy?

(c)  Assume you have two boxes and you have to place a coin into each box.  You toss a coin and place it into box 1 without changing its orientation.  The side of the coin facing up can be either head (blue) or tail (red).  Then you toss a second coin and place it into box 2 without changing its orientation.  You are interested in how many coins end up "head up" or blue.  There are 4 different microstates.

• How many macrostates are there and what is the multiplicity Ω of those macrostates, i.e.  how many microstates correspond to each macrostate?

The entropy of a macrostate is defined as S = kB lnΩ, where kB is the Boltzmann constant.  Let us measure S in units of kB and use S = lnΩ.

• What is the entropy of each macrostate?

(d)  Now assume that you have a 6 by 6 array of squares.  You number the squares from 1 to 36.  You toss a coin and place it onto square 1 without changing its orientation.  The side of the coin facing up can be either head (blue) or tail (red).  You keep tossing coins and placing them in succession onto squares 2 through 36 without changing their orientation.  Each square holds exactly one coin.
To specify the microstate of the array after you finished this process you have to list the color of each square.  Many different patterns (microstates) are possible, such as the ones shown on the right.  Every particular pattern is equally likely to occur.  There are 236 = 6.87*1010 possible patterns and the probability of observing any particular pattern is 1/236 = 1.46*10-11.

Suppose you are only interested in the macrostate, i.e.  you are only interested in how many squares are blue.  Most macrostates correspond to many microstates.  A few of the allowed microstate for the "15 blue squares" macrostate are shown on the right.

The graphs on the right show the multiplicity Ω for each "n out of 36" macrostate, the probability of observing this macrostate and its entropy S = lnΩ.

Which statement below best describes the relationship between probability and entropy for the macrostates we are considering?

1. The most probable macrostates have the lowest entropy.
2. The most probable macrostates have the highest entropy.
3. There is no clear relationship between entropy and probability for these macrostates.

If we examine the probabilities for all of the macrostates in the system we find that the probability is

1. spread evenly among all of the macrostates.
2. spread out among a large fraction of the macrostates, but with some macrostates having slightly higher probabilities than others.
3. concentrated within a small fraction of the macrostates centered on a most probable macrostate.
4. entirely concentrated into a single most probable macrostate.

How does this behavior scale with the number of coins N?

• If you have N squares and N coins, the most probable number "head up" coins is N/2.  The spread of the distribution increases a N½, so the percentage spread decreases as N-1/2
• For 36 square, the spread approximately plus or minus 6, the percentage spreads is (1/6)*100% or 16.6%.  That means that if n differs by more than 16.6% from 18, then it is not very likely to occur.
• If there were N = 1024 squares and coins (just as there might be 1024 molecules in a gas), then the percentage spread is (10-12)*100% = 10-10%, that means in n differs from N/2 by more than 10-10% from 0.5*1024, it is not very likely to occur.

(e) Consider approximately 1024 gas molecules in a box.  A divider which can conduct heat, for example a rubber diaphragm, separates the box into two chambers of equal size.  Each chamber contains half of the molecules.  Assume that the internal energy of the gas is (approximately) fixed.

• Describe some possible macrostates of the gas in the two chambers of the box.
• Describe a low entropy macrostate.
• Describe a high-entropy macrostate.
• In which situation described below is the entropy of the gas greatest?
1. When the energy is spread nearly evenly among all the molecules and all the molecules are clustered in small regions of each chamber.
2. When the energy is concentrated in a small number of molecules that are spread nearly evenly throughout the box.
3. When the energy is spread nearly evenly among all the molecules and all the molecules are spread nearly evenly throughout the box.
4. When the energy is concentrated in a small number of molecules which are clustered in a small region of one chamber of the box.

Newton's laws work equally well backwards in time as they do forwards.  If a sequence of motions and interactions leads to an increase in entropy, then the time-reversed sequence will lead to a decrease in entropy.  Both sequences are allowed by Newton's laws.

• Why does heat, of itself, not flow from a cold to a hot object, if it is allowed by Newton's laws?
• What, do you think, determines, the rate of change of the entropy of a system?

Entropy can decrease locally (in a subsystem).  Heat can be moved from a cold place to a hot place.  (We all are familiar with refrigerators and heat pumps.)  But it takes ordered energy that comes from outside the subsystem.  For the entropy of a subsystem to decrease, the system as a whole must not yet have reached a state of maximum entropy.