Studio session 10

In this studio session you will analyze the performance of a simple heat engine and review the various ways to state the second law of thermodynamics.

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.


A heat engine

How can we do work with heat?  How can we convert some disordered energy back into ordered energy?

In this exercise you will analyze data obtained with is a "real" thermal engine that can be taken through a four-stage expansion and compression cycle and that can do useful mechanical work by lifting small masses from one height to another.  You will determine the useful mechanical work done by the engine by measuring the vertical distance y a mass m is lifted.  You will compare this mechanical work Wmech = mgy to the net thermodynamic work done during a cycle.  The pressure as a function of volume is recorded for one cycle, and the net thermodynamic work done by the engine equals the enclosed area on the PV diagram.
W = ∑jPjΔVj.  We sum the work done during a large number of small changes of the volume.

imageThe PASCO TD-8572 Heat Engine/Gas Law Apparatus is used to obtain the data.  The heart of this apparatus is a nearly friction-free piston-cylinder system.  The graphite piston fits snugly into a precision-ground Pyrex cylinder so that the system produces almost friction-free motion and negligible leakage.  The Heat Engine/Gas Law Apparatus is designed with two pressure ports with quick-connect fittings for connecting to an air chamber and a pressure sensor with tubing.

Your instructor will take you through a cycle of the PASCO TD-8572 Heat Engine.  (For a short video of data acquisition click here.)

After you have made your measurements, the instructor will also demonstrate two different types of real working heat engines to you.

Experiment

Procedure:

With no mass on the platform, the piston is raised ~4 cm, and the air chamber with tubing is connected to the engine.  The piston stays at a height of ~4 cm.

The computer is set up to record the pressure with respect to some reference pressure (in units of kPa), the volume of the gas in the cylinder with respect to some reference volume (in units of cm3), and the position of the cylinder above its starting position (in units of m) as a function of time and to produce plots of pressure versus volume and position versus time.

Measurement:

Data analysis:

mass
 
Work done by heat engine
from P-V diagram (J)
Change in potential
energy of mass (J)
% difference
 
0.1 kg      

Paste your table into your Word document and answer the following questions.


Exercise

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

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.

(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.

image(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.

image(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.

 

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

 

image
image
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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.
(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.)

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?

(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.

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. 

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.


Convert your log into a lab report.

Name:
E-mail address:

Laboratory 10 Report

Save your Word document (your name_lab10.docx), go to Canvas, Assignments, Lab 10, and submit your document.