Download FYSA241/1 Thermodynamic Research

1
FYSA241/1
Thermodynamic Research
In this laboratory work, a metal wire is loaded and/or heated and its temperature and geometric
changes are measured. The purpose is to introduce the student to the different thermo physical
effects and processes.
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The Effect of Adiabatic Stretching on Temperature
In thermal physics the following relation for the temperature dependence of pressure has been
derived for reversible adiabatic processes:
 P 
 S 

  

 T  S
 T  P
 S 
 
 P T
(1)
 S 
By using the definition of the heat capacity C P  T 
 and Maxwell's thermal physics equation
 T  P
 V 
 S 
 equation (1) can be written as:
   
 T  P
 P  T
 C   V 
 P 

  P  

 T  S  T   T  P
(2)
And thus
dT 
T
CP
 V 

 dP
 T  P
(S = constant)
(3)
This relation describes a system's temperature change as a function of pressure in an adiabatic
process. This result can be applied to an experiment in which a wire is under load.
Transformations P   F and V  L can be done, where F is the stretching force and L is the
length of the wire. The negative sign of the force is justified, since the volume of gas reduces with
increasing pressure while the length of the wire increases when the stretching force increases.
Defining that dP   F A and dV  Adl , where A is the cross section of the wire, we get a relation
 T
dT  
 CF
 L 

 F
 T  F
(S = constant)
(4)
The minus sign indicates that when a metal wire is stretched, its temperature decreases.
Correspondently, the wire warms, if its load is made smaller. The total temperature change is of the
order of 0.1 K which is tiny compared to the room temperature T  293K  . This allows one to
write
2
 T  L 

T  
 F
 C F  T  F
(5)
The temperature coefficient of length is defined as

1  L 


L  T  F
(6a)
 L 

 T  F
(6b)
or
L  
Additionally, it is sufficiently accurate to assume that C F  C P , thus equation (5) becomes
T  
TLF
CP
(S = constant)
(6c)
For experimental purposes, this relation can be expressed as
T  
 T 
TLF
 F ,
 
cP m
 c P m1 
(7)
where C P  c P m , c P is the specific heat capacity of the wire at constant pressure, m is mass of
wire  r 2 L  m1 L ,  is the wire’s density and r is the radius of the wire’s cross section. For
the purpose of observations, the temperature coefficient of length  can be expressed as
 L  1

 T  L
 
2
(8)
Work Done in the Stretching of a Wire
In a reversible process, work dW done on the system has a magnitude dW   PdV and in the case
of the wire dW  FdL . Note that elastic limit of the wire may not be exceeded and that the force
has to be kept harmonic. In the following, it is assumed that all other changes in the wire can be
neglected.
There are two forces acting on the wire, a constant force trying to stretch the wire and an
approximately harmonic tension force. Total work done by these forces equal to
W
F
L
L  F
2
2
(9)
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In the opposite situation, when the load is removed, work is negative. According to its definition,
L dF
, where A is the cross section of the wire. On the basis of observed
elastic coefficient  
A dL
values,  can be obtained from the relation

Thus L 
L F 


A  L 
(10)
LF
and work is
A
W 
LF 2
2 A
(11)
We see that the work is proportional to the force squared.
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Thermodynamic Quantities
During an adiabatic process a temperature difference is created between the wire and the
environment. This will stabilize due to heat conduction. An amount of heat energy
Q  c P m1 LT
(12)
is transferred to the wire.
According to the sign agreement, heat energy absorbed by the system is positive. Thus, the heat
energy is positive if the temperature of the system increases in a stabilizing process.
The entropy change S which occurs simultaneously is found from relation dQ  TdS . When T
here is practically constant, then
S 
Q
.
T
(13)
The first law of thermodynamics governs the total change of the systems internal energy during the
process
E  Q  Wint  Q  W ,
(14)
where Wint is the work done by the system during the adiabatic phase and Q is the heat absorbed
subsequently. That is, the change in the internal energy is equal to the difference between the heat
energy absorbed and the work done by the system. When load is added, it holds that E  Q  W .
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4
The Experimental Set-Up
The experimental set-up is shown in figure 1. Two identical metal wires are suspended on a holder
and placed inside acrylic tubes to slow down thermal effects. Wire B is loaded with a constant load
L and wire A can be loaded using the lever and a single mass of 2.5 kg which can be placed in one
of the three notches on the lever bar. These three notches correspond to an applied force of 98, 74,
and 49 Newton.
Figure 1. Schematic view of the experimental setup
A thermocouple is attached between the wires and allows one to measure the temperature
difference between the wires as a function of voltage. In order to be able to heat the wire A, it is
connected via resistor E to the isolated voltage source D. The wire heats up max 25 K over the
room temperature in the experiment. Wire B is at the reference temperature, that is, T = Tr = room
temperature (≈ 293 K). The room temperature should remain constant during the experiment or else
the temperature change must be taken into account. The voltage supplied by the thermal element is
directed to a computer via a DC amplifier and a micro-voltmeter.
The change in the length of wire A is about one millimetre. It is detected with an optical reading
device described in figure 2 which shows the relationship between the laser spot's vertical position
(y1) and the change in length of wire A. The reading device is an optical system consisting of a
laser and a mirror. The mirror is attached to the wires and is swivelled as the length of wire A
changes. The laser beam is aimed to the mirror and from the mirror it is reflected to a reading scale
with a magnification ratio of 100:1. The reflection angle is twice the position angle of the mirror.
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As one can see in figure 2, that
y / d  tan 2   2 tan  /(1  tan 2  )  2 tan  (when β < 6º),
where tan   L / e . The mirror’s base e = 21.0 mm and the distance between the mirror and the
reading scale is 50 × e = 105 cm (check this!). Thus it holds
e
y


L  
y 
100
 2  50  e 
(15)
Figure 2. The method of measuring the change in length of the wire.
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Measurements
5.1
Preparations
When necessary, focus the laser beam. Aim the beam to the mirror so that the reflected light falls
on the screen. Check that the distance between the mirror and the screen is right. Check the zero
level of the lever-bar. You can load the lever pushing it with your hand but be careful not to cut off
the wire. The laser spot should return to its initial position after loading. To stabilise the measuring
equipment, turn it on before the actual measurement and let it warm for a while.
The laser is connected to a 4.5 V DC battery and positioned so that the laser spot is visible on the
height scale located on the adjacent wall. The thermocouple leads are connected to the Keithley
microvoltmeter, which should be set to have a full-scale reading of 3 µV. The output of the
Keithley meter is connected to a ANALOG input of a data logging box using a voltage sensor and
banana cables. The data-logging box is interfaced to a PC. Turn the PC on and start the
DATASTUDIO software. Set the software to measure voltage (Voltage Sensor). By clicking the
START key the software should begin recording data and present it as a graph. The voltage scale in
DATASTUDIO is 10 V.
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Before conducting actual measurements, values displayed using DATASTUDIO need to be
calibrated. This is done by starting to log data with the Keithley microvoltmeter set to 3 µV scale
and using the zero knob. The needle is kept at the –3 µV level and then held there for about one
minute. Then the needle is positioned to the –2 µV level for another minute. This is repeated up
until the +3 µV level. The resulting step graph gives the relationship between the values displayed
and those measured by the Keithley meter.
Take into account that the amplifier at 3 µV scale is so sensitive that even the smallest changes in
the surroundings, ex. movement and electrical effects may disturb the experiment. However, high
sensitivity is essential for successful measurements because the temperature changes during
adiabatic stretching are very small.
5.2
Adiabatic Stretching
The following is the procedure to follow for all three of the applied forces 98, 74 and 49 N.
1. The zero knob on the Keithley microvoltmeter is adjusted so that the no-load output is about +2
µV (the scale is kept at 3 µV).
2. Start collecting "no-load" data for about 1 minute.
3. Record the "no-load" height of the laser spot (y0)
4. Gently place the mass on the relevant notch (start with the notch corresponding 98 N) on the
lever bar and release it gradually, but not too slowly, without knocking or disturbing the system.
5. Wait until the recorded voltage signal has stabilised (3-4 min) and returned to initial signal
magnitude. On the computer screen you should have something similar to fig. 3.
6. Stop collecting data and save file.
7. Record final height of laser spot (y1).
8. Remove mass from lever bar and allow wire to shrink back to an equilibrium length (10
minutes).
Figure 3. Adiabatic stretching with a load of 98 N. The data has been modified so that all the data
points are on the positive side of y-axis.
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5.3
Determination of the Temperature Coefficient of Length
The following procedure is followed for this section of the experiment.
Remove bolt and all masses from the lever bar.
Set Keithley microvoltmeter to the 1 mV scale.
Allow the wire to reach an equilibrium state.
Record the room temperature.
Start collecting data for 1 minute.
Record the position of the laser spot (y0) and the amplifier reading.
Connect a 200V voltage to the heating circuit of wire A.
When a stationary state has been reached, record the position of the laser spot (y1) and the
corresponding voltage reading from the microvoltmeter.
9. The room temperature is recorded again.
1.
2.
3.
4.
5.
6.
7.
8.
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Data analysis
Determining T 's value in adiabatic stretching graphically
Fit an exponential curve through the data points of the measured voltage curve. Extrapolate the fit
function to time t = t0 when the load was set on the lever. The change in the voltage reading with
respect to zero level gives (after calibration) the change in EMF in the thermal element. In the
copper-constantan thermal element a change of 1 K in temperature corresponds to a change of 41.7
µV in EMF.
The following are characteristics of the copper wire used in the experiment.
r = 0.56 mm
L = 1.44 m
Using the observed values and the theoretical relations, calculate the following values along with
their experimental errors.
1.
2.
3.
4.
5.
6.
7.
8.
The temperature coefficient of length of the wire ()
Theoretical prediction for T of the wire during adiabatic stretching
The determination of T graphically from measurement
Work done in stretching of the wire
Heat energy transferred to the wire after stretching
Entropy change
Change in the system’s internal energy
Coefficient of elasticity (for 98N trial only)