Introduction to a gas. The formula for this


The objective for this experiment
was to calculate the latent heat of vaporisation of Nitrogen and then find the
specific heat capacities for Aluminium, Copper, Graphite and Lead. This was
achieved largely by calorimetry, which is to measure the changes in a dynamical
system to derive the heat transfer associated with a body’s change of state due
to outside variables. Joseph Black was the founder of this science in 1761 when
he deduced that heat and temperature are different entities. His theory of
latent heat paved the way for a new science, thermodynamics, and he also found
that different substances had different specific heats. However Joseph thought
of heat as an invisible fluid called the ‘caloric’ where objects can hold a
certain amount of the fluid, hence the term heat capacity was coined. It was
not until later, in the 18th-19th centuries, that
scientists abandoned the idea of the caloric and instead viewed heat as the
measure of internal energy of a body.1

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In this experiment we obtained
latent heats and specific heat capacities of various substances. Latent heat is
the thermal energy change in a body when undergoing a phase change at constant
temperature. In our experiment we will be measuring the latent heat of
vaporisation which is the energy required for a liquid to turn to a gas. The
formula for this is quite simple:



Where Q is the amount
of energy absorbed by the liquid during the phase change, L is the specific
heat capacity for particular substance and m is the mass of the substance.

Specific heat is the
heat capacity per unit mass of material. Heat capacity is the ratio of the heat
added to or removed from a substance to the resulting temperature change. The
formula for this is:



Where c(T) is the
specific heat capacity of a substance depending on temperature, Tf and
Ti are Temperature initially and finally respectively and Q and m
are the same as before.


Measuring the Latent
Heat of Vaporisation of Liquid Nitrogen:

For this part of the experiment
we only knew the mass of the Nitrogen from equation 1 but if we take the
differential of both sides with respect to time we can get:




the rate at which heat is transferred to the liquid Nitrogen’s surroundings and

the rate at which mass of the Nitrogen is lost due to this heat transfer.
Finding the rate of heat transfer would be very difficult and an easier
variable to measure would be to use an electrical heater with a known power to
disperse energy into the nitrogen. This turns equation 3 into:



Where P is the power of
the electrical heater and

the rate of mass lost due to the heat transfer caused by the heater. Now
subtracting equations 4 from 3 and rearranging we come to the final equation
needed for finding the latent heat of vaporisation of nitrogen:



P can be measured from
the current and voltage of the heater with the simple relation P=VI. So the
latent heat of nitrogen can be found by measuring the rate of mass loss first caused
by the surroundings and then by adding the heater to the liquid nitrogen and
measuring the new rate of mass loss.

Figure 1: Latent heat Calorimeter

Digital Scales

Liquid Nitrogen

Heating Coil
Lascar psu 130




The apparatus was set up as shown in figure 1 and the scales were
zeroed with the flask on it so that we would only be measuring the mass of the
liquid nitrogen. The flask was filled to approximately half way with liquid
nitrogen and then it was left for at least three minutes so that the flask and
the nitrogen could reach thermal equilibrium. Then measurements of the
decreasing mass on the scales were taking using a computer program to record
the readings from the scale roughly 4 times a second. Once sufficient data had
been taken for the mass loss due to the surroundings, 5-10 minutes is more than
enough data points, it was time to add the heating coil to the nitrogen, simply
inserting the electrical heater through the cork hole and setting the power on
the heater, again taking measurements for 5-10 minutes. As only the rate of
mass loss was being measured it was not of concern that the coil may add some
weight to the scales. The coil was added during the recording of measurements
so we would be able to see the change in rate of mass loss on one graph. Two
electrical power settings were used in this experiment: 10V and 5V so when the
first run of measurements finished, the flask was replenished with liquid
nitrogen to roughly the same level and the process was repeated.

Measuring the specific heat capacities of
solid samples:

If it is assumed that specific heat capacity doesn’t vary
significantly with temperature, then equation 2 can be solved and rearranged to



Then by substituting in equation 1 we get:



 represents the mass lost from the liquid
nitrogen sample which will give us values for the heat energy change in the
system. So when we submerge our sample materials we need to measure the mass
change. L is now known from the previous experiment and so all that is needed
is a thermometer to measure the initial room temperature, the final temperature
being -196°C. When the liquid nitrogen was poured into the flask and sufficient
time had been given for them to be in thermal equilibrium, measurements started
recording for 5 minutes before inserting the solid sample and keeping the
measurements going for another 5-10 minutes or so. The mass lost due to the
solid sample was obtained from the plots of mass of liquid nitrogen vs time and
the mass of the sample had to be taken into account as the sample were dropped
into the flask.

Additionally the
specific heat capacity was worked out by putting the samples in dry ice (solid
CO2) until it was at thermal equilibrium and then placing the sample
into the Dewar flask with Nitrogen. For the Lead sample, it was cooled to -30°C
and 0°C as well. These additional measurements were made to see if there is any
relationship with the temperature difference and the specific heat capacity.

It was the same
setup as before except the heating coil was replaced by solid samples of
aluminium, lead, graphite and copper.


Latent heat of
vaporisation of Nitrogen:

Our data wat input inot origin and turned into graphs
showing mass of liquid nitrogen vs time. Two linear fits were put on the graph
before and after the heater was added to get



Figure 2: Plots of Mass of Nitrogen
vs Time with 5V and 10V heater
Note that
for all of the graphs there are too many data points for the error bars to
be worthwhile showing











Then these graphs were differentiated using origin and the
linear function of the background mass loss was taken and then in excel
subtracted from every point of the rate of mass loss from the heater. The
average of all the values was then found graphically by plotting the new points
against time, taking a linear plot and setting the gradient to zero.

Figure 3: Differentiated plots of
mass vs time where a linear fit has been plotted on each of the background
mass loss data points










Figure 4: New plots with dM/dt
representing the difference between the heater and background values and
the gradient for the linear fit set to zero in order to find the average
value from the intercept












So with the values being averaged simply take the positive
value of the y intercept (gradient for mass loss is going to be a negative so
doesn’t matter) as a replacement for

 and L was found to be:

L = 231 ± 6 Jg-1                        10V: L = 230 ± 10 Jg-1

value: 199 Jg-1 3

our values are consistent with each other but only within 5? and 3?
respectively of the literature value so not particularly accurate.                                                                                                                                                                                                       


Specific Heat

Again the data taken was plotted in origin to give us lots of graphs!






















































Figure 5: Several plots showing
various samples at differing initial temperatures in liquid nitrogen. Graphs
are plotting the rate of mass lost before and after the samples were added.
The slope of the mass loss after the sample has been added was set to zero
as theoretically the rate of mass loss shouldn’t change.



















The specific heat capacities were therefore calculated by finding

each graph by taking the difference in intercepts and accounting for the mass
of the sample. Then adding that to equation 7 we get our values:

From which the average specific heat capacity was taken for
each sample to get:

Lead: c =
0.14 ± 0.01 Jg-1K-1

Copper: c = 0.35 ± 0.02 Jg-1K-1

Aluminium: c
= 0.71 ± 0.04 Jg-1K-1

Graphite: c = 0.42 ± 0.07 Jg-1K-1

Compared to
literature values of:

Copper, Aluminium, Graphite: c = 0.13, 0.38, 0.91, 0.71 Jg-1K-1


All of the samples went into a roughly 20 Kelvin flask so
perhaps lead and copper aren’t affected as much by the lower temperatures as
aluminium and graphite as these literature values are for higher temperatures
like room temperature.


In this experiment the latent heat of vaporisation of
nitrogen was measured using a heating element to measure the boil off rate of
the nitrogen, subtracting the background rate of mass loss and getting two
values of L that while weren’t particularly close to the literature value but
were however very similar results suggesting perhaps there was some sort of
systematic error involved. We then measured the specific heat capacity of
various samples using the liquid nitrogen to create a temperature. The values
were lower for Aluminium and Graphite likely due to specific heat’s
relationship with temperature at very low temperatures. Looking at Debye’s
theory of specific heat it says that at very low temperatures; for non-metallic
materials, i.e. graphite, specific heat is proportional to the temperature
cubed.2 The temperature of liquid nitrogen is about 20 Kelvin so
it is possible that the Debye T3 law was being observed and hence
would differ from the value we obtained. A different law applies for metals
however in which it was found that electrons contribute to the specific heat,
but is only important for low temperatures in metals where it becomes
significant enough to be added to the T3 law contribution.2
So again it’s possible that the aluminium sample was affected by this, however copper
is one of the best heat exchangers which causes some confusion as to why copper
was not affected along with lead.

During the experiments we were conscious of the fact that
the evaporation of nitrogen is proportional to the distance from the top of the
Dewar flask, so we tried to keep the level of nitrogen at the same place but
during the individual measurement runs the level would obviously go down and so
therefore the rate of mass loss would have decreased as the nitrogen level got
further away from the Dewar flask opening so this could be a source of
systematic error. Another source of possible error could have come from the
heating element. We checked the voltage and current before adding it to the
flask but didn’t check throughout to see if it was the same and so it could
have fluctuated without us knowing. Also when the solid samples were dropped
into the flask they would have sunk to the bottom and some of the surface area
would have been touching the edge of the flask which would have been slightly
warmer than the centre of the nitrogen.

So if these experiments were to be tried again, things to
improve would be: to measure the heating elements voltage and current
throughout to make sure it is constant and to suspend the solid samples from a
clamp stand so that it is in the centre of the nitrogen and makes measuring the
mass loss easier as you don’t have to account for the mass of the sample. Also
using more power settings for the heater could have been interesting to see if
varying voltage affected the final value of L.


When measuring latent heat of vaporisation using two power
values of a heating element, 5V and 10V respectively, the values of L obtained
were:        which differ from the
literature value of 199 J/g. We then measured the specific heat capacity of
Lead, Copper, Aluminium and Graphite where we obtained values: (0.14 ± 0.01) Jg-1K-1, (0.35 ± 0.02) Jg-1K-1,
(0.71 ± 0.04) Jg-1K-1 and (0.42 ± 0.07) Jg-1K-1
which differed from the literature values of 0.128, 0.385, 0.897 and 0.717 Jg-1K-1
respectively where the differences in aluminium and graphite are likely due to
Debye specific heat at low temperature.



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