Introduction

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

Theory

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:

(1)

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:

(2)

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.

Methods

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:

(3)

Where

is

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

is

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:

(4)

Where P is the power of

the electrical heater and

is

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:

(5)

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

KERN EW4200-2NM

Liquid Nitrogen

Heating Coil

Lascar psu 130

Dewar

flask

Cork

Wires

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

get:

(6)

Then by substituting in equation 1 we get:

(7)

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.

Results

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

and

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

N

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:

5V:

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

Literature

value: 199 Jg-1 3

So

our values are consistent with each other but only within 5? and 3?

respectively of the literature value so not particularly accurate.

Specific Heat

Capacities:

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

in

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:

Lead,

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.

Discussion

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.

Conclusion

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.