Latent Heat of Vaporization and Thermal Expansion

Abstract

Thermodynamics is the branch of science that studies the movement of energy between
various systems and how it transforms from one form to another. It mainly focuses on the
connections between temperature, heat, work, and energy. Systems typically possess the
capacity to transfer energy, encompassing both heat and work, with their surroundings. The
transfer of energy often hinges on temperature disparities, although it can also take place
during shifts in the phase of a substance, even when the temperature remains constant. In our
forthcoming experiments, we will ascertain the latent heat of vaporization of distilled water
and the coefficient of linear expansion of a metal.

Introduction

2.1) Heat - Heat capacity and specific heat

Primarily, heat is defined as the quantity of energy transferred across the boundary of a system
due to the presence of a temperature difference between the system and its surroundings.
When we heat a substance, energy moves into it from a source of higher temperature. For
example, if you place two objects with different temperatures next to each other, heat will
transfer from the hotter object to the cooler one until they both reach the same temperature.

In the past, scientists considered heat as a fluid known as "caloric," and they used the unit
"calorie" (cal) to measure it. A calorie represented the amount of energy needed to heat 1 gram
of water from 14.5°C to 15.5°C. However, in recent times, the unit "joule" (J) has become the
standard for describing thermal processes worldwide.

When we add energy to a substance without performing any work, and there is no change in
the substance's state (meaning it doesn't transition between solid, liquid, gas, or plasma), its
temperature typically rises. To quantify this heating process, we introduce a value called heat
capacity, denoted as C. Heat capacity represents the amount of energy required to increase the
substance's temperature by 1°C. Following this definition, we can express the relationship as
follows:

Q = CT
Where the temperature change ΔT results from the input of heat Q, it's clear that heat capacity
is expressed in units of J/°C (or J/K). If we aim to calculate the heat capacity per unit mass, we
obtain a distinct parameter called specific heat, denoted as c. When combined with the earlier
definition, this relationship can be expressed as follows:

Q = mcT

This implies that a higher specific heat indicates a greater amount of heat is needed to raise the
temperature. Therefore, specific heat can be thought of as a measure of how resistant a
material is to temperature changes when energy is added. Put simply, this value varies
depending on the substance; for example, it's 4186 J/kg∙°C for water, but just 387 J/kg∙°C for
copper.

Furthermore, it's worth mentioning that specific heat can vary with temperature. Nevertheless,
when the temperature range is not too extensive, this variation is so minimal that it can be
considered nearly constant. Water serves as an excellent example of this situation, as its
specific heat only changes by 1% from 0°C to 100°C at standard atmospheric pressure.

2.2. Conservation of energy

In the previously described situation involving two objects at varying temperatures, we

determined that energy flows from the hotter object to the cooler one. In line with the
principle of energy conservation, the energy entering the cooler object must be equivalent to
the energy departing from the hotter object. This connection can be mathematically
represented as:

Qcold =−Qhot or Qcold +Qhot =0.

The negative sign here accounts for the heat leaving the hotter sample.

Now, let's explore an example. Imagine we have two samples with masses m1 and m2, initially
at different temperatures, T1 and T2, respectively, and they are in contact with each other.
Over time, they both reach a common equilibrium temperature, Tf. Suppose we have
knowledge of the specific heat, c1, for one of these samples, and we aim to determine the
specific heat, c2, for the other material. If the system is isolated, the law of energy conservation
provides us with the following outcome

m1c1 (Tf−T1)
m1c1 (Tf−T1)+m2 c2(Tf−T2) ; Thus c2 = -
m2 (Tf−T2)
2.3. Latent heat

Heat transfer doesn’t always necessitate on a temperature differences; it can also transpire
when the temperature remains constant. These occurrences are typically linked to phase
transitions, where the physical properties of a substance shift from one state to another. Two
well-known phase transitions include melting (transitioning from a solid to a liquid) and
vaporization (changing from a liquid to a gas), with their reverse processes being freezing and
condensation. All of these phase changes involve a modification in internal energy. For
instance, when water boils, heat is supplied to disrupt the intermolecular bonds within the
liquid, allowing the molecules to disperse further in the gaseous state, which results in an
increase in intermolecular potential energy.

To quantify these processes, we define a measure of the quantity of heat needed to induce a
phase transition in 1 kilogram of substance as "latent heat," so named because it doesn't
necessitate a temperature difference. Its unit is joules per kilogram (J/kg). Similar to specific
heat, the latent heat is contingent on the properties of the material involved and also varies
according to the specific phase transition. In fact, there are two distinct types of latent heat: the
latent heat of fusion and the latent heat of vaporization, corresponding to the two
aforementioned familiar phase transitions.

𝑄
L =𝑚

2.4. Thermal expansion

We often notice instances where a glass can shatter when hot water is poured into it, or during
summer, metal railway tracks can deform on their own. These are examples of thermal
expansion resulting from an increase in temperature.

In a solid, atoms are organized in a regular pattern and are connected by electrical forces,
which can be modelized to a network of springs linking the atoms together. Consequently,
these atoms are in a constant state of oscillation around their equilibrium positions within the
structure, regardless of the temperature. When the material is heated, the atoms begin to
vibrate with greater intensity, causing their average separation to increase. This in turn leads to
the expansion of the entire solid.

Linear expansion pertains to alterations in any of a solid's dimensions, including its length,
width, and thickness. If we designate the initial dimension as "L" and the increase in that
dimension ΔL due to a temperature increment of "ΔT," empirical observations indicate that
within a limited temperature range, the elongation of length is directly proportional to the
temperature increase and the original dimension. In other words, "L" can be represented as:
 L =  LT

Here, α represents the coefficient of linear expansion. By manipulating this equation, it's
possible to derive an expression for the coefficient of linear expansion.

𝛥𝐿/𝐿
⍺= 𝛥𝑇

This value varies among different materials. However, it is also influenced by the actual
temperature and the chosen reference point for determining "L," which, for practical purposes,
is often negligible in measurements.

3. Measurement of the vaporization heat

We are provided with the following equipment to carry on the experiment: (A) a water
reservoir with heating element and distilled water, (B) water condenser, (C) beakers, (D)
ampere- and voltmeters, (E) water tap, toroid transformer (beneath the desk), digital
scale, and stopwatch.

Figure 1 – Experiment setup (source: M12_guide.pdf (unideb.hu))

 Figure 2 – Toroid transformer (source: M12_guide.pdf (unideb.hu))

To begin with, it is essential to perform an initial equipment inspection. Commence by filling up

the reservoir with distilled water, ensuring that the water level is maintained at a height of 10-
12cm above the heating element. Subsequently, initiate the flow of cold water into the
condenser by opening the tap. As a precautionary measure for safety, it is necessary to seek the
instructor's permission before connecting the toroid transformer to an AC voltage power
source. Following this, accurately measure the mass of an empty and dry beaker, which should
be 53.35g ± 0.01g. Finally, place this beaker at the distal end of the condenser to collect the
condensed water.

The transformer knob serves the purpose of adjusting the current, thereby influencing the
heating power. By turning it in a clockwise direction, you can raise the current. Adjust the knob
until the current reaches 2.5A to facilitate the heating of the water to its boiling point.

After the water has reached a full boil, reduce the current back to 1A and allow approximately 3
minutes to pass by for equilibrium to be established. Following this, gradually increase the
current in increments of 0.25A until it reaches 2.5A. At each step, wait for 1 minute to ensure
that the system attains a new equilibrium, and then measure the mass of the condensing water
over a period of 3-15 minutes (at the lowest current, measure for 15 minutes).

It's important to note that higher heating power leads to quicker water vaporization, resulting
in a greater amount of condensed water in a given time. Therefore, as the current increases,
the measurement time can be reduced.
The energy produced by the transformer, P = UI, is utilized to convert liquid water into steam.
However, some heat inevitably escapes into the surrounding environment. Assuming a constant
rate of heat loss for all heating powers, the principle of energy conservation can be applied,
yielding the following relationship.

𝑚
Pt = Lm + wt = P = L 𝑡 + w ,

In this context, where L represents the latent heat and w signifies the rate of heat dissipation to
the nearby surroundings, we observe a linear connection. Hence, the latent heat of
vaporization can be determined by fitting all the gathered data using the least squares method.

mass of Beaker + Mass of

Current(A) Voltage(V) water water(g) time (s) M/t(kg/s) Power(w)
1 92.5 65.08 65.08 15min30s 1.260E-05 92.5
1.25 120 71.15 71.15 10min 2.960E-05 150
1.5 145 83.1 83.1 8min 6.19E-05 217.5
1.75 170 87.87 87.87 6min 9.58E-05 297.5
2 197.5 93.25 93.25 5min 1.33E-04 395
2.25 217.5 95.6 95.6 4min 1.76E-04 489.375
2.5 247.5 94.77 94.77 3min 2.30E-04 618.75

The heat of vaporization is the slope: L = 2390706.42 J/kg ≈ 23.9∙105 J/kg.

The rate of heat loss is w = 70.59 W.

4. Measurement of coefficient of linear expansion of metal

The following equipment is provided for the experiment: (A) the metal tube of interest
mounted on a suitable stand, (B) a dial indicator, (C) thermostat with thermometer, (D)
water tap, and tape measure.

Figure 3 (source: M12_guide.pdf (unideb.hu))

To commence, verify the adequacy of the distilled water present in the thermostat's water
reservoir for our measurement. Subsequently, activate the water tap to enable the flow of cold
water into the thermostat, serving the purpose of cooling. Power on the thermostat and allow
it to run for approximately 5 minutes without applying heat, enabling an equilibrium to form
between the thermostat and the metal tube. Simultaneously, measure the length of the tube
and adjust the dial indicator to read zero.

Next, activate the thermostat's heating function and systematically elevate the tube's
temperature to reach 45°C. It's crucial to increase the temperature gradually, maintaining a
maximum rate of change at 0.5°C per minute, which can be adjusted using the thermostat's
knob. Throughout this process, record the expansion of the tube at intervals of 3°C. Once the
tube has attained the intended maximum temperature.All the acquired data should be
meticulously recorded and entered into the table on the subsequent page.
Ultimately, by applying linear fitting to the relationship between the relative elongation and the
temperature variation, it becomes possible to determine the coefficient of linear expansion of
𝛥𝐿/𝐿
the tube, ⍺ = .
𝛥𝑇

Initial Tempreature ( C) 21.3

Original Length 60.1 601000 μm

Process Tempreature Elongation(μm) Change in temp (C) Relative Elongation

25 47 3.7 7.82E-05
28.4 83 7.1 0.000138103
31.2 109 9.9 0.000181364
33.9 128 12.6 0.000212978
36 165 14.7 0.000274542
Heating 39.2 200 17.9 0.000332779
42.3 231 21 0.000384359
45.2 260 23.9 0.000432612

The coefficient of linear thermal expansion of the metal tube is the slope of the fitting
line:

⍺ = 1.78 * 10^-5 1/K

5. Conclusion

The heat of vaporization of water was accurately determine with the calculated value being
5.7% higher ( 23.9•10^5 J/Kg) than the expected book value(22.6•10^5 J/Kg). Additionally most
data points on the graph closely follow the fitted line, indicating precision.

The second experiment aimed to measure the coefficient of thermal linear expansion and was
successfully achieved. The instructor advised repeating the experiment for the cooling process
would be redundant as the slopes for both the processes of heating and cooling would be
approximately the same.