Experiment 106: Specific Heat

Analysis

Energy can manifest in many different ways, such as potential,

kinetic, thermal, electrical, chemical, nuclear, and other various forms.

However, one specific form of energy is thoroughly observed and

examined in this experiment, and that is heat energy.

Heat is simply referred to as the energy transferred from one body or

substance to another in physical contact with each other as the result of a

difference in temperature, but to fully grasp and understand the concept of

heat, we must first observe and analyze how it works and how it is applied

to our surroundings. We must first study the relationship between heat

and other forms of energy, particularly how thermal energy is converted to

and from other forms of energy and how it affects matter. To be precise, the

study of thermodynamics.

To begin, thermodynamics is the branch in physical science that deals

with heat and temperature, and their relation to energy, work, radiation,

and other properties of matter. In broad terms, thermodynamics deals with

the transfer of energy from one place to another and from one form to

another.

In this experiment, the focus will be about specific heat and

calorimetry. Specific heat capacity is the amount of heat energy

required to raise the temperature of a substance per unit of mass. The

specific heat capacity of a material is a physical property. It is also an

example of an extensive property since its value is proportional to the

size of the system being examined. In SI units, specific heat capacity

(symbol: c) is the amount of heat in joules required to raise 1 gram of a

substance 1 Kelvin. It may also be expressed as J/kg·K. Specific heat

capacity may be reported in the units of calories per gram degree Celsius,

too. Related values are molar heat capacity, expressed in J/mol·K, and

volumetric heat capacity, given in J/m3·K. Heat capacity is defined as the

ratio of the amount of energy transferred to a material and the change in

temperature that is produced:

C = Q / ΔT

where C is heat capacity, Q is energy (usually expressed in joules), and

ΔT is the change in temperature (usually in degrees Celsius or in

Kelvin). Alternatively, the equation may be written:

 Q = CmΔT

Specific heat and heat capacity are related by mass:

C=m*S

Where C is heat capacity, m is mass of a material, and S is specific heat.

Note that since specific heat is per unit mass, its value does not change,

no matter the size of the sample. So, the specific heat of a gallon of water

is the same as the specific heat of a drop of water. It's important to note

the relationship between added heat, specific heat, mass, and

temperature change does not apply during a phase change. The reason

for this is because heat that is added or removed in a phase change does

not alter the temperature.

Moreover, the specific Heat of a substance is the amount of heat needed

to raise the temperature of a unit mass of a substance by 1°. The table

shows the Specific Heats of substances.

Figure 1. Coffee-Cup Calorimeter

The second focus of the experiment is calorimetry. Calorimetry is a

method of measuring the heat transfer within a chemical reaction or

other physical processes, such as a change between different states of

matter. The term "calorimetry" comes from the Latin calor ("heat") and

Greek metron ("measure"), so it means "measuring heat." Devices used to

perform calorimetry measurements are called calorimeters. Refer to

Figure 1 of the coffee-cup calorimeter above. Since heat is a form of

energy, it follows the rules of conservation of energy. If a system is

contained in thermal isolation (in other words, heat cannot enter or leave

the system), then any heat energy that is lost in one part of the system

has to be gained in another part of the system. If you have a good,

thermally isolating thermos, for example, that contains hot coffee, the

coffee will remain hot while sealed in the thermos. If, however, you put

ice into the hot coffee and re-seal it, when you later open it, you will find

that the coffee lost heat and the ice gained heat ... and melted as a result,

thus watering down your coffee. Now let's assume that instead of hot

coffee in a thermos, you had water inside a calorimeter. The calorimeter

is well insulated, and a thermometer is built into the calorimeter to

precisely measure the temperature of the water inside. If we were to

then put ice into the water, it would melt - just like in the coffee

example. But this time, the calorimeter is continually measuring the

temperature of the water. Heat is leaving the water and going into the
ice, causing it to melt, so if you watched the temperature on the

calorimeter, you'd see the temperature of the water dropping.

Eventually, all of the ice would be melted and the water would reach a

new state of thermal equilibrium, in which the temperature is no longer

changing. From the change in temperature in the water, you can then

calculate the amount of heat energy that it took to cause the melting of

the ice.

Furthermore, calorimetry is the science associated with

determining the changes in energy of a system by measuring the heat

exchanged with the surroundings. In physics class (and for some, in

chemistry class), calorimetry labs are frequently performed in order to

determine the heat of reaction or the heat of fusion or the heat of

dissolution or even the specific heat capacity of a metal. These types of

labs are rather popular because the equipment is relatively inexpensive,

and the measurements are usually straightforward. In such labs, a

calorimeter is used. A calorimeter is a device used to measure the

quantity of heat transferred to or from an object. Most students likely do

not remember using such a fancy piece of equipment known as a

calorimeter. Fear not; the reason for the lack of memory is not a sign of
early Alzheimer's. Rather, it is because the calorimeter used in high

school science labs is more commonly referred to as a Styrofoam cup. It

is a coffee cup calorimeter - usually filled with water. The more

sophisticated cases include a lid on the cup with an inserted

thermometer and maybe even a stirrer.

We learned that water will change its temperature when it gains or

loses energy. And in fact, the quantity of energy gained or lost is given

by the equation.

Q = m water • Cwater • ΔTwater

Where Cwater is 4.18 J/g/°C. So, if the mass of water and the temperature

change of the water in the coffee cup calorimeter can be measured, the

quantity of energy gained or lost by the water can be calculated.

The assumption behind the science of calorimetry is that the energy

gained or lost by the water is equal to the energy lost or gained by the

object under study. So, if an attempt is being made to determine the

specific heat of fusion of ice using a coffee cup calorimeter, then the

assumption is that the energy gained by the ice when melting is equal to

the energy lost by the surrounding water. It is assumed that there is a

heat exchange between the ice and the water in the cup and that no

other objects are involved in the heat exchanged.

Q ice =-Q

surroundings = - Q calorimeter

This statement could be placed in equation form as the role of the

Styrofoam in a coffee cup calorimeter is that it reduces the amount of

heat exchange between the water in the coffee cup and the surrounding
air. The value of a lid on the coffee cup is that it also reduces the amount

of heat exchange between the water and the surrounding air. The more

that these other heat exchanges are reduced, the more true that the

above mathematical equation will be. Any error analysis of a calorimetry

experiment must take into consideration the flow of heat from system to

calorimeter to other parts of the surroundings. And any design of a

calorimeter experiment must give attention to reducing the exchanges of

heat between the calorimeter contents and the surroundings.

In thermodynamics, there are three modes of heat transfer-

conduction, convection, and radiation, a representation of all three

modes of heat transfer is shown in Figure 2.

Figure 2. Modes of Heat Transfer

 First is conduction, it is the mode of heat transfer particularly in

solids and also for liquid at rest. In this mode of transfer, the heat

transfers from one atom to its neighboring atom through molecular

vibrations. At the molecular level, first heat energy of a higher energy

level molecule converts to vibrating kinetic energy and this kinetic

energy is transferred to neighboring atoms and so on, such that the

process repeats until the temperature difference between two

neighboring atoms is zero.

The second mode of heat transfer is through convection which

particularly occurs in fluids in motion. That is in both liquids and gases

that are in motion. This mode of heat transfer occurs due to the transfer

of energy through bulk mass. In detail, whenever there is a temperature

difference in a fluid, density difference occurs, and motion of fluid starts

as lower density fluid attempts to reach the top of the fluid. During this

motion, mass and energy transfer occurs; thus, heat transfer takes place

which results to convection being the most efficient way to transfer heat.

Lastly, radiation is a mode of heat transfer that enables heat

transfer through a vacuum, or empty space. It is a method of heat

transfer that does not rely upon any contact between the heat source and
the heated object. Heat can be transmitted though empty space by

thermal radiation. Thermal radiation (often called infrared radiation) is a

type electromagnetic radiation (or light). Radiation is a form of energy

transport consisting of electromagnetic waves traveling at the speed of

light. No mass is exchanged, and no medium is required. When visible

light is absorbed by an object, the object converts the short wavelength

light into long wavelength heat. This causes the object to get warmer.

The main difference of the three modes of heat transfer may be visible

through an in-depth explanation of the three modes of heat transfer.

Conduction is the transfer of heat in a material due to molecular motion,

such that energy transfers through matter from particle to particle. In

this mode of heat transfer, a temperature gradient must exist to act as

the potential for the flow of heat. Heat will always flow from high

temperatures to cooler temperatures. The rate of heat transfer for

conduction is governed by Fourier’s Law of Conduction, which

expresses the rate of heat transfer from each end of a medium.

In Figure 3, the conduction rate equation for the Fourier’s Law of

Conduction is given; wherein, Q is the heat flow rate by conduction, k is

the thermal conductivity of body material, A is the cross-sectional area

normal to direction of heat flow, and dT/dx is the temperature gradient.

Figure 3. Fourier’s Law of Conduction

It is important to remember that the negative sign in Fourier’s

equation indicates that the heat flow is in the direction of negative

gradient temperature and that serves to make heat flow positive. The

thermal conductivity k is one of the transport properties like the

viscosity associated with the transport of momentum, diffusion

coefficient associated with the transport of mass. In addition, it provides

an indication of the rate at which heat energy is transferred through a

medium by conduction process. A few assumptions for the use of

Fourier equation: it is in steady state heat conduction, one directional

heat flow, bounding surfaces are isothermal in character that is constant

and uniform temperatures are maintained at the two faces, isotropic and

homogenous material and thermal conductivity k is constant, constant

temperature gradient and linear temperature profile, and no internal

heat generation. Some features of Fourier equation are as follows: it is

valid for all matter solid, liquid, and gas, the vector expression

indicating that heat flow is normal to an isotherm and it is in the

direction of decreasing temperature, it cannot be derived from first

principle, and it helps to define the transport property k.

When it comes to convection, it is the transfer of heat between a

solid surface and the adjacent fluid that is in motion by the actual

movement of the warmed matter. The faster the fluid motion, the greater

the amount of heat transferred via convective heat transfer. The idea of

forced convection occurs when the fluid is forced into motion by a fan,

pump, moving object, or another form of external energy introduced

into the system that results in the fluid flower over the solid surface. On

the other hand, forced convection, or also known as natural convection,

occurs when the fluid is forced into motion by buoyancy forces that are
induced as a result of changes in density due to changes in the

temperature of the fluid. The rate of heat transfer for convection is

governed by Newton’s Law of Cooling.

Newton’s Law of Cooling

Reynold’s number

Wherein, T(t) is the temperature of an object at a certain time, t is

the time, TS is the temperature of the surroundings, T0 is the starting

temperature of the object, and k is a cooling constant that is specific to

the object. All temperatures are in Kelvin, and the k indicated here has a

unit of s-1. The convective heat transfer coefficient “h” is a function of

the Reynold’s number. The more turbulent the flow, the higher the

convective heat transfer coefficient will be. In Figure 3, the equation for

Reynold’s number is given as the ration of inertia forces all over the

viscous forces; wherein, δ is the characteristic length, ρ is the density of

the fluid, V is the volume of the fluid, and μ is the viscous forces acting

onto the fluid.

Meanwhile, radiation is the transfer of heat in the form of

electromagnetic waves that directly transport energy through space.

Radiation heat transfer is temperature dependent, with the fourth

power, and can occur without a medium, or simply in empty space. A

black body is defined as a body that absorbs all energy incident upon it.

It also emits radiation at the maximum rate for a body of a particular

size at a particular temperature. Black bodies are perfect emitters and

absorbers. Black bodies are also known as cavity radiation that refers to

an object or system which absorbs all radiation incident upon it and re-

radiates energy which is characteristic of this radiating system only, not

dependent upon the type of radiation which is incident upon it. The

radiated energy can be considered to be produced by standing wave or

resonant modes of the cavity which is radiating. The amount of

radiation emitted in a given frequency range should be proportional to

the number of modes in that range. The best of classical physics

suggested that all modes had an equal chance of being produced, and

that the number of modes went up proportional to the square of the

frequency. As mentioned earlier, cavity mode is a mode for an

electromagnetic wave in a cavity that must satisfy the condition of zero

electric field at the wall. If the mode is of shorter wavelength, there are
more ways you can fit it into the cavity to meet that condition. Through

the careful analysis by Rayleigh and Jeans showed that the number of

modes was proportional to the frequency squared. From the assumption

that the electromagnetic modes in a cavity were quantized in energy

with the quantum energy equal to Planck’s constant times the frequency,

Planck derived a radiation formula.

Given the Einstein-Bose distribution function in Equation 1, the

average energy per “mode” or “quantum” times the density of such

states is expressed in terms of Equations 2 and 3. From equations 2 and 3

Planck was able to derive the radiation formulas in Equations 4 and 5;

wherein, h is the Planck’s Constant, c is the speed of light, λ is the

wavelength, T is the period, and k is the Boltzmann’s constant.

(1)

(2)

(3)

(4)

(5)
 Now that we understand the theories and concepts of

thermodynamics, heat, calorimetry, and specific heat that were stated

above, we can now move on to the experiment. In this experiment, we were

tasked to demonstrate the change in temperature upon mixing the

substance in a calorimeter and be able to calculate the specific heat of solid

metal.

The objectives of the experiment is to use the principles of

calorimetry, specifically the law of heat exchange, when combining objects

with different temperatures, and to use the law of heat exchange in

determining the specific heat of a solid metal. Proceeding to the

experiment, we first filled a beaker with water ¾ full and heat it using the

electric stove. Then, we immersed the metal into the beaker with boiling

water for heating by holding the string and make sure that the metal will

not touch the beaker. While boiling the metal, we also recorded the weight

of the inner cup of the calorimeter and filled it with tap water and recorded

its weight again. We now then prepared for the set-up by putting the

calorimeter cup into the outer shell with the cover, thermometer, and
stirrer. Then, we recorded the temperature of the water in thermal

equilibrium inside the calorimeter. After 20 minutes of boiling the metal,

we immediately add it into the calorimeter that we’ve prepared and then

we stirred it gently and waited for the reading of the thermometer to

become stable and record it. To calculate for the experimental value of

specific heat of the metal, we used the equation:

QLOSS (metal) + QGAINED (calorimeter & water) = 0

We then compared the experimental value and the accepted value of

specific heats of metal and computed the percentage error.

Table 1. Determining the Specific Heat of a Metal Specimen

Mass of Metal 45.60 g
Mass of Calorimeter 46.60 g
Mass of Water 292.63 g
Initial Temperature of Metal 100 °C
Initial Temperature of Calorimeter 27 °C
Initial Temperature of Water 27 °C
Final Temperature of Mixture 29 °C
Experimental Specific Heat of Metal 0.1835 cal/g*°C
Actual Specific Heat of Metal 0.2174 cal/g*°C
Percentage of Error 15.59 %

Conclusion
 Based on the data that we have gathered in the experiment; we can

conclude that the specific heat of the brass metal is quite low and will only

need a few amounts of heat energy to change its temperature by 1°C.

Yielding a result of 15.59% percent error, we were satisfied with the result

for the reason that there are external factor that were inevitable that

affected the percentage error during the experiment, such as room

temperature, the loss of electricity during the experiment which made heat

inconsistent, etc.

Since the specific heat is related to the excitation of the atoms and

how well they hold heat, we can conclude that the experiment is similar to

how a refrigerator function. According to the second law of

thermodynamics heat cannot spontaneously flow from a colder location to

a hotter area. If so then work is required to achieve this. Similarly, a

refrigerator moves heat from inside the cold icebox (the heat source) to the

warmer room-temperature air of the kitchen (the heat sink). The fan pulls
air through the evaporator fins where it is cooled by conduction and

circulated through the refrigerator by forced convection to cool the food.

That heat from the food is transferred into the refrigerant through the tube

walls from the fins by conduction. That heat plus the heat of compression is

rejected by conduction through condenser tubes and fins into the room and

is carried away by natural convection.