# Physics Tutorial: Temperature. The Zeroth Law of Thermodynamics

In this Physics tutorial, you will learn:

• What is temperature?
• How the speed of molecules is related to temperature?
• What are the devices used to measure the temperature?
• What are the units of temperature?
• How is the temperature related to the thermal energy of objects?
• What is thermal equilibrium?
• What does the Zeroth Law of Thermodynamics say?

## Introduction

All us are quite familiar with the term "temperature". When we touch a hot object, we say "the object's temperature is high". On the other hand, we consider the temperature of ice as low, as for us "the ice is cold." In a certain sense, we define temperature as "the degree of hotness in an object". But this definition is intuitive, and it makes reference to our senses. We cannot use this approach to determine correctly the degree of hotness, i.e. the assign an exact numerical value to temperature. A cup of tea can be "warm" for one person who is inside the room and "very hot" for another person who is just entering house during a winter day. Thus, we must find a universally applicable way to measure the numerical value of temperature, i.e. of the degree of hotness in an object.

In this tutorial, we will explain in detail the meaning of temperature along with its units and measuring devices. Therefore, by the end of this tutorial, every doubt on this issue will be clarified.

## The Molecular Meaning of Temperature

All particles of any object in whatever state it may be are in unceasing motion. In general, they vibrate around some fixed points known as "equilibrium positions". It is obvious that faster the particles' vibration, greater their kinetic energy.

It is impossible to measure the kinetic energy of all particles of an object due to the very big number of particles (but also because such motion is very irregular). Molecules of a objects may have a wide range of velocities, and furthermore, the speed of a molecule changes millions of times in a second due to the collisions with other molecules. At a given instant one molecule may nearly be at rest while another molecule moves with almost the speed of light during vibrations. Consequently, the kinetic energies of molecules of an object are quite different. But an average value of their kinetic energy gives us an idea about the behavior of an object's particles, especially in gases. Therefore, we use an indirect way to estimate the kinetic energy of an object's particles (in other words its thermal energy), which is one of the main components of their internal energy (the other is the chemical energy generated during chemical reactions). The only way to draw a correct conclusion regarding the objects' degree of hotness is by finding an appropriate way to measure their warmth. This is achieved by introducing a more direct measurement than particles' average kinetic energy. As a result, a few centuries ago the concept of temperature was introduced.

By definition, temperature represents a measure of the ability of a substance, or more generally of any physical system, to transfer heat energy to another physical system.

Temperature as a concept is closely related to the average kinetic energy of all particles of the object or system. This means temperature is a more suitable quantity related to the measurements of objects' warmth than heat energy, because its value can be measured directly through devices called thermometers. A thermometer uses the expansive of contractive properties of liquids such as alcohol or mercury to show different values. But first, a thermometer needs to be graded. For this, a lower fixed point and an upper fixed point are needed. The process of assigning different values to different heights of liquid column in a thermometer is known as "calibration of thermometer".

## Units of Temperature

Anders Celsius, a Swedish scientist introduced (in the eighteenth century) a practical method of thermometer calibration. He used the freezing and boiling points (temperatures) of water as a lower and upper fixed points respectively and then he decided to divide the range of temperatures between these two fixed points in 100 equal parts. Each division is called 1-Celsius degree (1°C)and such a way of thermometer calibration is known as Celsius Scale. Obviously, such a calibration can be extended even beyond these two fixed points as we know that in winter air temperatures can go below zero or melting process of metals needs temperatures much higher than 100°C.

On the other hand, Daniel Gabriel Fahrenheit, a German scientist, used the temperatures in the coldest and the hottest days in his country to calibrate thermometers. In this way, he invented the Fahrenheit Scale. The conversion formula between Celsius and Fahrenheit scales is:

t (°F) = 1.8 × t (°C) + 32°

### Example 1

The weather forecast shows a value of 86° F for the next day. What is the corresponding value in Celsius degree?

### Solution 1

From the formula

t (°F) = 1.8 × t (°C) + 32°

we obtain after substituting the values:

86 = 1.8 × t (°C) + 32
1.8 × t (°C) = 86 - 32 = 54
t (°C) = 54 / 1.8
t (°C) = 30°C

However, the official SI unit of temperature is none of the above but Kelvin Scale. It is named after the Lord Kelvin, alias William Thompson, the Scottish famous scientist. Kelvin degree is the SI unit of temperature as it offers a great advantage compared to the other units: it has only positive (or zero) values because the lower fixed point of this scale refers to the lowest temperature in the universe, i.e. the temperature in which particles of matter stop vibrating around their equilibrium positions. When measured in Celsius degree, this minimum temperature is equal to -273.16°C. Therefore, we can write:

0 K = - 273.16°C ≈ - 273°C

or more generally,

T(K) = t (°C) + 273°

The division method is the same in both Celsius and Kelvin degrees, only the lower fixed points are different. This means an increase in temperature by 5°C for example, represents an increase in temperature by 5 K as well.

### Example 2

Convert the following temperatures into the required ones:

1. 5 °F = . . . . . °C = . . . . . K
2. 373 K = . . . . . °C = . . . . . °F

### Solution 2

a) Given that

t (°F) = 1.8 × t (°C) + 32°
5 = 1.8 × t (°C) + 32
1.8 × t (°C) = 5 - 32 = -27
t (°C) = -27 / 1.8
t (°C) = -15°C

When converted this value into Kelvin degree, it becomes

T(K) = t (°C) + 273°
T(K) = -15°C + 273°
= 258 K

b) First, let's convert the temperature from Kelvin to Celsius Scale. Thus, given that

T(K) = t (°C) + 273°

we obtain, after substitutions for the temperature in Celsius scale,

373° = t (°C) + 273°
t (°C) = 373° - 273°
= 100°C

Remark!

Unlike in the other two scales, in Kelvin scale temperature is denoted by capital T instead of t. Also, the symbol of degree (°) is not written in Kelvin scale.

## Temperature and Thermal Equilibrium. The Zeroth Law of Thermodynamics

As written in the definition, temperature represents a measure of the ability of an object or a system of objects to transfer heat energy to another object or system. But before explaining what this definition really means, it is necessary to explain a few concepts, such as heat energy, thermal energy, internal energy and heat transfer.

Objects possess energy in a variety of forms. Some types of energy are easily visible and measurable such as kinetic energy, gravitational potential energy and elastic potential energy we have explained earlier in Section 5. However, there are some other types of energy the objects possess in a microscopic form, which are not easy to be identified and calculated. These energies belong to the category of internal energy. The two main subcategories of internal energy are chemical and thermal energy. The first involves the energy generation or absorption during chemical reactions. The later involves the energy generation during the local movements (like vibrations) of objects' particles due to their hotness, as explained earlier. Therefore, it is easy to conclude that thermal energy is somehow related to temperature, i.e. hot objects possess more thermal energy than cold objects as their particles vibrate more rapidly.

The part of this thermal energy that is transferred to another object, is known as heat energy. As explained in the Physics tutorial 5.1 "Work and Energy. Types of Energy", heat can be transferred from one object into another when the proper conditions are created. One condition would be putting a hot and a cold object in contact to each other and thus, paving the way to the heat energy to transfer from the hottest object to the coldest one. There is not any matter transfer during this process; the only thing that is transferred is the heat energy due to the collision pf particles in the bordering parts between the two objects. Faster particles in the outer layer of the hot object collide with the slower particles of the outer layer of the cold object. In this way, the hot object loses some heat energy as its particles get slower during the collision and the cold object gains some heat energy because its particles become faster during the collision with more energetic particles. As a result, there is a heat exchange between objects which eventually bring a change in the internal energy of both objects.

The heat "flow" eventually stops when both objects have reached the same temperature. This means the average speeds of particles vibration in both objects are already equal. In other words, the heat flow stops when thermal equilibrium is established. In this way, we obtained the meaning of thermal equilibrium, i.e. a condition in which all parts of a system are at the same temperature.

It is precisely on this concept that the Zeroth Law of Thermodynamics is based. It states that:

If a thermodynamic system A is in thermal equilibrium with another thermodynamic system B and the thermodynamic system B is in thermal equilibrium with a third thermodynamic system C, then the thermodynamic system A is also in thermal equilibrium with the thermodynamic system C.

A thermodynamic system is an object of a group of objects with the same temperature. For simplicity, we will consider the system as a single object. Thus, in simpler words, the Zeroth Law of Thermodynamics says:

If an object A has the same temperature with another object B and the object B has the same temperature with a third object C, then the object A has the same temperature with the object C.

For example, the object in the factory in which a thermometer was originally placed in contact to do its calibration represents the system A, the thermometer itself is the system B and the patient's body is the system C.

The Zeroth Law of Thermodynamics is schematically shown below. It is called the "Zeroth Law of Thermodynamics" as it was formulated after the First and the Second Laws of Thermodynamics, for which we will discuss in the next tutorials.

## Summary

All particles of any object in whatever state it may be are in unceasing motion. In general, they vibrate around some fixed points known as "equilibrium positions." Faster the particles' vibration, greater their kinetic energy.

We use an indirect way to estimate the kinetic energy of an object's particles (in other words its heat energy), which is one of the main components of their internal energy (the other is the chemical energy generated during chemical reactions). A more direct measurement than the measurement of particles' average kinetic energy was therefore invented a few centuries ago. In this way, the concept of temperature was introduced.

By definition, temperature represents a measure of the ability of a substance, or more generally of any physical system, to transfer heat energy to another physical system.

Temperature as a concept is closely related to the average kinetic energy of all particles of the object or system. This means temperature is a more suitable quantity related to the measurements of objects' warmth than heat energy, because its value can be measured directly through devices called thermometers. But first, a thermometer needs to be graded. For this, a lower fixed point and an upper fixed point are needed. The process of assigning different values to different heights of liquid column in a thermometer is known as "calibration of thermometer."

Celsius Scale (invented by the Swedish scientist Anders Celsius) uses the melting and boiling points of water as a lower and upper fixed points respectively. The range of temperatures between these two fixed points is divided in 100 equal parts. Each division is called 1-Celsius degree (1°C).

Daniel Gabriel Fahrenheit, a German scientist, used the temperatures in the coldest and the hottest days in his country to calibrate thermometers. In this way, he invented the Fahrenheit Scale. The conversion formula between Celsius and Fahrenheit scales is:

t (°F) = 1.8 × t (°C) + 32°

However, the official SI unit of temperature is the Kelvin Scale. It is named after the Lord Kelvin, alias William Thompson, the Scottish famous scientist. Kelvin degree is the SI unit of temperature as it offers a great advantage compared to the other units: it has only positive (or zero) values because the lower fixed point of this scale refers to the lowest temperature in the universe, i.e. the temperature in which particles of matter stop vibrating around their equilibrium positions. When measured in Celsius degree, this minimum temperature is equal to -273.16°C. Thus, we have:

0 K = - 273.16°C ≈ - 273°C

or more generally,

T(K) = t (°C) + 273°

Internal energy is composed by two main subcategories: chemical and thermal energy. The first involves the energy generation or absorption during chemical reactions. The later involves the energy generation during the local movements (like vibrations) of objects' particles due to their hotness. Thermal energy is somehow related to temperature, i.e. hot objects possess more thermal energy than cold objects as their particles vibrate more rapidly. The part of this thermal energy that is transferred to another object, is known as heat energy.

The heat "flow" eventually stops when two objects in contact have reached the same temperature. This means the average speeds of particles vibration in both objects are already equal. This is the base of the Zeroth Law of Thermodynamics, which says:

If a thermodynamic system A is in thermal equilibrium with another thermodynamic system B and the thermodynamic system B is in thermal equilibrium with a third thermodynamic system C, then the thermodynamic system A is also in thermal equilibrium with the thermodynamic system C.

In simpler words, the Zeroth Law of Thermodynamics says:

If an object A has the same temperature with another object B and the object B has the same temperature with a third object C, then the object A has the same temperature with the object C.

For example, the object in the factory in which a thermometer was originally placed in contact to do its calibration represents the system A, the thermometer itself is the system B and the patient's body is the system C.

## Physics Revision Questions for Temperature. The Zeroth Law of Thermodynamics

1) A TV weather forecast gives the value of local temperature in Fahrenheit scale for a city. It shows the value 23°F. What is the value of this temperature in Celsius scale?

1. 23°C
2. 5°C
3. -5°C
4. 30°C

2) For what value of temperature the thermometer in Celsius scale is equal to that in Fahrenheit scale?

1. -40 degree
2. 0 degree
3. 32 degree
4. 100 degree

3) An object A at 27°C is in thermal equilibrium to an object B. What is the value of temperature in Kelvin degrees for the object B?

1. 27 K
2. 80.6 K
3. 245 K
4. 300 K