Thermal Energy & Heat

Syllabus Content

Temperature difference determines the direction of the resultant thermal energy transfer between bodies
Quantitative analysis of thermal energy transfers Q with the use of specific heat capacity c and specific latent heat of fusion and vaporization of substances L as given by Q = mcΔT and Q = mL

Introduction:

In our everyday lives, we often encounter situations where objects exchange energy in the form of heat. Whether it's the warmth we feel from the sun, the heat generated by a cooking stove, or the coldness we feel when touch ice.

Thermal Energy

Thermal energy is a form of internal energy possessed by a substance due to the motion of its particles. It is directly related to the temperature of the object. The higher the temperature of an object, the greater its thermal energy. Conversely, a lower temperature corresponds to lower thermal energy.

Thermal energy can be transferred from one object to another in the form of heat through various mechanisms, such as conduction, convection, and radiation. These transfer processes occur due to the temperature difference between the objects involved.

Heat Transfer & The Heating Equation

Heat always flows from hotter objects to colder objects; and never the other way around. This is explained by the second law of thermodynamics. (Photo by A. Forgue)

Heat is the transfer of thermal energy between objects that are at different temperatures. The direction of heat flow is always from hotter objects to colder objects. This principle is known as the Second Law of Thermodynamics. Heat transfer continues until thermal equilibrium is reached, which occurs when the temperatures of the objects are equal.

To understand heat transfer, let's consider an example of a cup of hot coffee placed in a room at a lower temperature. The hot coffee will gradually lose thermal energy to the surrounding air until both the coffee and the air reach the same temperature. This transfer of heat occurs through the process of conduction, where the particles of the coffee collide with the particles of the air, transferring energy in the process.

Thermal Equilibrium

Thermal equilibrium is the state at which two or more objects are at the same temperature and no further heat transfer occurs between them. Once thermal equilibrium is reached, the objects are said to be in thermal contact, and their temperatures are equal. This is a fundamental concept in thermodynamics and is often used as a reference point for analyzing heat transfer processes.

Consider a scenario where a metal rod is heated at one end and kept in contact with a colder object at the other end. Initially, heat flows from the hot end to the cold end until both ends reach the same temperature. At this point, thermal equilibrium is achieved, and the heat transfer ceases.

Rate of Heat Transfer:

he rate at which heat is transferred between objects depends on several factors. One of the key factors is the temperature difference between the objects involved. The greater the temperature difference, the faster the rate of heat transfer.

The rate of heat transfer, also known as the heat flow, can be quantified using the equation:

Q = mc∆T

Where:

Q is the amount of heat transferred (measured in joules, J),
m is the mass of the substance (measured in kilograms, kg),
c is the specific heat capacity of the substance (measured in joules per kilogram per degree Celsius, J/kg°C), and
∆T is the change in temperature (measured in degrees Celsius, °C).

This equation shows that the heat transfer is proportional to the mass of the substance, the specific heat capacity, and the temperature difference. The specific heat capacity is a property of the substance and represents the amount of heat required to raise the temperature of a unit mass of the substance by one degree Celsius.

The rate at which heat is transferred is dependent on the temperature difference between the objects. From the graph, we see that the temperature of the objects changes at a faster rate when the temperature difference between the two objects is greater. As the objects begin to reach thermal equilibrium, the rate at which heat is transferred becomes slower. (Photo by A. Forgue)

Specific Heat Capacity:

Specific heat capacity, denoted by the symbol c, is a measure of a substance's ability to absorb or release heat. It is defined as the amount of heat required to raise the temperature of a unit mass of the substance by one degree Celsius. The specific heat capacity is a characteristic property of the substance and varies from one material to another.

Different substances have different specific heat capacities due to variations in their atomic or molecular structures. For example, water has a relatively high specific heat capacity compared to metals like copper. This means that it takes more heat to raise the temperature of water compared to an equal mass of copper.

The specific heat capacity can be experimentally determined by measuring the amount of heat transferred to a substance and the resulting temperature change. This information allows us to calculate the specific heat capacity using the equation:

c = Q / (m∆T)

Where:

c is the specific heat capacity (measured in joules per kilogram per degree Celsius, J/kg°C),
Q is the amount of heat transferred (measured in joules, J),
m is the mass of the substance (measured in kilograms, kg), and
∆T is the change in temperature (measured in degrees Celsius, °C).

The specific heat capacity is an essential parameter in heat transfer calculations, as it determines the amount of heat required or released during a temperature change.

Example Problems Using The Heating Equation:

Example 1:

Suppose we have 2 kilograms of water at an initial temperature of 20°C. We want to calculate the amount of heat energy required to raise the temperature of the water to 80°C.

Solution:

Given:

m = 2 kg c (specific heat capacity of water) = 4186 J/kg°C
∆T = 80°C - 20°C = 60°C

Using the equation Q = mc∆T, we can calculate the heat energy transferred:

Q = (2 kg) * (4186 J/kg°C) * (60°C) Q = 502,320 J

Therefore, it would require 502,320 joules of heat energy to raise the temperature of 2 kilograms of water from 20°C to 80°C.

Example 2:

Suppose we have 500 grams (0.5 kg) of aluminum at an initial temperature of 25°C. We want to calculate the amount of heat energy required to raise the temperature of the aluminum to 100°C.

Solution:

Given:

m = 0.5 kg c (specific heat capacity of aluminum) = 900 J/kg°C
∆T = 100°C - 25°C = 75°C

Using the equation Q = mc∆T, we can calculate the heat energy transferred:

Q = (0.5 kg) * (900 J/kg°C) * (75°C) Q = 33,750 J

Therefore, it would require 33,750 joules of heat energy to raise the temperature of 500 grams of aluminum from 25°C to 100°C.

Example 3:

Now, let's consider a scenario where we want to find the change in temperature (∆T) when a known amount of heat energy is transferred.

Suppose we have 1 kilogram of iron with a specific heat capacity of 450 J/kg°C. We want to determine the change in temperature when 20,000 joules of heat energy is transferred to the iron.

Solution:

Given: m = 1 kg c (specific heat capacity of iron) = 450 J/kg°C

Q = 20,000 J

Rearranging the equation Q = mc∆T, we get:

∆T = Q / (mc)

∆T = 20,000 J / (1 kg * 450 J/kg°C) ∆T = 44.44 °C

Therefore, when 20,000 joules of heat energy is transferred to 1 kilogram of iron, the temperature of the iron would increase by 44.44°C.

Limitations of the Heating Equation

While the equation Q = mc∆T provides a useful framework for calculating heat transfer, there are certain limitations and considerations to keep in mind when applying it to real-life situations. Some of the common problems encountered with this equation are:

Phase Changes:

The equation Q = mc∆T assumes that the substance remains in the same phase throughout the temperature change. However, during phase changes (e.g., solid to liquid, liquid to gas), the substance undergoes a change in internal structure, which requires additional heat energy. This additional energy is not accounted for in the equation. To address phase changes, an additional term called the latent heat is introduced.; we'll cover latent heat in another section.

Heat Loss to Surroundings:

In real-life scenarios, it is often challenging to prevent heat loss to the surroundings. The equation Q = mc∆T assumes that all the heat is transferred solely between the objects in question, without any loss to the environment. However, in practice, heat can be lost through conduction, convection, and radiation to the surrounding environment.

Note: Students need to know that heat loss to the surroundings through conduction, convection, and radiation will cause a difference between the theoretical-calculated value and the measured value.

To account for heat losses, an additional term called the heat loss factor can be introduced. This factor represents the fraction of heat that is lost to the surroundings and is subtracted from the total heat transfer calculated using the equation Q = mc∆T.

The heat loss factor can be experimentally determined or estimated based on the specific conditions of the system. It is an important consideration in real-world heat transfer applications, where minimizing heat loss is crucial for efficiency.

Variable Specific Heat Capacity:

The specific heat capacity of a substance can vary with temperature. The equation Q = mc∆T assumes a constant specific heat capacity throughout the temperature range. However, in reality, the specific heat capacity may change significantly with temperature.

To account for this variation, an average specific heat capacity can be used. This involves calculating the specific heat capacity at different temperature intervals and taking the average value. While this approach provides a better estimation, it is still an approximation and may introduce some errors in the calculations.

Summary:

Heat is the thermal energy between objects that are at different temperatures.
The direction of heat flow is always from hotter objects to colder objects.
The rate of heat transfer depends on the temperature difference between the objects, the mass of the objects, and the specific heat capacity of the objects.
The specific heat capacity is a property of a substance that measures its ability to absorb or release heat.
The heating equation Q = mc∆T has limitations, but provides a close estimate of the amount heat exchanged between two objects at different temperatures

Comprehension Questions:

1. What is thermal energy and how is it related to the temperature of an object?

2. Explain the difference between the terms temperature, thermal energy, and heat?

3. Define thermal equilibrium and state what happens when two objects reach thermal equilibrium

4. An ice cube is dropped in a hot coffee cup. Explain why the ice cube cools the coffee down in terms of thermal energy transfer.

5. A student in a lab calculates that providing 8000 J of energy to a beaker of water will increase the water temperature by 2°C. When the student measures the change in temperature, the water is only increased by 1.5°C. Explain the difference between the calculated value and the measured value.

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