Thermal Energy Formula

As the advancements took place in understanding the conservation of energy many new and interesting aspects of energy were discovered. During that time, we understood that when the objects undergo motion they encounter energy loss, and also they increase the temperature of the object. When this was mysterious and the question arose was what is happening with the lost energy? 

Over a period of time scientists were able to conclude that the energy has not been lost but it got converted into thermal energy. Basically, thermal energy is the internal energy of the system which is responsible for the change in temperature. In this article, we will learn about the thermal energy formula, heat energy formula, thermal energy equation with solved examples for a better understanding.

With careful experimentation, scientists were able to understand the fact that the change in temperature or increase in temperature of an object and its surroundings as a result of friction is directly proportional to the amount of mechanical energy lost. In the 1840s,  James Joule found that elevating the temperature of a kilogram of water by one degree Celsius required a loss in mechanical (gravitational) energy of approximately 4200 joules. Joule, therefore, suggested that this mechanical energy is not lost, but converted into a new type of energy, thermal energy. 

Thus, thermal energy refers to the total energy enclosed within a system that is accountable for its temperature change. We know that heat is the flow of thermal energy. One of the important branches of physics, thermodynamics, deals with how heat is transferred (i.e., dynamics of the heat flow) between several systems. The thermal energy formula is often also known as the heat energy formula. The thermal energy formula or the heat energy formula shows that the thermal energy is directly proportional to the change in temperature. 

Thermal Energy Equation

Now, let us have a look at the thermal energy equation. According to the definition of thermal energy, we know that it is the total internal energy stored inside the system which is responsible for the change in temperature. Thus, thermal energy is directly related to the change in temperature. Therefore, mathematically we write:

\[ \Rightarrow E_{t}~\alpha~\Delta T...(1)\]

Where,

\[{E_{t}}\] - The thermal energy

\[{\Delta T}\] - The change in temperature

We know that the temperature change of the system is related to the mass of the object and hence thermal energy will also depend on the mass of the object under consideration.

Therefore, we write:

\[ \Rightarrow E_{t}~\alpha~m~\Delta T\]

\[ \Rightarrow E_{t} = m~C~\Delta T...(2)\]

Where,

m - The mass of the system or substance

C - The specific heat capacity of the substance

Equation (2) is known as the thermal energy equation or thermal energy formula, which can be used to determine the thermal energy of the system. In some articles and textbooks, thermal energy is even denoted by the letter Q, thus one should not get confused with the notations used.

The thermal energy formula or thermal energy equation can be written in terms of the mechanics as well. As we know, initially thermal energy was found as a result of the friction force. Thus, the heat energy formula in terms of friction coefficient is given by:

\[\Rightarrow E_{t} = \mu_{k}~N~d.....(3)\]

Where,

\[\mu_{k}\] - The coefficient of kinetic friction

N - The normal force acting on the object

d - The total displacement

Both equations (3) and (2) are known as thermal energy equations or thermal physics formulas. Equation (2) is the thermal energy formula in terms of thermodynamics, whereas equation (3) is the thermal energy formula in terms of mechanics.

Examples:

1. Calculate the thermal energy of a substance whose mass is 10kg and the corresponding specific heat of the substance is\[0.050 J/kg^{\circ}\] and the temperature difference of this thermodynamic system is given at \[30^{\circ}c\].

Sol:

Given,

The mass of the substance = m = 10kg

The specific heat capacity of the given substance =\[C = 0.050 J/kg^{\circ}\]

The temperature difference of the given thermodynamic system =\[\Delta T = 30^{\circ}c\]

We are asked to determine the thermal energy of the given substance. Thus, the thermal energy of the given substance can be estimated using the heat energy equation given by:

\[ \Rightarrow E_{t} = m C \Delta T...(1)\]

Where,

m - The mass of the system or substance

C - The specific heat capacity of the substance

\[\Delta T\] - The change in temperature

Substituting all the values in the thermal energy equation i.e., equation (1) and simplifying it further for the required answer. Therefore, we write:

\[ \Rightarrow E_{t} = m C \Delta T\]

\[ \Rightarrow E_{t} = (10)~(0.050)~(30) = 15 joules\]

Therefore, the total amount of thermal energy of the given substance is \[15 joules\].

2. Suppose that a person pushes the box kept on the ground with a constant velocity. The mass of the box is 150kg and displaced by a distance of 100m. The coefficient of kinetic friction between the box and floor is \[\mu_{k} = 0.4\]. Then, calculate the amount of thermal energy transferred as a result of the friction between the floor and the box system?

Sol:

Given,

The mass of the box = m = 150kg

The total displacement as a result of applied force = d = 100m

The coefficient of kinetic friction \[=\mu_{k} = 0.4\]

Now, we are asked to determine the total amount of thermal energy transferred as a result of the friction between the floor and the box system. Since the box was being pushed with the constant velocity the acceleration of the box will be zero, and hence the only force acting on it will be the normal force. Therefore, the normal force is given by:

\[\Rightarrow = mg = 150 \times 9.81 = 1471.5~N.....(2)\]

Now, the thermal energy of the system can be calculated using the thermal energy formula given by:

\[\Rightarrow E_{t} = \mu_{k}~N~d.....(2)\]

Where,

\[\mu_{k}\] - The coefficient of kinetic friction

N - The normal force acting on the object

d - The total displacement

Substituting the values from the given data in equation (2) and simplify. We get:

\[\Rightarrow E_{t} = \mu_{k}~N~d\]

\[\Rightarrow E_{t} =(0.4)~(1471.5)~(100)\]

\[\Rightarrow E_{t} = 58.86 kj\]

Therefore, the total amount of thermal energy transferred as a result of the friction between the floor and the box system is 58.86 kj.