Thermal Energy Formula: Explanation and Derivation

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Thermal energy is referred to as the energy produced from heat. Usually, the heat produced from thermal energy is due to the movements of tiny particles within the system. It is the energy present within a system, allowing heat to flow. Typically, as particles move within an object, the heat is consequently generated.

Also Read: Kinetic energy

Table of Content

What is Thermal Energy?
Thermal Energy Nature
Thermal Energy Formula
Derivation of Thermal Energy
Things to Remember
Sample Questions

Key Takeaways: Thermal energy formula, thermal energy, thermodynamics, kinetic energy, mechanical energy, energy

What is Thermal Energy?

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Thermal energy, also otherwise known as random or internal kinetic energy, can be defined as the energy present either within a system or an object. It is the energy that we generate from heat. The heat is often generated from the movement of small particles present within a system. Simply, thermal energy is broadly responsible for the temperature flow in systems or objects.

Thermodynamics, which is a broad study of physics, specifically deals with the relation between heat and different forms of energy, much like mechanical energy and others. It especially studies heat transfers between objects. In other words, the process of conduction is seen to take place when thermal energy transfers via the interaction between solid particles.

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Thermal Stress	Latent Heat of Water
Types of Generators and its Application	Conservative Force

Thermal Energy Nature

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The formula of thermal energy can be denoted by,

Q = mc Δ T

As per the expression, “Q” signifies thermal energy

“m” denotes the mass of the respective substance

c denotes specific heat capacity, and

Δ T signifies the difference in temperature.

Derivation of Thermal Energy Formula

Specific Heat Capacity = \({\text{thermal energy input} \over (\text{mass}) \times (\text {temperature change})}\)

However, since the expression has to be signified with specific symbols, we can say that C denotes specific heat capacity, T denotes temperature, while Et denotes thermal energy. But, it is important to remember that T is not utilized as a standalone expression, but signifies the change in T during the process. The symbol we typically use to represent change is Δ, which is to say, ΔT. In the meantime, the input we can assume is the amount determining change in thermal energy, represented simply as ΔEt.

Thus, the equation now stands as: \(C = {\bigtriangleup E_1 \over m. \bigtriangleup T}\)

In simple terms, the specific heat capacity of a respective substance can be defined as the amount of total energy necessary to raise the temperature of 1 gram a by a degree (e.g., in celsius).

For instance, when raising the temperature of a 10 kg table from about 20â?¦C to 25â?¦C, the energy necessary would be,

ΔE_t = m. C. ΔT

(now, as per the equation, we need to substitute the following values),

= (10 kg) (1700 J/ kg. Celcius) (5 degree Celsius)

= 85,000 J.

Read More:

Impact of temperature

Latent heat

Thermodynamics

Things to Remember

Thermal energy can be defined as the energy present either within a system or an object.
Thermal energy is typically heat flow within a system.
The heat produced due to thermal energy occurs mainly due to the movement of small particles within the system. The more the movement, the more heat will seemingly flow.
The equation of thermal energy stands as: Q = mc Δ T.
The equation of specific heat capacity = \(C = {\bigtriangleup E_1 \over m. \bigtriangleup T}\)

Sample Questions

Ques. Assume a man pushing a box, yet continuously having a steady velocity. The mass of the box is 100 kg, while it moves a distance of about 100 meters. Considering the coefficient of kinetic friction between the floor and the box, μκ is 0.3. Determine the amount of thermal energy transferred between the box and floor system. (3 marks)

Ans: The box maintains a constant velocity since there is no acceleration, while the force also remains in the same direction of motion without any vertical component.
Simply, the net force to the man is also simultaneously in equilibrium by the force produced due to friction. Thus, it can be denoted, as per the given equation,
\(\bigtriangleup E_T\) = 0.3.9.81 m/s².100 kg. 100m

= 29.43 k J (kilo-joules)
Therefore, the amount of thermal energy transferred is 29.43 kj.

Ques. Assume a wheel is being rotated via an electric motor, rated with an output power of about 10 W. The time until it stops is about 30 minutes. Determine the amount of thermal energy transferred to water. (2 marks)

Ans: As per the system, it can be said that the energy finally travels to the thermal energy of the water. It can be further determined after assuming the heat capacity being slightly negligible. Therefore, it can be said,
E_t= P * D (power and distribution)
= 10 W * (30 * 60 s)
= 18 kJ (kilo-joules)

Ques. Consider a tank containing 1 L of water is at 10 degree celsius. Determine the temperature of water after a motor, of 10 W, suddenly halts. (5 marks)

Ans: When determining the change of temperature from thermal energy, we need to know the specific heat of the fluid.
Now, since we already know the specific heat of water, it can be said: c = 4.186 J/g °C. While we know that, we need to rearrange the equation,
Et = c. m Δ T
Δ T = Et / c. m

T = T0 + Δ T
Thus, = 10 °C + 4.3 °C
= 14.3 °C
Therefore, the answer is 14.3°C

Ques. The mass of a substance is given as 6 kg, while the specific heat reads 0.030 J/kg °C. The difference of temperature is specified as 20°C. As per the data, determine the thermal energy of the substance.(5 marks)

Ans: As per the given equation, we can list the following data,
Mass (m) = 6 kg,
The specific heat (c) = 0.030 J/kg°c,
The Temperature difference (ΔT) = 20°c
Since we already know the formula denoted by thermal energy, we can mention that, Q = m. c. Δ T
Therefore, after replacing the values of the equation, we get,
Hence, Q = 3.6 Joules
The thermal energy of the substance is 3.6 Joules.

Ques. Assume that a substance of about 5 kg experiences a temperature change of 60 ° C, while the specific heat is at 0.07 J/kg °C. As per the following data, determine the thermal energy of the substances. (3 marks)

Ans: As per the given equation, we can list the following data,
The mass of the substance, m = 5 kg
Specific heat of the substance, c = 0.07 J/kg °C
And, the temperature difference, ΔT = 60°C
Since we already know the formula denoted by thermal energy, we can mention that, Q = m. c. Δ T
Therefore, by replacing the values, we get,
= 5 * 0.07 * 60
Hence, Q = 21 Joules

Ques. Determine the amount of energy required to hike the temperature of about 200 grams of copper at least by 20 degree celsius, with the specific heat of the copper mentioned as 386 J/kg degree celsius? (3 marks)

Ans: To determine the value of the required energy, it is important to use the following equation: Q = m * cp (Tf – Ti). clearly, if we cannot determine the initial or final temperature of the substance, copper as of the equation, then the difference of temperature can alternatively be denoted as:
(Tf – Ti) = 20 ºC
Now, as per the equation, the following data can be reverted,
Mass, (m) = 200 grams
(By simplifying and transforming it to kilo, we get, g = 0.2 kg)
Specific heat of the substance, copper, cp = 386 J/kg ºC.
Thus, Q = m*cp(Tf-Ti)
= 0.2 kg * 368 J/kg º C * 20 º C
= 1,472 Joules.
Thus, the amount of energy required is 1,472 Joules.

Ques. Assume that a man joins an iron ring on the very rim of the wooden wheel of a bullock carriage. Considering, at about 27 degree celsius, the diameter of the rim, alongside the iron ring are both 5.243 meter and 5.231 meter respectively. Assuming the data, determine the extent until which the temperature needs to rise to allow the rim to fit the wheel. (4 marks)

Ans: As per the given question above, the following data can be interpreted,
T1 (temperature) = 27 º C
LT1 (length one) = 5.231 meter
L T2 (length two) = 5.243 meter
Therefore, LT2 = LT1 [ 1αl (T2 –T1 )]

Thus, the extent until which the temperature needs to rise to allow the rim to fit the wheel = 218 º C

Ques. With the specific heat of water at 4180 J/kgºC, determine the extent until which the temperature of 200 grams of water shall rise after transferring about 2,500 Joules of energy. (4 marks)

Ans: As per the given question,
The Mass of water is, (m) = 200 grams (Now, like once did prior, we need to simplify and transform the value to kilo. Hence, we get, g = 0.2 kg)
The amount of energy, Q = 2,500 Joules
Specific heat of water is mentioned as, cp = 4180 J/kg º C
Thus, Q = m*cp(Tf-Ti) (we need to rearrange the equation to favor the question)
Thus, we get, (Tf-Ti) = Q / m * cp
\((T_f - T_i) = 2500 J / (0.2 kg * 4180 J/ kg^oC)\)
Hence, we get, (Tf – Ti) = 2.99 degree celsius.
Therefore, the temperature should at least rise by 2.99 degree celsius.

Also Read:

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Stokes’ Law	Moment of inertia	Cyclic process
Heat engines	Celsius to Kelvin	Thermodynamic processes
Blackbody Radiation	Colorimeter	Difference between heat and temperature

Thermal Energy Formula: Explanation and Derivation

What is Thermal Energy?

Thermal Energy Nature

Derivation of Thermal Energy Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field \( \mathbf{B} \) at a point \( (1 \, \text{m}, 1 \, \text{m}, 0) \).

3.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

4.
Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

5.
Write any two features of nuclear forces.

6.
Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.

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Thermal Energy Formula: Explanation and Derivation

What is Thermal Energy?

Thermal Energy Nature

Derivation of Thermal Energy Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field \( \mathbf{B} \) at a point \( (1 \, \text{m}, 1 \, \text{m}, 0) \).

3.What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

4.Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

5. Write any two features of nuclear forces.

6. Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Write the expression for the magnetic field due to a current element in vector form. Consider a 1 cm segment of a wire, centered at the origin, carrying a current of 10 A in positive x-direction. Calculate the magnetic field \( \mathbf{B} \) at a point \( (1 \, \text{m}, 1 \, \text{m}, 0) \).

3.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

4.
Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

5.
Write any two features of nuclear forces.

6.
Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.