7.6 Internal Energy and Temperature of a Substance

We have learned that temperature is a measure of the average kinetic energy of particles. But particles in a substance also have potential energy due to the forces between them. The total energy of all the particles in a substance—both kinetic and potential—is called internal energy. Understanding internal energy is essential for understanding heat, work, and changes of state.

Definition

Internal Energy is the total energy possessed by the particles of a substance. It is the sum of the total kinetic energy and the total potential energy of all the particles in the substance.

Internal Energy = Total Kinetic Energy + Total Potential Energy

Main Idea

The main idea is that the internal energy of a substance is the sum of the kinetic and potential energies of all its particles. The kinetic energy is due to the motion of the particles (translational, rotational, vibrational), and the potential energy is due to the attractive forces between them. When a substance is heated, its internal energy increases because the kinetic energy of its particles increases (they move faster) and/or the potential energy increases (they move apart against attractive forces). Temperature is a measure of the average kinetic energy, while internal energy is the total energy of all particles.

Core Concept

Definition of Internal Energy The text states: "Internal energy of a substance is the total energy possessed by the particles of the substance. Internal energy is the total kinetic and potential energy of the particles of a substance."

• Kinetic Energy: The energy due to the motion of particles. It includes:
• Translational Kinetic Energy: The energy due to the straight-line motion of particles.
• Rotational Kinetic Energy: The energy due to the rotation of particles.
• Vibrational Kinetic Energy: The energy due to the vibration of particles.
• Potential Energy: The energy due to the attractive forces between particles.

Internal Energy of an Ideal Gas The text states: "In case of ideal gas, it has only translational kinetic energy of particles. Therefore, its internal energy is only due to kinetic energy of particles."

In an ideal gas, the particles are assumed to have no intermolecular forces (no potential energy), so the internal energy is purely kinetic.

Relationship Between Internal Energy and Temperature The text states: "In the previous topic, we have studied that temperature is directly proportional to average kinetic energy of the particles of a substance. When we heat a substance, it speeds up the particles and increases the kinetic energy of its particles. Hence, we can say that internal energy of the substance also increases. By increasing the temperature of a substance, its internal energy also increases."

• Heating a substance: Increases the kinetic energy of particles → increases internal energy.
• Temperature: A measure of the average kinetic energy. An increase in temperature means an increase in internal energy.

Internal Energy and Changes of State The text states: "When a material is heated, one of the two things may happen: (1) Strength of attractive force between particles can decrease and bonds between particles may break (as it happens during melting and boiling processes of a material) (2) it can speed up the particles and hence increase K.E. of the particles."

- During a change of state (melting, boiling): The added heat energy is used to overcome the attractive forces between particles (increasing potential energy), not to increase their kinetic energy. Therefore, the temperature remains constant during a change of state, even though the internal energy is increasing.

Detailed Explanation

Internal energy is a fundamental concept in thermodynamics. It is the total energy of all the particles in a system.

• Kinetic Energy: The energy of motion. Particles in a substance are constantly moving (translating, rotating, vibrating). The higher the temperature, the faster they move, and the greater their kinetic energy.
• Potential Energy: The energy stored due to the position of particles relative to each other. It depends on the strength of the attractive forces between particles and the distance between them. When particles are pulled apart (e.g., during expansion or a change of state), their potential energy increases.

• Solids: Particles vibrate about fixed positions. Internal energy is mostly vibrational kinetic energy and potential energy from strong interparticle forces.
• Liquids: Particles can move past each other. Internal energy includes translational, rotational, and vibrational kinetic energy, plus potential energy from weaker interparticle forces.
• Gases: Particles move freely. Internal energy is mostly translational kinetic energy (and some rotational/vibrational for polyatomic gases). Potential energy is negligible (for ideal gases).

Internal Energy and Temperature Temperature is a measure of the average kinetic energy of particles. When temperature increases, the average kinetic energy increases, so the internal energy increases. However, internal energy also depends on the number of particles (mass) and the potential energy (interparticle forces).

Changes of State and Internal Energy During a change of state (e.g., melting or boiling), the temperature remains constant even though heat is being added. This is because the added energy is used to overcome the attractive forces between particles (increasing potential energy), not to increase their kinetic energy. The internal energy increases, but the temperature does not change until the change of state is complete.

Intermediate Level

In physics, internal energy is the sum of the kinetic and potential energies of all the particles in a system. It is a state function, meaning it depends only on the current state of the system, not on how that state was reached.

For an ideal gas, the internal energy is purely kinetic (no intermolecular forces), and it depends only on temperature: U = 32nRT (for a monatomic ideal gas). For real gases, liquids, and solids, the internal energy also includes potential energy from interparticle forces.

• Increasing kinetic energy: The particles move faster, and the temperature rises.
• Increasing potential energy: The particles move apart against attractive forces, which occurs during expansion or a change of state (melting, boiling). During a change of state, the temperature remains constant because the added energy goes into increasing potential energy, not kinetic energy.

The first law of thermodynamics relates the change in internal energy (Δ U) to heat added (Q) and work done (W): Δ U = Q - W

Important Points

• - Internal Energy: Total kinetic + potential energy of all particles.
• - Kinetic Energy: Due to particle motion (translational, rotational, vibrational).
• - Potential Energy: Due to interparticle forces.
• - Ideal Gas: Internal energy is purely kinetic.
• - Heating: Increases internal energy (kinetic and/or potential).
• - Temperature: Measure of average kinetic energy.
• - Change of State: Temperature remains constant, but internal energy increases (potential energy increases).

Remember

• - Internal Energy = Total KE + Total PE of particles.
• - Heating increases internal energy.
• - Temperature is average KE.
• - Change of state: Temperature constant, internal energy increases.
• - Ideal gas: Internal energy is purely kinetic.

Common Mistakes

• - Thinking internal energy is the same as temperature: Temperature is a measure of the average kinetic energy of particles. Internal energy is the total energy (kinetic + potential) of all particles. A large object at the same temperature as a small object has more internal energy because it has more particles.
• - Thinking temperature always increases when heat is added: During a change of state (melting, boiling), the temperature remains constant even though heat is being added and internal energy is increasing.
• - Forgetting about potential energy: Internal energy includes both kinetic and potential energy. In solids and liquids, potential energy is significant.

Quick Revision

Internal energy is the total energy (kinetic + potential) of all the particles in a substance. Kinetic energy is due to particle motion, and potential energy is due to interparticle forces. When a substance is heated, its internal energy increases. During a change of state (melting or boiling), the temperature remains constant even though the internal energy increases, because the added energy is used to overcome interparticle forces (increasing potential energy) rather than increasing kinetic energy.

60-Second Revision

• Heating: Increases internal energy.
• Temperature: Average KE.
• Change of state: Temperature constant, internal energy increases.
• Ideal gas: Internal energy is purely kinetic.
• Formula: Δ U = Q - W.

Frequently Asked Questions (FAQs)

Q1. What is internal energy? Answer: Internal energy is the total energy possessed by the particles of a substance. It is the sum of the total kinetic energy and the total potential energy of all the particles.

Q2. What is the difference between internal energy and temperature? Answer: Temperature is a measure of the average kinetic energy of the particles in a substance. Internal energy is the total energy (kinetic + potential) of all the particles. A large object at the same temperature as a small object has more internal energy because it has more particles.

Q3. How does heating affect internal energy? Answer: Heating a substance increases its internal energy by increasing the kinetic energy of its particles (they move faster) and/or their potential energy (they move apart against attractive forces).

Q4. Why does the temperature remain constant during a change of state? Answer: During a change of state (melting, boiling), the added heat energy is used to overcome the attractive forces between particles (increasing potential energy), not to increase their kinetic energy. Therefore, the average kinetic energy (and thus the temperature) remains constant until the change of state is complete.

Q5. What is the internal energy of an ideal gas? Answer: For an ideal gas, the internal energy is purely kinetic because there are no intermolecular forces (no potential energy). It depends only on the temperature and the number of particles.

• Kinetic energy: Due to the translational, rotational, and vibrational motion of particles.
• Potential energy: Due to the attractive forces between particles.

Q7. How is internal energy related to the first law of thermodynamics? Answer: The first law of thermodynamics states that the change in internal energy (Δ U) is equal to the heat added to the system (Q) minus the work done by the system (W): Δ U = Q - W.

Multiple Choice Questions (MCQs)

• Q1. Internal energy is the sum of:
• A. Kinetic energy and mass
• B. Kinetic energy and potential energy of particles
• C. Potential energy and volume
• D. Temperature and pressure

• Correct Answer: B
• Explanation: The text states that internal energy is the total kinetic and potential energy of the particles of a substance.

• Q2. The kinetic energy of particles includes:
• A. Only translational motion
• B. Only rotational motion
• C. Translational, rotational, and vibrational motion
• D. Only vibrational motion

• Correct Answer: C
• Explanation: The text states that particles can have three forms of kinetic energy: translational, rotational, and vibrational.

• Q3. For an ideal gas, the internal energy is:
• A. Purely potential energy
• B. Purely kinetic energy
• C. Both kinetic and potential energy
• D. Zero

• Correct Answer: B
• Explanation: The text states that in the case of an ideal gas, it has only translational kinetic energy, so its internal energy is only due to kinetic energy.

• Q4. When a substance is heated, its internal energy:
• A. Decreases
• B. Increases
• C. Remains constant
• D. Becomes zero

• Correct Answer: B
• Explanation: The text states that when we heat a substance, the kinetic energy of its particles increases, and the internal energy also increases.

• Q5. During a change of state (e.g., melting), the temperature:
• A. Increases
• B. Decreases
• C. Remains constant
• D. Becomes zero

• Correct Answer: C
• Explanation: The text implies that during melting and boiling, the energy is used to break bonds (potential energy), so the temperature remains constant.

• Q6. The potential energy of particles is due to:
• A. Their motion
• B. The attractive forces between them
• C. Their mass
• D. Their color

• Correct Answer: B
• Explanation: The text states that potential energy of the particles is due to attractive force between them.

• Q7. Which of the following is NOT a form of kinetic energy of particles?
• A. Translational kinetic energy
• B. Rotational kinetic energy
• C. Vibrational kinetic energy
• D. Potential energy

• Correct Answer: D
• Explanation: Potential energy is not a form of kinetic energy; it is a separate component of internal energy.

• Q8. The internal energy of a substance depends on:
• A. Only temperature
• B. Only mass
• C. Both temperature and mass (number of particles)
• D. Only volume

Correct Answer: C Explanation: Internal energy is the total energy of all particles, so it depends on both the average kinetic energy (temperature) and the number of particles (mass).

• Q9. When a substance is heated and its temperature rises, the internal energy:
• A. Decreases
• B. Increases
• C. Remains constant
• D. Becomes zero

• Correct Answer: B
• Explanation: The text states that by increasing the temperature of a substance, its internal energy also increases.

• Q10. The first law of thermodynamics relates the change in internal energy to:
• A. Heat added and work done
• B. Mass and volume
• C. Pressure and temperature
• D. Force and distance

• Correct Answer: A
• Explanation: The first law of thermodynamics states that Δ U = Q - W, where Q is heat added and W is work done.

Frequently Asked Questions (FAQs)

Q1. What is the difference between internal energy and heat? Answer: Internal energy is the total energy (kinetic + potential) of the particles in a substance. Heat is the transfer of energy from a hotter object to a colder one due to a temperature difference. Heat is a process of energy transfer, while internal energy is a state property.

Q2. Why does the temperature remain constant during boiling? Answer: During boiling, the added heat energy is used to overcome the attractive forces between liquid particles and convert them into gas. This increases the potential energy of the particles, not their kinetic energy. Since temperature is a measure of average kinetic energy, the temperature remains constant.

Q3. What is the internal energy of a solid? Answer: For a solid, the internal energy includes the vibrational kinetic energy of the particles (they vibrate about fixed positions) and the potential energy due to the strong attractive forces between them.

Q4. How does the internal energy of a gas differ from that of a solid? Answer: The internal energy of a gas is mostly kinetic (particles move freely at high speeds), with negligible potential energy (for an ideal gas). In a solid, the internal energy includes both vibrational kinetic energy and significant potential energy due to strong interparticle forces.

Q5. What happens to the internal energy of a substance when it is cooled? Answer: When a substance is cooled, its internal energy decreases. The particles lose kinetic energy (they move slower) and/or potential energy (they move closer together). This is the opposite of heating.

Q6. Can internal energy be measured directly? Answer: Internal energy cannot be measured directly. Only changes in internal energy (Δ U) can be measured, typically through changes in temperature, heat transfer, or work done.

Q7. What is the relationship between internal energy and temperature for an ideal gas? Answer: For an ideal gas, the internal energy is directly proportional to the absolute temperature: U = 32nRT (for a monatomic gas). As temperature increases, internal energy increases proportionally.

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