Internal energy

1. Overview

Internal energy is the total energy contained within a thermodynamic system at the microscopic level. It is a fundamental state function, which means its value depends exclusively on the current state of the system (defined by parameters such as pressure $P$, volume $V$, and temperature $T$) and is entirely independent of the process or "path" taken to reach that state.

In the context of A-Level Physics, internal energy is understood as the microscopic counterpart to macroscopic mechanical energy. While a moving object has macroscopic kinetic energy, its internal energy is concerned only with the random motion and arrangements of the individual atoms and molecules that make up the object.

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Key Definitions

• Internal Energy ($U$): The sum of a random distribution of kinetic and potential energies associated with the molecules of a system.
• State Function: A property of a system that depends only on its current equilibrium state and not on how that state was achieved.
• Thermodynamic (Kelvin) Scale: An absolute temperature scale that does not depend on the properties of any specific substance. It starts at absolute zero.
• Absolute Zero ($0\text{ K}$): The theoretical temperature at which a system possesses minimum internal energy. At this point, the kinetic energy of the molecules is at its absolute minimum.
• Thermal Equilibrium: A condition where two objects in contact have the same temperature, resulting in no net transfer of thermal energy between them.

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Content

The Microscopic Components of Internal Energy

The internal energy ($U$) of any substance is composed of two distinct microscopic energy contributions:

1. Microscopic Kinetic Energy ($\sum E_k$) This energy arises from the random motion of the particles. It is not "ordered" motion (like a ball flying through the air), but rather the chaotic movement of molecules. It includes:

• Translational Kinetic Energy: Molecules moving from one location to another in space (dominant in gases and liquids).
• Rotational Kinetic Energy: Molecules spinning around their center of mass (relevant for polyatomic molecules).
• Vibrational Kinetic Energy: Atoms oscillating about their equilibrium positions (dominant in solids).

Crucial Principle: The average kinetic energy of the molecules is directly proportional to the thermodynamic temperature ($T$) of the system. If the temperature increases, the molecules move faster on average, and the total internal energy increases.

2. Microscopic Potential Energy ($\sum E_p$) This energy arises from the intermolecular forces (electrostatic forces) between the molecules.

• In solids and liquids, molecules are close together and experience significant attractive forces.
• To move molecules further apart (e.g., during melting or boiling), work must be done against these attractive forces.
• This work is stored as an increase in the potential energy of the molecules.
• By convention, potential energy is zero when molecules are infinitely far apart. Since intermolecular forces are generally attractive, the potential energy of molecules in solids and liquids is negative. As they move apart, the potential energy becomes "less negative" (i.e., it increases).

Internal Energy in Different States of Matter

The balance between $E_k$ and $E_p$ changes significantly depending on the phase of the substance:

• Solids: Particles vibrate about fixed positions. They have low $E_k$ (depending on $T$) and very high negative $E_p$ because the particles are packed closely together in a lattice.
• Liquids: Particles have enough $E_k$ to slide past one another. The $E_p$ is still significant but higher (less negative) than in solids because the particles are slightly further apart and the lattice structure is broken.
• Gases: Particles move rapidly and randomly. They have high $E_k$. In a real gas, $E_p$ is small but non-zero.
• Ideal Gases: In the kinetic theory of gases, we assume there are no intermolecular forces between molecules. Consequently, the potential energy is zero. Therefore, the internal energy of an ideal gas is entirely kinetic and depends only on its temperature.

Temperature and Internal Energy

A rise in the temperature of an object is always associated with an increase in its internal energy.

1. Heating an object increases the thermal energy supplied to the molecules.
2. This energy increases the mean square speed of the molecules.
3. The average kinetic energy of the molecules increases.
4. Since $U = \sum E_k + \sum E_p$, an increase in $\sum E_k$ leads to an increase in $U$.

Phase Changes (Latent Heat)

During a change of state (e.g., boiling water at $100\text{ }^\circ\text{C}$):

• Energy is supplied to the system, but the temperature remains constant.
• Because the temperature is constant, the average kinetic energy of the molecules remains constant.
• The energy supplied is used to break intermolecular bonds and increase the separation between molecules.
• This results in a significant increase in the potential energy of the molecules.
• Therefore, the internal energy increases during a phase change, even though the temperature does not change.

The First Law of Thermodynamics

While topic 16.2 covers this in detail, it is the primary tool for calculating changes in internal energy ($\Delta U$). The law states that the increase in internal energy is equal to the sum of the heat supplied to the system and the work done on the system: $$\Delta U = q + w$$

• $q$ is positive if thermal energy is added to the system.
• $w$ is positive if work is done on the system (e.g., compression).

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Worked Example 1 — Heating a Solid

A $2.0\text{ kg}$ block of aluminum is heated from $300\text{ K}$ to $350\text{ K}$. The specific heat capacity of aluminum is $900\text{ J kg}^{-1}\text{ K}^{-1}$. Calculate the change in internal energy of the block, assuming thermal expansion is negligible.

Step 1: Identify the process The temperature is rising, so the kinetic energy of the molecules is increasing. Since expansion is negligible, we assume the potential energy change is zero and no work is done by the block against the atmosphere ($w = 0$).

Step 2: State the relevant equation The energy supplied ($q$) is calculated using: $$q = mc\Delta\theta$$ Since $w = 0$, $\Delta U = q$.

Step 3: Substitute the values $m = 2.0\text{ kg}$ $c = 900\text{ J kg}^{-1}\text{ K}^{-1}$ $\Delta\theta = 350 - 300 = 50\text{ K}$

$$q = 2.0 \times 900 \times 50$$ $$q = 90,000\text{ J}$$

Step 4: Final Answer $$\Delta U = 9.0 \times 10^4\text{ J}$$ The internal energy increases by $9.0 \times 10^4\text{ J}$.

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Worked Example 2 — Internal Energy during Melting

A $0.15\text{ kg}$ sample of pure ice at $0\text{ }^\circ\text{C}$ melts completely to form water at $0\text{ }^\circ\text{C}$. The specific latent heat of fusion of ice is $3.34 \times 10^5\text{ J kg}^{-1}$. Describe the changes to the kinetic and potential energies of the molecules and calculate the change in internal energy.

Step 1: Analyze the molecular energy changes

• Kinetic Energy: The temperature remains constant at $0\text{ }^\circ\text{C}$ ($273.15\text{ K}$). Therefore, the average kinetic energy of the molecules does not change.
• Potential Energy: Energy is required to break the rigid hydrogen bonds in the ice lattice. The molecules become slightly more disordered (though in the case of water, they actually get closer, the "bonds" are broken/rearranged). In general, for melting, work is done to break the lattice, so potential energy increases.

Step 2: Calculate the energy supplied $$q = mL_f$$ $$q = 0.15 \times (3.34 \times 10^5)$$ $$q = 50,100\text{ J}$$

Step 3: Relate to internal energy Assuming the volume change is negligible (or work done is minimal), the energy supplied equals the change in internal energy. $$\Delta U \approx 5.0 \times 10^4\text{ J}$$

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Worked Example 3 — Ideal Gas Compression

An ideal gas is contained in a cylinder with a frictionless piston. The gas is compressed suddenly, such that $450\text{ J}$ of work is done on the gas. During this process, $150\text{ J}$ of thermal energy is lost to the surroundings. Calculate the change in internal energy and state what happens to the temperature of the gas.

Step 1: Identify the signs for the First Law

• Work done on the system: $w = +450\text{ J}$
• Heat lost by the system: $q = -150\text{ J}$

Step 2: Apply the First Law of Thermodynamics $$\Delta U = q + w$$ $$\Delta U = -150 + 450$$ $$\Delta U = +300\text{ J}$$

Step 3: Relate $\Delta U$ to Temperature For an ideal gas, internal energy is only the sum of the kinetic energies of the molecules ($U = \sum E_k$). Since $\Delta U$ is positive, the total kinetic energy has increased. Because average $E_k \propto T$, the temperature of the gas must have increased.

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Key Equations

Equation	Description	Data Sheet?
$U = \sum E_k + \sum E_p$	Definition of internal energy as the sum of random microscopic energies.	No
$\Delta U = q + w$	The First Law of Thermodynamics (change in internal energy).	Yes
$T/\text{K} = \theta/^\circ\text{C} + 273.15$	Conversion between Celsius and Thermodynamic scales.	Yes
$E_k \propto T$	Relationship between average molecular KE and Kelvin temperature.	No

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Common Mistakes to Avoid

• ❌ Confusing Macroscopic and Microscopic Energy: Students often think that if a gas cylinder is moving on a train, its internal energy increases.
  • ✓ Right: Internal energy only includes microscopic energies. The bulk motion of the container (macroscopic KE) does not affect the internal energy ($U$).
• ❌ Omitting the word "Random": In definitions of internal energy, the word "random" is a required keyword.
  • ✓ Right: Always define $U$ as the sum of a random distribution of kinetic and potential energies.
• ❌ Assuming $U=0$ at $0\text{ K}$:
  • ✓ Right: At absolute zero, internal energy is at a minimum, but not necessarily zero (due to quantum mechanical zero-point energy, though "minimum" is the standard A-Level answer).
• ❌ Forgetting Potential Energy in Solids/Liquids:
  • ✓ Right: While $U$ is purely kinetic for an ideal gas, you must include potential energy for any real substance, especially during phase changes.
• ❌ Sign Errors in $\Delta U = q + w$:
  • ✓ Right: Always check if work is done on the gas (positive) or by the gas (negative), and if heat is supplied to (positive) or lost from (negative) the system.

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Exam Tips

1. The "State Function" Explanation: If asked why internal energy change is the same for two different paths, state that internal energy is a state function and depends only on the initial and final states (P, V, and T).
2. Describing Melting/Boiling: When explaining why temperature stays constant during a phase change:
   • State that the energy supplied is used to increase the separation between molecules.
   • State that this increases the microscopic potential energy.
   • Explicitly state that the average kinetic energy remains constant because the temperature is constant.
3. Ideal Gas Simplification: If a question mentions an "ideal gas," immediately note that $\sum E_p = 0$. This means any change in internal energy must result in a change in temperature.
4. Defining Absolute Zero: If asked to define absolute zero in terms of internal energy, say it is the temperature at which the system has minimum internal energy. If asked in terms of the ideal gas laws, say it is the temperature at which the volume/pressure of an ideal gas would extrapolate to zero.
5. Keywords for Definitions: When defining Internal Energy, ensure you hit these four points:
   • Sum of...
   • Random distribution of...
   • Kinetic and potential energies...
   • Associated with molecules/atoms.