Thermal equilibrium

1. Overview

Thermal equilibrium is the foundational principle of thermodynamics that defines the nature of temperature and the spontaneous direction of energy flow. The core physical law states that thermal energy (heat) always transfers from a region of higher temperature to a region of lower temperature. This process continues until the temperatures of the interacting systems are identical. At this point, the systems are in thermal equilibrium, and the net transfer of energy between them becomes zero. This concept is essential for the calibration of thermometers and the understanding of how energy distributes itself within a system.

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Key Definitions

• Temperature ($T$ or $\theta$): A scalar quantity that determines the direction of net thermal energy transfer between two objects in thermal contact. On a microscopic level, it is a measure of the average kinetic energy of the particles within a substance.
• Thermal Energy (Heat): The energy transferred between substances or systems due specifically to a temperature difference. It is measured in Joules ($\text{J}$).
• Thermal Equilibrium: A state in which two or more objects in thermal contact have the same temperature, resulting in no net transfer of thermal energy between them.
• Thermal Contact: Two objects are in thermal contact if energy can be exchanged between them via the processes of conduction, convection, or radiation.
• Internal Energy: The sum of the random distribution of kinetic and potential energies associated with the molecules of a system. While temperature relates to the average kinetic energy, thermal equilibrium relates to the equality of temperature, not necessarily the equality of internal energy.

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Content

3.1 The Direction of Thermal Energy Transfer

The transfer of thermal energy is a spontaneous process driven by a temperature gradient. If two systems are at different temperatures, they are not in equilibrium.

• The Microscopic Mechanism: In a high-temperature region, particles (atoms or molecules) possess higher average kinetic energies. When these particles collide with particles from a lower-temperature region (at the boundary of thermal contact), momentum and energy are transferred. Statistically, more energy is transferred from the "faster" particles to the "slower" particles than vice versa.
• The Macroscopic Result: We observe a bulk flow of energy from the "hot" object to the "cold" object. This is often referred to as "heating."

The Fundamental Rules of Transfer:

1. Direction: Energy flows from $T_{high}$ to $T_{low}$.
2. Rate: The rate of thermal energy transfer ($\frac{\Delta Q}{\Delta t}$) is generally proportional to the temperature difference ($\Delta T$). As the temperatures of two objects approach each other, the rate of energy transfer slows down.
3. Spontaneity: This flow occurs naturally without the need for external work.

3.2 Achieving and Defining Thermal Equilibrium

As thermal energy leaves the hotter object, its internal energy decreases, typically leading to a decrease in temperature. Conversely, the cooler object gains energy, and its temperature rises. This exchange continues until the temperature gradient is eliminated.

Characteristics of Thermal Equilibrium:

1. Equality of Temperature: $T_A = T_B$. This is the only condition required for equilibrium.
2. No Net Energy Transfer: It is a common misconception that energy transfer stops. In reality, energy is still being exchanged at the molecular level. However, the rate of energy flow from $A$ to $B$ is exactly equal to the rate of energy flow from $B$ to $A$. Therefore, the net transfer is zero.
3. Stability: Once in equilibrium, the temperatures will remain constant unless an external change is made to the system or its surroundings.

[DIAGRAM DESCRIPTION: Imagine two metal blocks, X and Y. In the first frame, X is at 100°C and Y is at 20°C. A thick arrow points from X to Y. In the second frame, after some time, both are at 60°C. Two identical, thin arrows point in opposite directions between the blocks, indicating that while individual molecular transfers occur, the net flow is zero.]

3.3 The Zeroth Law of Thermodynamics

The concept of thermal equilibrium allows us to define the "Zeroth Law." This law is the logical basis for all temperature measurements.

• Statement: If object $A$ is in thermal equilibrium with object $B$, and object $B$ is in thermal equilibrium with object $C$, then object $A$ must be in thermal equilibrium with object $C$.
• Application: This allows the use of a thermometer (object $B$). If the thermometer shows the same reading when placed in contact with $A$ and then $C$, we can conclude $A$ and $C$ are at the same temperature without ever placing them in direct contact.

3.4 Temperature Scales and Absolute Zero

In A-Level Physics, we distinguish between the Celsius scale and the Thermodynamic (Kelvin) scale.

• The Celsius Scale ($^\circ\text{C}$): Defined by the freezing point ($0^\circ\text{C}$) and boiling point ($100^\circ\text{C}$) of pure water at standard atmospheric pressure.
• The Thermodynamic Scale ($\text{K}$): An absolute scale that does not depend on the properties of any specific substance. It starts at Absolute Zero ($0 \text{ K}$), the temperature at which a system possesses minimum internal energy (and particles have minimum kinetic energy).

3.5 Worked Examples

Worked Example 1 — Qualitative Energy Flow

A lead sphere at $150^\circ\text{C}$ is dropped into an insulated container of oil at $25^\circ\text{C}$. Describe the changes in the average kinetic energy of the molecules in both the lead and the oil as they reach thermal equilibrium.

Answer:

1. Identify Temperature Difference: The lead sphere is at a higher temperature than the oil ($150^\circ\text{C} > 25^\circ\text{C}$).
2. Direction of Flow: Thermal energy will transfer from the lead to the oil.
3. Microscopic Change (Lead): As the lead loses thermal energy, the average kinetic energy of its molecules decreases, causing its temperature to drop.
4. Microscopic Change (Oil): As the oil gains thermal energy, the average kinetic energy of its molecules increases, causing its temperature to rise.
5. Equilibrium State: This process continues until the lead and oil reach the same temperature. At this point, they are in thermal equilibrium, and there is no net transfer of energy. The average kinetic energy of the lead molecules and oil molecules will be consistent with this final shared temperature.

Worked Example 2 — Calculating Equilibrium Temperature

A $0.20 \text{ kg}$ block of aluminum at $90^\circ\text{C}$ is placed in $0.50 \text{ kg}$ of water at $20^\circ\text{C}$. Calculate the final temperature of the mixture, assuming the container is perfectly insulated. (Specific heat capacity of aluminum $c_{Al} = 900 \text{ J kg}^{-1}\text{ K}^{-1}$; Specific heat capacity of water $c_w = 4200 \text{ J kg}^{-1}\text{ K}^{-1}$)

Working:

1. State the Principle: At thermal equilibrium, Energy Lost by Aluminum = Energy Gained by Water. $$\Delta Q_{loss} = \Delta Q_{gain}$$
2. Set up the Equation: Let $T_f$ be the final equilibrium temperature. $$m_{Al} c_{Al} (T_{initial, Al} - T_f) = m_w c_w (T_f - T_{initial, w})$$
3. Substitution: $$(0.20) \times (900) \times (90 - T_f) = (0.50) \times (4200) \times (T_f - 20)$$ $$180 \times (90 - T_f) = 2100 \times (T_f - 20)$$
4. Expansion and Rearrangement: $$16200 - 180T_f = 2100T_f - 42000$$ $$16200 + 42000 = 2100T_f + 180T_f$$ $$58200 = 2280T_f$$
5. Final Calculation: $$T_f = \frac{58200}{2280} = 25.526...$$ Answer: $T_f \approx 26^\circ\text{C}$ (to 2 significant figures, matching the input data).

Worked Example 3 — Power and Equilibrium

A $50 \text{ W}$ immersion heater is used to heat a $1.5 \text{ kg}$ block of metal. The block is in thermal equilibrium with its surroundings at $20^\circ\text{C}$. When the heater is switched on, the temperature of the block rises. After a long period, the temperature of the block stays constant at $45^\circ\text{C}$ even though the heater is still on. Explain this in terms of thermal equilibrium.

Answer:

1. Initial State: Initially, the heater provides energy, increasing the block's temperature above the surroundings.
2. Energy Loss: As the block's temperature increases, the temperature difference between the block and the surroundings increases. This increases the rate of thermal energy transfer from the block to the air.
3. Dynamic Equilibrium: At $45^\circ\text{C}$, the rate at which the block gains energy from the heater ($50 \text{ J s}^{-1}$) is exactly equal to the rate at which it loses energy to the surroundings.
4. Conclusion: The block has reached a state of thermal equilibrium with the energy source and surroundings combined, resulting in a constant temperature. The net energy change of the block is zero.

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Key Equations

Equation	Meaning	Data Sheet?
$T/\text{K} = \theta/^\circ\text{C} + 273.15$	Conversion between Celsius and Kelvin scales.	Yes
$T_A = T_B$	Condition for thermal equilibrium between two bodies.	No
$\text{Net } \Delta Q = 0$	No net heat flow occurs at thermal equilibrium.	No
$\Delta Q = mc\Delta\theta$	Energy required to change temperature (used to find equilibrium).	Yes

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Common Mistakes to Avoid

• ❌ Wrong: Stating that "no energy flows" between two objects at thermal equilibrium.
  • ✓ Right: Always state that there is no net flow of energy. Energy is still exchanged at the molecular level, but the rates in both directions are equal.
• ❌ Wrong: Assuming that objects in thermal equilibrium must contain the same amount of thermal energy (internal energy).
  • ✓ Right: Thermal equilibrium only requires equal temperatures. A large bucket of water at $20^\circ\text{C}$ has much more internal energy than a small cup of water at $20^\circ\text{C}$, but they are in thermal equilibrium.
• ❌ Wrong: Using Celsius in equations where a temperature ratio is required (though $\Delta \theta$ in Celsius is equal to $\Delta T$ in Kelvin).
  • ✓ Right: Always convert to Kelvin ($T$) if the equation involves temperature outside of a "change in temperature" ($\Delta T$) context.
• ❌ Wrong: Thinking the final equilibrium temperature is always the simple average of the two starting temperatures.
  • ✓ Right: The final temperature depends on the mass and the specific heat capacity of the substances involved.

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Exam Tips

1. The "Net" Keyword: In any definition of thermal equilibrium, the word "net" is usually a required mark in the Cambridge mark scheme. "No net transfer of thermal energy" is the gold-standard phrase.
2. Defining Temperature: If asked to define temperature, relate it to the direction of energy flow (macroscopic) or the average kinetic energy of molecules (microscopic).
3. Thermal Contact: If a question mentions two objects are "insulated from each other," they cannot reach thermal equilibrium because there is no path for thermal contact.
4. Significant Figures: In thermal calculations, students often lose marks by providing too many decimal places. If the masses and temperatures in the question are given to 2 s.f., provide your final equilibrium temperature to 2 s.f.
5. Directionality Questions: If asked why energy flows from A to B, your answer should always start with: "Because the temperature of A is higher than the temperature of B." Do not mention density, mass, or thermal conductivity as the reason for the direction; they only affect the rate.