Heat, Specific Heat, and Thermal Equilibrium: Physics with Calculus Study Notes

Heat and Thermal Energy

Definition and Concept of Heat

Heat (Q) is the amount of energy transferred between systems or objects due to a difference in temperature. It is measured in Joules (J) and is a fundamental concept in thermodynamics and thermal physics.

• Heat transfer occurs when two objects at different temperatures are brought into contact.

• Thermal equilibrium is reached when both objects attain the same temperature.

• Heat is not a substance, but a process of energy transfer.

Thermal Expansion and Constraints

When materials are heated, they expand. The total change in length (∆Ltot) of a system can be due to both thermal expansion and mechanical stress.

• Thermal expansion: $\Delta L_{thermal} = \alpha L_0 \Delta T$ where $\alpha$ is the coefficient of linear expansion, $L_0$ is the original length, and $\Delta T$ is the change in temperature.

• Mechanical stress: $\Delta L_{stress} = \frac{L_0 F}{A Y}$ where $F$ is force, $A$ is area, and $Y$ is Young's modulus.

• For a system of connected objects, the sum of individual expansions must satisfy the constraint: $\Delta L_1 + \Delta L_2 + \Delta L_3 + \Delta L_4 = 0$ (if rigid walls prevent separation).

Specific Heat and Heat Capacity

Specific Heat

The specific heat (c) of a material is the amount of heat required to raise the temperature of 1 kg of the material by 1 K.

• Units: J/(kg·K)

• Formula: $Q = m c \Delta T$ where $Q$ is heat added, $m$ is mass, $c$ is specific heat, and $\Delta T$ is temperature change.

• Materials with higher specific heat require more energy to change their temperature.

Molar Heat Capacity

The molar heat capacity (C) is the amount of heat required to raise the temperature of 1 mole of a substance by 1 K.

• Units: J/(mol·K)

• Formula: $Q = n C \Delta T$ where $n$ is the number of moles, $C$ is molar heat capacity.

• Relationship: $c$ is for mass, $C$ is for moles.

Table: Specific Heats and Molar Heat Capacities

The table below compares specific heat and molar heat capacity for various substances at constant pressure.

Substance	Specific Heat, c (J/kg·K)	Molar Mass, M (kg/mol)	Molar Heat Capacity, C (J/mol·K)
Aluminum	910	0.0270	24.6
Beryllium	1970	0.00901	17.7
Copper	390	0.0635	24.8
Ethanol	2428	0.0461	111.9
Ethylene glycol	2386	0.0620	148.0

Heat Transfer and Thermal Equilibrium

Thermal Equilibrium Between Two Objects

When two objects of equal mass but different initial temperatures are placed in thermal contact and insulated from the environment, heat flows from the hotter to the cooler object until equilibrium is reached.

• The final equilibrium temperature depends on the masses and specific heats of the objects.

• If both objects have the same mass and specific heat, the final temperature is the average of the initial temperatures.

• If specific heats differ, the final temperature is weighted by the heat capacities.

Calculation Example: Final Equilibrium Temperature

Given two 1-kg objects, one at 100ºC (c1 = 300 J/kg·K) and one at 0ºC (c2 = 900 J/kg·K), the final equilibrium temperature can be found by setting the heat lost by the hot object equal to the heat gained by the cold object:

• Let $T_f$ be the final temperature.

• Heat lost: $Q_1 = m_1 c_1 (T_1 - T_f)$

• Heat gained: $Q_2 = m_2 c_2 (T_f - T_2)$

• At equilibrium: $Q_1 = Q_2$

• Set up the equation: $m_1 c_1 (T_1 - T_f) = m_2 c_2 (T_f - T_2)$

• Solve for $T_f$:

Equation: $T_f = \frac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2}$

Factors Affecting Heat Capacity

Microscopic Explanation

The heat capacity of a material depends on its molecular structure and the types of energy storage available. For simple materials, most energy goes into translational motion. For more complex materials, energy is also stored in rotational and vibrational modes, increasing the heat capacity.

• Temperature is proportional to the average kinetic energy of molecules.

• More degrees of freedom (rotations, vibrations) mean more energy is needed to raise temperature.

Summary Table: Key Formulas

Concept	Formula	Units
Specific Heat	$Q = m c \Delta T$	J/(kg·K)
Molar Heat Capacity	$Q = n C \Delta T$	J/(mol·K)
Thermal Expansion	$\Delta L = \alpha L_0 \Delta T$	m
Mechanical Stress	$\Delta L = \frac{L_0 F}{A Y}$	m

Applications and Simulations

PhET Simulation: States of Matter

Interactive simulations, such as the PhET States of Matter, help visualize how heat affects molecular motion and phase changes.

• Heating increases molecular kinetic energy.

• Phase changes (solid, liquid, gas) occur as energy is added.

Conclusion

Understanding heat, specific heat, and heat capacity is essential for analyzing energy transfer and temperature changes in physical systems. These concepts are foundational for further study in thermodynamics and thermal properties of matter.