4.3 Specific heats of ideal gases, solids, and liquids
6 min read•july 30, 2024
is crucial for understanding how substances respond to temperature changes. It's the amount of heat needed to raise a material's temperature by one degree, varying based on composition and structure. This property is key for energy analysis in closed systems.
Knowing specific heat helps in designing thermal systems, calculating energy balances, and predicting material behavior. It's used in HVAC, food processing, and energy storage. For gases, we distinguish between constant pressure and constant volume specific heats, linked by the Mayer relation.
Specific heat capacity
Definition and importance
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Specific heats of solids vary with temperature, especially at low temperatures
At high temperatures, specific heat approaches a constant value called the Dulong-Petit limit (about 3R per mole of atoms for many solids)
Temperature dependence is explained by the Einstein and Debye models, which consider the quantized nature of lattice vibrations (phonons)
Example: The specific heat of copper increases from about 0.1 J/(g·K) at 10 K to about 0.4 J/(g·K) at 300 K, approaching the Dulong-Petit limit
Liquids
Specific heats of liquids are generally higher than those of solids and gases due to the additional energy required to overcome intermolecular forces
Temperature dependence is usually less pronounced than that of solids but can be significant near phase transitions
Empirical correlations (polynomial functions or tabulated data) are often used to describe the temperature dependence
Example: The specific heat of water decreases from about 4.22 J/(g·K) at 0°C to about 4.18 J/(g·K) at 20°C, showing a slight temperature dependence
Energy balance problems
First law of thermodynamics
The first law of thermodynamics, or the principle of energy conservation, is the basis for solving energy balance problems
For a closed system with no work interaction, the heat added to the system (Q) equals the change in the system's internal energy (ΔU): Q = ΔU
The change in internal energy is related to the specific heat and :
For an ideal gas, ΔU = m × Cv × ΔT, where m is the mass of the gas
For solids and liquids, ΔU = m × C × ΔT, where m is the mass of the substance
Problem-solving approach
Identify the system boundaries, initial and final states, and any heat transfer or work interactions with the surroundings
Apply the conservation of energy principle to each component in the system
Sum the individual contributions to obtain the total energy balance
Example: A 2 kg aluminum block (C ≈ 900 J/(kg·K)) is heated from 20°C to 80°C. Calculate the heat added to the block.
System: Aluminum block
Initial temperature: 20°C, Final temperature: 80°C
Heat added (Q) equals the change in internal energy (ΔU): Q = ΔU = m × C × ΔT
Q = 2 kg × 900 J/(kg·K) × (80°C - 20°C) = 108,000 J or 108 kJ
Applications in multi-component systems
In problems involving multiple substances or phases, apply the conservation of energy principle to each component
Sum the individual energy balances to obtain the total energy balance for the system
Example: A 1 kg iron block (C ≈ 450 J/(kg·K)) at 100°C is placed in 2 kg of water (C ≈ 4184 J/(kg·K)) at 20°C in an insulated container. Calculate the final equilibrium temperature.
System: Iron block + Water
Energy balance: Heat lost by iron = Heat gained by water