Chapter 14: Heat, Internal Energy, and Heat Transfer Mechanisms

Heat as Energy Transfer

• Nature of Heat: Heat is not a physical material or substance that flows between objects. It is a form of energy.
• Common Units of Heat:
  • Calorie (cal): Defined as the amount of heat necessary to raise the temperature of $1.0\,g$ of water by $1.0\,Celsius$ degree.
  • Kilocalorie (kcal or Calorie): This is the unit typically found on food labels. It is the amount of heat necessary to raise the temperature of $1.0\,kg$ of water by $1.0\,Celsius$ degree.
  • $1.0\,kcal = 1000\,cal$.
• Mechanical Equivalent of Heat: Since heat is energy, it can be equated to mechanical work. Experiments involving falling weights to heat water established the following equivalents:
  • $4.186\,J = 1.0\,cal$
  • $4.186\,kJ = 1.0\,kcal$
• Strict Definition of Heat: Heat is the energy transferred from one object to another specifically because of a difference in temperature between those objects.
• Temperature vs. Heat: The temperature of a gas is a measurement of the average kinetic energy of its molecules, whereas heat refers to the transfer of that energy.
• SI Unit: The standard International System unit for heat is the Joule ($J$), because heat is a form of energy.

Internal Energy

• Definition: Internal energy (also referred to as thermal energy) is the sum total of all the energy of all the molecules within a substance.
• Distinctions in Terminology:
  • Temperature: Measures the average kinetic energy of individual molecules.
  • Internal Energy: The total energy (sum) of all molecules in the object.
  • Heat: The actual transfer of energy resulting from a temperature gradient.
• Internal Energy of an Ideal Atomic Gas: It is calculated by multiplying the average kinetic energy per molecule by the total number of molecules ($N$). Given the relationship between average kinetic energy and temperature, internal energy ($U$) can be expressed in terms of temperature.
• Molecular vs. Atomic Gases: If a gas is molecular (multi-atomic) rather than atomic, internal energy must also account for rotational and vibrational kinetic energy in addition to translational kinetic energy.

Specific Heat and Heat Capacity

• Heat Capacity (C): The proportionality constant relating the heat ($Q$) added to an object to the resulting increase in temperature ($\Delta T$):
  • $Q = C\Delta T$
• Specific Heat Capacity (c): Heat capacity can be rewritten to account for the mass ($m$) of the substance. Specific heat is an inherent material property:
  • $C = mc$
• Heat Transfer Equation: The quantity of heat transferred is calculated as:
  • $Q = mc\Delta T$
• Units of Specific Heat (c): Expressed in SI units as $J/(kg\cdot C^{\circ})$.
• Gaseous Specific Heats: Specific heats for gases are more complex and are typically measured under two different conditions:
  • $c_P$: Specific heat at constant pressure.
  • $c_V$: Specific heat at constant volume.

Calorimetry and Problem Solving

• System Classifications:
  • Closed System: A system where no mass enters or leaves, but energy exchange with the environment is possible.
  • Open System: A system where both mass and energy may be transferred into or out of the system.
  • Isolated System: A closed system where no energy in any form is transferred.
• Conservation of Energy in Isolated Systems: In an isolated system, the energy leaving one part must equal the energy entering another part:
  • $\text{heat lost} = \text{heat gained}$
• Calorimeters: Instruments used to make quantitative measurements of heat exchange.
  • Standard Calorimeter: A heated sample is placed into water; the resulting equilibrium temperature allows for the calculation of the sample's specific heat.
  • Bomb Calorimeter: Specifically used to measure the thermal energy released during combustion. This is the standard method for measuring the caloric content of food.
• Conservation Example (Aluminum and Water):
  • Consider a $100\,g$ aluminum block at $100\,^{\circ}C$ ($c_{Al} = 900\,J/kg\cdot C^{\circ}$) placed in a Styrofoam cup with $100\,g$ of water at $0\,^{\circ}C$ ($c_{water} = 4186\,J/kg\cdot C^{\circ}$). The final equilibrium temperature is determined by setting the heat lost by aluminum equal to the heat gained by water.

Phase Diagrams and Latent Heat

• Phase Changes: Adding heat can increase temperature or cause a change in physical state (phase).
• Temperature and Pressure Effects: The specific temperature at which phase changes occur depends on the external pressure.
• Sublimation: Under specific pressure conditions, a substance can transition directly from a solid to a gas.
• Critical Points:
  • Critical Pressure: Above this pressure, the distinction between gas and liquid disappears, and the substance is described as a "fluid."
  • Critical Temperature: Above this temperature, gas and vapor are indistinguishable because the gas can no longer be condensed back into a liquid regardless of pressure.
• Latent Heat (L): The energy required to change the phase of a material without changing its temperature.
  • Heat of Fusion ($L_F$): Heat required to transition $1.0\,kg$ of material from solid to liquid.
  • Heat of Vaporization ($L_V$): Heat required to transition $1.0\,kg$ of material from liquid to vapor.
• Calculation for Phase Change:
  • $Q = mL$
  • This energy is either added to or removed from the system at a constant temperature.

Calorimetry Problem Solving Strategy

1. System Identification: Determine if the system is isolated and if all energy sources are known.
2. Conservation of Energy: Set up the energy balance equations.
3. Sensible Heat: If no phase change occurs, use $Q = mc\Delta T$.
4. Phase Change Terms: If phase changes are possible, include $mL$ terms. Estimate the final phase of the system.
5. Sign Conventions: Ensure all $\Delta T$ values are positive and terms are correctly placed on the "lost" or "gained" side of the equation.
6. Equilibrium: Recognize there is only one final temperature ($T$) for the entire system at equilibrium.
7. Solve: Algebraically isolate the unknown variable.

Case Study: Ice Melting in Tea (Example 14-7)

• Scenario: A $0.50\,kg$ chunk of ice at $-10\,^{\circ}C$ is placed in $3.0\,kg$ of tea (water) at $20\,^{\circ}C$.
• Method 1 ($\sum Q = 0$):
  • [Heat to raise ice from $-10\,^{\circ}C$ to $0\,^{\circ}C$] + [Heat to melt ice at $0\,^{\circ}C$] + [Heat to raise melted ice water from $0\,^{\circ}C$ to $T$] + [Heat lost by tea cooling from $20\,^{\circ}C$ to $T$] = 0.
  • $10,500\,J + 167,000\,J + (0.50\,kg)(4186\,J/kg\cdot C^{\circ})(T - 0\,^{\circ}C) + (3.0\,kg)(4186\,J/kg\cdot C^{\circ})(T - 20\,^{\circ}C) = 0$
• Method 2 ($\text{Heat Gained} = \text{Heat Lost}$):
  • $10,500\,J + 167,000\,J + (0.50\,kg)(4186)(T - 0) = (3.0\,kg)(4186)(20 - T)$
• Result: Final temperature $T = 5.0\,^{\circ}C$.

Heat Transfer: Conduction

• Mechanism: Conduction occurs through molecular collisions where kinetic energy is transferred between particles.
• Formula for Heat Flow: The heat flow per unit time (power) is given by:
  • $\frac{Q}{t} = kA\frac{T_1 - T_2}{l}$
• Thermal Conductivity (k): A material-specific constant.
  • Conductors: Materials with a large value for $k$.
  • Insulators: Materials with a small value for $k$.
• R-values: Building materials are often rated by R-values, which relate to the thickness ($l$) and conductivity ($k$):
  • $R = \frac{l}{k}$

Heat Transfer: Convection

• Mechanism: Heat flow occurs via the mass movement of molecules from one location to another.
• Types of Convection:
  • Natural Convection: Occurs due to density differences (e.g., hot air rising).
  • Forced Convection: Driven by external means like a fan or pump (e.g., forced hot-air heating systems).
• Biological Application: The human body regulates temperature via blood flow. Blood carries heat to the skin surface, where it is released through convection, evaporation, and radiation.

Heat Transfer: Radiation

• Mechanism: Transfer of energy via electromagnetic waves. The most prominent example is the Sun radiating energy at approximately $6000\,K$.
• Stefan-Boltzmann Law: The energy radiated per unit time ($P$) is proportional to the fourth power of the absolute temperature:
  • $\frac{Q}{t} = e\sigma AT^4$
• Constants and Variables:
  • Stefan-Boltzmann Constant (\sigma): $\sigma = 5.67 \times 10^{-8}\,W/m^2\cdot K^4$
  • Emissivity (e): A dimensionless number between $0$ and $1$.
    • Shiny objects have an emissivity near $0$.
    • Dark/Black objects have an emissivity near $1$.
• Solar Radiation Absorption: The rate of absorption depends on the angle relative to the sun's rays:
  • $\frac{Q}{t} = (1000\,W/m^2)eA\cos(\theta)$
• Environmental and Medical Impacts:
  • Seasons: The $\cos(\theta)$ effect caused by the Earth's tilt is responsible for changing seasons.
  • Metabolism: Shivering occurs when the body radiates more heat than it produces, necessitating an increased metabolic rate or clothing.
  • Thermography: Medical imaging used to detect radiation from the body. Warmer areas can indicate infection or tumors, while cooler areas may indicate poor circulation.