Heat and Calorimetry I

Overview of Topics

• Heat

• Internal Energy

• Specific Heat

• Calorimetry

• Heat Transfer Mechanisms

  • Conduction

  • Convection

  • Radiation

Internal Energy and Temperature

Kinetic Theory and Molecular Motion

The kinetic theory of gases relates the temperature of a substance to the average kinetic energy of its particles. For a monatomic ideal gas, the average kinetic energy per molecule is given by:

• Average Kinetic Energy:

• Root Mean Square Speed:

Here, is Boltzmann's constant, is temperature in Kelvin, and is the mass of a molecule.

Maxwell-Boltzmann Distribution

The distribution of molecular speeds in a gas depends on temperature. At higher temperatures, the distribution broadens and shifts to higher speeds.

• Example: The Maxwell distribution for O2 at 300 K and 1100 K shows higher average speeds and a wider spread at higher temperature.

• Escape Velocity: The speed required for a molecule to escape Earth's gravity is about 11,200 m/s.

Phase Changes and Phase Diagrams

Behavior of Gases and Liquids

Phase diagrams show the state of a substance (solid, liquid, gas) at different temperatures and pressures. The transition between phases involves energy changes.

• Critical Point: The point at which the liquid and gas phases become indistinguishable.

• Triple Point: The unique set of conditions where all three phases coexist.

Phase Change Processes

• Condensing: Gas to liquid

• Boiling: Liquid to gas

• Melting: Solid to liquid

• Freezing: Liquid to solid

• Sublimating: Solid to gas

• Depositing: Gas to solid

Evaporation, Saturation, and Humidity

Evaporation and Equilibrium

Evaporation occurs when molecules at the surface of a liquid gain enough energy to escape into the gas phase. Saturation is reached when the rate of evaporation equals the rate of condensation.

• Humidity: The ratio of the partial pressure of water vapor to the saturation pressure, expressed as a percentage:

Heat and Heat Transfer

Units of Heat

• Calorie (cal): The amount of heat required to raise 1 g of water by 1°C.

• Kilocalorie (kcal):

• Mechanical Equivalent of Heat:

Heat flows spontaneously from regions of high temperature to regions of low temperature.

Joule's Experiment

Joule demonstrated the equivalence of mechanical work and heat by measuring the temperature increase of water stirred by falling weights.

• Application: The work done by the weights is converted into heat, raising the water's temperature.

Internal Energy of an Ideal Gas

Definition and Calculation

Internal energy is the total energy contained within a system, including kinetic and potential energies of all particles.

• For a monatomic ideal gas: where is the number of molecules, is the number of moles, is the gas constant.

Specific Heat and Calorimetry

Heat Capacity and Specific Heat

Specific heat is the amount of heat required to raise the temperature of 1 kg of a substance by 1 K.

• Heat added or removed: where is heat (J), is mass (kg), is specific heat (J/kg·K), is temperature change (K).

• Heat Capacity:

Heat Transfer Mechanisms

Conduction

Conduction is the transfer of heat through a material without the movement of the material itself. The rate of heat transfer by conduction is given by:

• Conduction Equation: where is thermal conductivity, is cross-sectional area, and are temperatures, is thickness.

• Good Conductors:

  • High thermal conductivity

  • Large temperature difference

  • Large contact area

Convection and Radiation

• Convection: Heat transfer by the movement of fluids (liquids or gases).

• Radiation: Transfer of energy by electromagnetic waves, does not require a medium.

Worked Examples

Problem 1: Work and Heat Equivalence

An object of mass 12.0 kg descends 1.25 m at constant speed, repeated 20 times. Calculate work done by gravity and equivalent heat in calories. What mass of water could this heat raise by 1°C?

• per descent

• Total work:

• Convert to calories:

• Mass of water raised by 1°C:

Problem 2: Heating Water

5.0 kg of water is heated from 20°C to 30°C. Calculate the increase in internal energy and final temperature if heated for an additional 5 minutes.

• Final temperature after additional heating: 35°C

Problem 2+: Cost of Heating Water

Calculate the cost to heat 160 L of water from 10°C to 70°C at per kW·hr.

• Use to find energy required, then convert to kW·hr and multiply by cost.

• Additional info: 1 L water ≈ 1 kg.

Problem 3: Heating a Gas at Constant Volume

Calculate heat required to raise temperature of 250 L xenon gas from 20°C to 50°C at constant volume, treating Xe as a monatomic ideal gas.

• Use where for monatomic gases.

Problem 4: Heat Transfer through Insulation

Calculate the rate at which heat must be removed from a freezer to keep the inside at -20°C, given insulation thickness and thermal conductivity.

• Given J/(s·m·K), m2, , , m

• Calculated rate:

Summary Table: Key Formulas

Additional info: These notes cover foundational concepts in thermodynamics and heat transfer, including phase diagrams, calorimetry, and practical applications. The problems illustrate real-world calculations and reinforce the theoretical framework.