Thermodynamics and the Macroscopic Description of Matter

Chapter 18: A Macroscopic Description of Matter

Solids, Liquids, and Gases

Macroscopic matter can be classified into three primary states: solids, liquids, and gases. Each state is characterized by the arrangement and motion of its constituent particles.

• Solids: Rigid structures where atoms vibrate around fixed equilibrium positions, held together by molecular bonds.

• Liquids: Nearly incompressible, with molecules free to move and flow, allowing the liquid to take the shape of its container.

• Gases: Highly compressible fluids where molecules move freely and independently, colliding occasionally with each other or the container walls.

Density and Number Density

Density is a fundamental property describing how much mass is contained in a given volume.

• Mass Density (\( \rho \)): \( \rho = \frac{M}{V} \), where M is mass and V is volume. SI units: kg/m3.

• Number Density: The number of particles per unit volume, \( n = \frac{N}{V} \), where N is the number of particles.

Temperature and Temperature Scales

Temperature quantifies the thermal state of a system and is measured using various scales.

• Celsius Scale: Defined by the freezing (0°C) and boiling (100°C) points of water.

• Kelvin Scale: Absolute scale with zero at absolute zero; conversion: \( T_K = T_C + 273 \).

• Fahrenheit Scale: Used mainly in the US; conversion: \( T_F = \frac{9}{5}T_C + 32 \).

Temperature is an intensive property and has an absolute zero, the lowest possible temperature (~ -273°C).

Thermal Expansion

Most materials expand when heated. The change in length or volume is proportional to the temperature change.

• Linear Expansion: \( \frac{\Delta L}{L} = \alpha \Delta T \), where \( \alpha \) is the coefficient of linear expansion.

• Volume Expansion: \( \frac{\Delta V}{V} = \beta \Delta T \), where \( \beta \) is the coefficient of volume expansion. For solids, \( \beta = 3\alpha \).

Material	\( \alpha \) (°C-1)	\( \beta \) (°C-1)
Aluminum	2.3 × 10-5
Brass	1.9 × 10-5
Concrete	1.2 × 10-5
Steel	1.1 × 10-5
Invar	0.09 × 10-5
Gasoline		9.6 × 10-4
Mercury		1.8 × 10-4
Ethyl alcohol		1.1 × 10-4

Chapter 19: Work, Heat, and Thermodynamics

The Ideal-Gas Law

The ideal-gas law relates the pressure, volume, temperature, and number of particles in a gas:

• \( pV = nRT \) or \( pV = Nk_B T \)

• p: Pressure (Pa)

• V: Volume (m3)

• n: Number of moles

• N: Number of particles

• R: Universal gas constant (8.314 J/mol·K)

• kB: Boltzmann constant (1.38 × 10-23 J/K)

Work in Ideal-Gas Processes

Work done on or by a gas during a process can be interpreted geometrically as the area under the p-V curve.

• Work done on a gas: \( W = - \int_{V_i}^{V_f} p \, dV \)

• For isothermal (constant T) expansion: \( W = -nRT \ln\left(\frac{V_f}{V_i}\right) \)

• For isobaric (constant p) process: \( W = -p(V_f - V_i) \)

• For isochoric (constant V) process: \( W = 0 \)

Heat, Temperature, and Thermal Energy

Heat (Q) is energy transferred due to a temperature difference. Thermal energy is the energy associated with the random motion of atoms and molecules. Temperature quantifies the average kinetic energy of particles.

• Specific Heat (c): The energy required to raise 1 kg of a substance by 1 K. \( Q = mc\Delta T \)

• Molar Specific Heat (C): The energy required to raise 1 mol of a substance by 1 K. \( Q = nC\Delta T \)

Units of Heat

• SI unit: Joule (J)

• 1 calorie (cal) = 4.186 J

• 1 food Calorie (Cal) = 1000 cal = 4186 J

Phase Change and Latent Heat

During a phase change, heat is absorbed or released without a change in temperature. The energy required is called latent heat.

• Heat of Fusion (Lf): Energy to melt 1 kg of a solid.

• Heat of Vaporization (Lv): Energy to vaporize 1 kg of a liquid.

• \( Q = mL \) for phase changes.

Calorimetry

Calorimetry involves measuring heat transfer between substances. When two systems at different temperatures interact thermally, heat flows until thermal equilibrium is reached.

• \( \sum Q_i = 0 \) for an isolated system.

• Example: Mixing hot iron and cold water, the final temperature is found by setting the heat lost by iron equal to the heat gained by water.

Specific Heat of Gases: Cv and Cp

For gases, the specific heat depends on the process:

• Cv: Specific heat at constant volume (isochoric process).

• Cp: Specific heat at constant pressure (isobaric process).

• \( C_p > C_v \) because work is done during expansion at constant pressure.

• For an ideal gas: \( C_p - C_v = R \)

Summary of Thermodynamic Processes

Process	Constant	Work (W)	Heat (Q)
Isochoric	V	0	\( nC_v\Delta T \)
Isobaric	p	\( -p\Delta V \)	\( nC_p\Delta T \)
Isothermal	T	\( -nRT \ln\left(\frac{V_f}{V_i}\right) \)	\( Q = -W \)

Examples and Applications

• Heating water in a microwave: Calculate Q using \( Q = mc\Delta T \), then determine time using power and efficiency.

• Mixing substances at different temperatures: Use calorimetry equations to find final temperature.

• Phase change: Calculate mass vaporized using \( Q = mL \).

Ideal-Gas Law Applications

• Doubling temperature at constant volume doubles pressure.

• At constant pressure, increasing the number of moles increases volume.

Key Equations

• \( pV = nRT \)

• \( Q = mc\Delta T \)

• \( Q = nC\Delta T \)

• \( Q = mL \)

• \( \frac{\Delta L}{L} = \alpha \Delta T \)

• \( \frac{\Delta V}{V} = \beta \Delta T \)

Additional info:

Some clicker questions and examples in the source material reinforce the application of these concepts, such as calculating work in different thermodynamic processes, determining the final temperature after heat exchange, and understanding the relationship between specific heats for gases.