Work, Heat, and the First Law of Thermodynamics – Chapter 19 Study Notes

Work, Heat, and the First Law of Thermodynamics

Introduction to Thermodynamics

Thermodynamics is the study of energy, its transformations, and its relation to matter. The first law of thermodynamics is a fundamental principle describing the conservation of energy in physical systems.

• Thermal energy refers to the energy due to the motion of atoms and molecules within a system.

• Work (W) is energy transferred by mechanical means, such as forces causing displacement.

• Heat (Q) is energy transferred due to a temperature difference between a system and its environment.

The first law of thermodynamics states:

$\Delta E_{th} = W + Q$

where $\Delta E_{th}$ is the change in thermal energy of the system.

Energy Transfer: Work and Heat

Energy can be transferred into or out of a system as work or heat. The sign conventions are important:

• Work (W): Positive when energy is added to the system (compression), negative when energy is extracted (expansion).

• Heat (Q): Positive when energy is added (environment hotter than system), negative when energy is extracted (system hotter than environment).

Mechanisms of Heat Transfer

Heat can be transferred between a system and its environment by several mechanisms:

• Conduction: Direct transfer through a material.

• Convection: Transfer by the movement of fluids.

• Radiation: Transfer by electromagnetic waves.

• Evaporation: Transfer due to phase change from liquid to gas.

Thermal Properties of Matter

Thermal energy can cause temperature changes or phase changes in matter. The response of materials is governed by:

• Specific heat (c): Energy required to raise the temperature of 1 kg of a substance by 1 K.

• Heat of fusion (Lf): Energy required for a solid to become a liquid.

• Heat of vaporization (Lv): Energy required for a liquid to become a gas.

• Thermal conductivity (k): Measure of a material's ability to conduct heat.

Calorimetry is the process of determining the final temperature when two or more systems interact thermally.

The First Law of Thermodynamics

Statement and Importance

The first law of thermodynamics is a general statement of energy conservation: energy cannot be created or destroyed, only transferred or transformed. This law underpins the operation of engines, power plants, and many other technologies.

Energy Review

The total energy of a system consists of macroscopic mechanical energy and microscopic thermal energy:

$E_{total} = E_{mech} + E_{th}$

For an isolated system (no work done by external forces), the total energy remains constant.

Work in Ideal-Gas Processes

Work Done on a Gas

Consider a gas in a cylinder with a movable piston. Work is done on the gas when the piston compresses or expands the gas:

$W = -p \Delta V$

The negative sign indicates that work done by the external force is opposite to the force caused by the gas pressure.

Work on a pV Diagram

On a pressure-volume (pV) diagram, the work done on a gas is the negative of the area under the pV curve between initial and final volumes.

• Expansion (left to right): Negative work.

• Compression (right to left): Positive work.

Special Ideal-Gas Processes

• Isochoric process (constant volume): $W = 0$

• Isobaric process (constant pressure): $W = -p \Delta V$

• Isothermal process (constant temperature): $W = -nRT \ln\left(\frac{V_f}{V_i}\right)$

• Adiabatic process (no heat transfer): $Q = 0$, so $\Delta E_{th} = W$

Heat, Temperature, and Thermal Energy

Definitions

• Thermal energy (Eth or U): Energy due to the motion of atoms and molecules.

• Heat (Q): Energy transferred due to temperature difference.

• Temperature (T): State variable quantifying the hotness or coldness of a system.

For ideal gases:

$E_{th} = \frac{3}{2} nRT = \frac{3}{2} Nk_B T$

where $n$ is the number of moles, $R$ is the universal gas constant, $N$ is the number of molecules, and $k_B$ is Boltzmann's constant.

Units of Heat

• SI unit: Joule (J)

• Calorie: $1 \text{cal} = 4.186 \text{J}$

• Food calorie: $1 \text{Cal} = 1000 \text{cal} = 4186 \text{J}$

Specific Heat and Calorimetry

Temperature Change and Specific Heat

The energy required to change the temperature of a substance is given by:

$Q = mc \Delta T$

where $m$ is mass, $c$ is specific heat, and $\Delta T$ is the temperature change.

Molar specific heat ($C$):

$Q = nC \Delta T$

where $n$ is the number of moles.

Table: Specific Heats of Various Materials

Material	Specific Heat (J/kg·K)	Molar Specific Heat (J/mol·K)
Aluminum	900	24.3
Copper	385	24.4
Iron	449	25.1
Gold	129	25.4
Lead	128	26.5
Ethyl alcohol	2400	104
Mercury	140	28.1
Water	4186	75.4

Additional info: Water has a very high specific heat, which moderates climate near large bodies of water.

Phase Change and Heat of Transformation

During a phase change, thermal energy changes without a change in temperature. The heat required for a phase change is:

$Q = mL$

where $L$ is the heat of transformation (fusion or vaporization).

Table: Melting/Boiling Temperatures and Heats of Transformation

Substance	Melting Point (°C)	Heat of Fusion (J/kg)	Boiling Point (°C)	Heat of Vaporization (J/kg)
Nitrogen	-210	0.26 × 105	-196	1.99 × 105
Ethyl alcohol	-114	1.09 × 105	78	8.79 × 105
Mercury	-39	0.11 × 105	357	2.96 × 105
Water	0	3.33 × 105	100	22.6 × 105

Additional info: Water's high heat of vaporization is important for climate and biological processes.

Specific Heats of Gases

Constant Volume and Constant Pressure

For gases, specific heat depends on whether the process occurs at constant volume ($C_v$) or constant pressure ($C_p$):

• At constant volume: $Q = nC_v \Delta T$

• At constant pressure: $Q = nC_p \Delta T$

For ideal gases:

$C_p = C_v + R$

where $R = 8.31 \text{ J/mol·K}$ is the universal gas constant.

Table: Molar Specific Heats of Gases

Gas Type	$C_v$ (J/mol·K)	$C_p$ (J/mol·K)
Monatomic	12.5	20.8
Diatomic	20.8	29.1

Additional info: The difference between $C_p$ and $C_v$ arises because work is done during expansion at constant pressure.

Adiabatic Processes

Adiabatic Compression and Expansion

In an adiabatic process, no heat is transferred ($Q = 0$). The relationship between pressure and volume is:

$p V^{\gamma} = \text{constant}$

where $\gamma = \frac{C_p}{C_v}$ (1.67 for monatomic gases, 1.40 for diatomic gases).

• Adiabats are steeper than isotherms on a pV diagram.

• Temperature rises during adiabatic compression and falls during adiabatic expansion.

Heat-Transfer Mechanisms

Conduction

Heat transfer by conduction is described by:

$\frac{dQ}{dt} = kA \frac{\Delta T}{L}$

where $k$ is thermal conductivity, $A$ is cross-sectional area, $\Delta T$ is temperature difference, and $L$ is length.

Convection

Convection is the transfer of heat by the movement of fluids. It can be natural (due to buoyancy) or forced (using fans or pumps).

Radiation

All objects emit energy as electromagnetic radiation. The rate of heat transfer by radiation is:

$\frac{dQ}{dt} = e \sigma A T^4$

where $e$ is emissivity (0 to 1), $\sigma = 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$ is the Stefan-Boltzmann constant, $A$ is surface area, and $T$ is absolute temperature in kelvins.

Net radiative heat transfer between an object and its environment:

$\frac{dQ_{net}}{dt} = e \sigma A (T^4 - T_{env}^4)$

Evaporation and Condensation

• Evaporation: Fast-moving molecules escape from the liquid surface, cooling the remaining liquid.

• Condensation: Vapor molecules re-enter the liquid, releasing heat.

• Relative humidity affects the rate of evaporation and condensation.

Summary Table: Key Equations

Process	Equation
First Law of Thermodynamics	$\Delta E_{th} = W + Q$
Work (Isobaric)	$W = -p \Delta V$
Work (Isothermal)	$W = -nRT \ln\left(\frac{V_f}{V_i}\right)$
Heat (Temperature Change)	$Q = mc \Delta T$
Heat (Phase Change)	$Q = mL$
Conduction	$\frac{dQ}{dt} = kA \frac{\Delta T}{L}$
Radiation	$\frac{dQ}{dt} = e \sigma A T^4$

Example Application: The operation of an internal combustion engine involves adiabatic compression, isochoric heating, and isobaric expansion, all governed by the first law of thermodynamics.

Additional info: These principles are foundational for understanding engines, refrigerators, climate, and biological processes.