Work, Heat, and the First Law of Thermodynamics: Study Notes

Work, Heat, and the First Law of Thermodynamics

Introduction to Thermodynamic Energy Transfer

Thermodynamics studies the transfer of energy in the form of work and heat between a system and its environment. These processes are fundamental to understanding how energy is conserved and transformed in physical systems, particularly gases.

• Work (W): Energy transferred via mechanical interaction (force and displacement).

• Heat (Q): Energy transferred due to a temperature difference between system and environment.

• First Law of Thermodynamics: The change in a system's thermal energy is the sum of work done on the system and heat added to it.

Work in Ideal-Gas Processes

Work is done on a gas when its volume changes under the influence of an external force, such as a movable piston. The amount of work can be visualized and calculated using a pV diagram (pressure-volume diagram).

• Geometric Interpretation: The work done on a gas is the negative of the area under the pV curve between the initial and final volumes.

The mathematical expression for work done on a gas is:

$W = - \int_{V_i}^{V_f} p \, dV$

• If the gas expands ($V_f > V_i$), the work is negative (energy leaves the system).

• If the gas is compressed ($V_f < V_i$), the work is positive (energy enters the system).

Special Ideal-Gas Processes

Different thermodynamic processes have characteristic work and heat relationships, which can be visualized on pV diagrams:

Isochoric Process (Constant Volume)

• Definition: Volume remains constant ($\Delta V = 0$).

• Work Done: $W = 0$ (no area under the curve).

• Heat Transfer: All energy transfer is as heat.

Isobaric Process (Constant Pressure)

• Definition: Pressure remains constant.

• Work Done: $W = -p \Delta V$

• pV Diagram: Area under the horizontal line represents work done.

$W = -p \Delta V$

Isothermal Process (Constant Temperature)

• Definition: Temperature remains constant.

• Work Done: $W = -nRT \ln\left(\frac{V_f}{V_i}\right)$

• pV Diagram: Area under the hyperbolic curve.

Heat and Its Measurement

Heat is energy transferred due to a temperature difference. The SI unit of heat is the joule (J), but the calorie (cal) is also used historically:

• 1 cal = 4.186 J

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

Heat, Temperature, and Thermal Energy

Thermal energy is the energy associated with the random motion of atoms and molecules. Temperature quantifies the average kinetic energy of particles in a system. Heat transfer requires a temperature difference between system and environment.

The Sign of Heat

• Q > 0: Heat flows into the system (system absorbs energy).

• Q < 0: Heat flows out of the system (system loses energy).

• Q = 0: No heat transfer (thermal equilibrium).

Understanding Work and Heat: Comparison Table

The following table summarizes the differences between work and heat as forms of energy transfer:

	Work	Heat
Interaction	Mechanical	Thermal
Requires	Force and displacement	Temperature difference
Process	Macroscopic pushes and pulls	Microscopic collisions
Positive value	$W > 0$ when a gas is compressed. Energy is transferred in.	$Q > 0$ when the environment is at a higher temperature than the system. Energy is transferred in.
Negative value	$W < 0$ when a gas expands. Energy is transferred out.	$Q < 0$ when the system is at a higher temperature than the environment. Energy is transferred out.
Equilibrium	Mechanical equilibrium: no net force or torque.	Thermal equilibrium: same temperature as environment.

The First Law of Thermodynamics

The First Law of Thermodynamics is a statement of energy conservation for thermodynamic systems:

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

• $\Delta E_{th}$: Change in thermal energy of the system

• W: Work done on the system

• Q: Heat added to the system

Thermal Energy for an Ideal Gas

The thermal energy of an ideal gas depends only on its temperature and the number of particles:

• Monatomic gas: $E_{th} = \frac{3}{2} nRT$

• Diatomic gas: $E_{th} = \frac{5}{2} nRT$

Summary Table: Work, Heat, and Thermal Energy in Ideal-Gas Processes

Process	Work (W)	Heat (Q)	Thermal Energy Change ($\Delta E_{th}$)
Isochoric	0	$Q = \Delta E_{th}$	$Q > 0$ (heating), $Q < 0$ (cooling)
Isobaric	$-p\Delta V$	$Q = \Delta E_{th} - W$	Depends on $\Delta V$ and $Q$
Isothermal	$-nRT \ln(V_f/V_i)$	$Q = -W$	0
Adiabatic	Equals $\Delta E_{th}$	0	$\Delta E_{th} = W$

Examples of Thermodynamic Processes

• Isochoric: Volume fixed by a locking pin; no work is done.

• Isothermal: Temperature kept constant by thermal contact with a heat reservoir (flame or ice).

• Adiabatic: Cylinder insulated so no heat is transferred in or out.

Path Dependence of Work and Heat

For a given change of state, the work and heat depend on the path taken in the pV diagram, but the change in thermal energy ($\Delta E_{th}$) depends only on the initial and final states (state function).

Summary of Key Equations

• General Work: $W = - \int_{V_i}^{V_f} p \, dV$

• Isobaric Work: $W = -p \Delta V$

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

• First Law: $\Delta E_{th} = W + Q$

• Thermal Energy (Monatomic): $E_{th} = \frac{3}{2} nRT$

• Thermal Energy (Diatomic): $E_{th} = \frac{5}{2} nRT$

Example Application

Suppose a gas expands isothermally from $V_i$ to $V_f$ at temperature $T$. The work done on the gas is:

$W = -nRT \ln\left(\frac{V_f}{V_i}\right)$

Since the process is isothermal, $\Delta E_{th} = 0$, so $Q = -W$ (all energy transferred as work is balanced by heat flow).