The First Law of Thermodynamics: Energy, Heat, and Work

Chapter 2: The First Law of Thermodynamics

Introduction to the First Law

The First Law of Thermodynamics states that energy in the universe is conserved and can be transferred only in two ways: as heat or as work. This foundational law is essential for understanding energy changes in physical and chemical processes.

• Mathematical Statement: $\Delta U = q + w$

• Heat (q): Transfer of thermal energy due to a temperature difference; path function.

• Work (w): Energy change due to movement against an opposing force; path function.

• Internal Energy (U): The sum of kinetic and potential energy of all particles in a system.

Types of Work in Thermodynamics

Mechanical work is just one of many types of work (e.g., electrical, chemical, osmotic, surface), but in thermodynamics, pressure–volume (PV) work is especially important, particularly for gases.

• PV Work: Work done by or on a gas during expansion or compression.

• Formula for Work: $w = - \int P_{ext} dV$

Reversibility and Irreversibility in Thermodynamic Processes

Thermodynamic processes can be classified as reversible or irreversible. The amount of work calculated for expansions or compressions depends on the reversibility of the process.

• Reversible Process: Proceeds infinitely slowly, always in equilibrium; maximizes work done by the system.

• Irreversible Process: Proceeds rapidly, not in equilibrium; less work is done compared to a reversible process.

Calculating Work in Thermodynamic Processes

To calculate the work carried out under reversible or irreversible conditions, always start from:

• $w = - \int P_{ext} dV$

Special Relationships for Heat (q) at Constant Volume and Pressure

• Isochoric (Constant Volume, dV = 0): No work is performed, so $q_v = \Delta U$

• Isobaric (Constant Pressure): Work is performed, so $q_p = \Delta H$

Enthalpy and Its Relation to the First Law

Enthalpy (H) is a state function defined as $H = U + PV$. The change in enthalpy is given by:

• $\Delta H = \Delta U + \Delta(PV) = \Delta U + nR\Delta T$ (for ideal gases)

State Functions and Exact Differentials

State functions are properties whose values depend only on the current state of the system, not on the path taken to reach that state. Their differentials are exact, meaning they can be related to other state functions in an equation of state (E.O.S.).

• Mathematical Test for Exactness:

$dz = \left( \frac{\partial z}{\partial x} \right)_y dx + \left( \frac{\partial z}{\partial y} \right)_x dy$ is exact if and only if:

$\left( \frac{\partial^2 z}{\partial x \partial y} \right) = \left( \frac{\partial^2 z}{\partial y \partial x} \right)$

Heat Capacities and Their Relationships

Heat capacity is the amount of heat required to change the temperature of a substance by one degree. There are two important types:

• Constant Volume: $C_v = \left( \frac{\partial U}{\partial T} \right)_V$ leads to $q_v = \Delta U = nC_v\Delta T$

• Constant Pressure: $C_p = \left( \frac{\partial H}{\partial T} \right)_P$ leads to $q_p = \Delta H = nC_p\Delta T$

Relationship Between $C_p$ and $C_v$

• Ideal Gases: $C_p = C_v + R$

• Monatomic Ideal Gases: $C_v^m = \frac{3}{2}R$, $C_p^m = \frac{5}{2}R$

• Diatomic Ideal Gases: $C_v^m = \frac{5}{2}R$, $C_p^m = \frac{7}{2}R$

• Liquids and Solids: $C_p \approx C_v$

Thermodynamic Processes: PV Diagrams

It is important to know how to sketch and label PV diagrams for the following processes:

• Isobaric: Constant pressure

• Isochoric: Constant volume

• Isothermal: Constant temperature

• Adiabatic: No heat exchange

Calculating $\Delta U$, $\Delta H$, $q$, and $w$ for Key Processes

Be able to calculate the change in internal energy ($\Delta U$), enthalpy ($\Delta H$), heat ($q$), and work ($w$) for isobaric, isochoric, isothermal, and adiabatic processes. Instead of memorizing all equations, focus on understanding and deriving them as needed.

Summary Table: Heat Capacities of Gases

Type of Gas	$C_v^m$	$C_p^m$
Monatomic Ideal Gas	$\frac{3}{2}R$	$\frac{5}{2}R$
Diatomic Ideal Gas	$\frac{5}{2}R$	$\frac{7}{2}R$
Liquids/Solids	$C_p \approx C_v$

Additional info: The notes reference handouts and summaries not included in the image. For a complete understanding, consult the referenced handouts on "Energy, Heat and Work" and "Summary of Thermodynamic Equations from Chapter 2" if available.