The ideal gas law connects pressure, volume, temperature, and moles of gas. It's crucial for understanding how gases behave under different conditions. This relationship helps explain everyday phenomena like balloon inflation and pressure cookers.

Thermodynamic systems follow energy conservation principles. The first law of thermodynamics shows how internal energy changes relate to heat and work. This concept is key to understanding energy transfers in various processes, from engines to chemical reactions.

Thermodynamic Systems and Properties

Ideal gas law relationships

• Relates pressure ($P$), volume ($V$), temperature ($T$), and number of moles ($n$) for an ideal gas using the equation $PV = nRT$, where $R$ is the universal gas constant
• In isothermal processes (constant $T$), $P$ and $V$ are inversely proportional ($P_1V_1 = P_2V_2$), so increasing $P$ decreases $V$ and vice versa (balloon compression)
• In isobaric processes (constant $P$), $V$ and $T$ are directly proportional ($V_1/T_1 = V_2/T_2$), so increasing $T$ increases $V$ and vice versa (hot air balloon)
• In isochoric processes (constant $V$), $P$ and $T$ are directly proportional ($P_1/T_1 = P_2/T_2$), so increasing $T$ increases $P$ and vice versa (pressure cooker)

Pressure-volume work calculations

• Work done by a gas during expansion or compression is $W = -\int_{V_1}^{V_2} P,dV$
  • Negative sign indicates work done by the system when expanding ($V_2 > V_1$) and work done on the system when compressing ($V_2 < V_1$)
• For isobaric processes, work simplifies to $W = -P\Delta V$, where $\Delta V$ is the change in volume ($V_2 - V_1$)
• For isochoric processes, no work is done ($W = 0$) since volume remains constant (rigid container)
• Work is a path-dependent process, meaning its value depends on the specific path taken between initial and final states

First law of thermodynamics

• States energy conservation: change in internal energy ($\Delta U$) equals heat added ($Q$) minus work done ($W$), expressed as $\Delta U = Q - W$
• $\Delta U$ depends only on initial and final states, not the path taken
• For isolated systems, $\Delta U = 0$ (no heat or work exchanged with surroundings)
• For cyclic processes, $\Delta U = 0$ (system returns to initial state) and net heat added equals net work done ($Q = W$)
• The first law is a manifestation of the principle of conservation of energy in thermodynamic systems

Applications of first law

1. Adiabatic processes: no heat exchanged ($Q = 0$), so $\Delta U = -W$

   • Examples: rapid gas compression/expansion (diesel engine, sound wave)

2. Isothermal processes: constant temperature, $\Delta U = 0$, and $Q = W$

   • Examples: slow gas compression/expansion (heat engine between thermal reservoirs)

3. Isobaric processes: constant pressure, $W = -P\Delta V$, and $\Delta U = Q - P\Delta V$

   • Examples: heating gas at constant pressure (hot air balloon)

4. Isochoric processes: constant volume, no work done ($W = 0$), and $\Delta U = Q$

   • Examples: heating gas in rigid container (pressure cooker)

Thermodynamic Processes and State Functions

• Thermodynamic equilibrium is a state where all macroscopic properties of a system remain constant over time
• State functions, such as internal energy, depend only on the current state of the system, not on how it reached that state
• Reversible processes are idealized processes where the system remains infinitesimally close to equilibrium at all times
• Irreversible processes involve finite changes and are more realistic representations of real-world thermodynamic processes