First law of thermodynamics Knowledge-Based Study

Study notes

1. Internal Energy, Heat, and Work

   The First Law of Thermodynamics is essentially the law of conservation of energy applied to thermodynamic systems. It establishes that energy cannot be created or destroyed, only transformed from one form to another.

   For any closed system transitioning from an initial state to a final state, the change in internal energy, denoted as $\Delta U$, is determined by the heat added to the system and the work done by the system:

   $$\Delta U = Q - W$$

   • $\Delta U$ (Change in Internal Energy): Represents the change in the total microscopic kinetic and potential energies of the system's particles. Internal energy is a state function, meaning its value depends solely on the current state of the system, not on the path taken.
   • $Q$ (Heat): The energy transferred across the system boundary due to a temperature difference. Heat is a path function.
   • $W$ (Work): The energy transferred across the system boundary via mechanical means, such as an expansion against external pressure. Work is also a path function.

   Standard Sign Conventions

   Applying the first law requires strict adherence to sign conventions:

   • Heat ($Q$): Positive ($Q > 0$) when heat enters the system from the surroundings (endothermic absorption). Negative ($Q < 0$) when heat leaves the system to the surroundings.
   • Work ($W$): Positive ($W > 0$) when the system performs work on its surroundings (expansion, $\Delta V > 0$). Negative ($W < 0$) when work is done on the system by the surroundings (compression, $\Delta V < 0$).
   • Internal Energy Change ($\Delta U$): Positive when internal energy increases (e.g., temperature rises for an ideal gas) and negative when it decreases.

   Pressure-Volume Work

   When a gas undergoes an infinitesimal volume change $dV$ under pressure $P$, the work done is given by:

   $$dW = P dV$$

   For a finite expansion or compression from volume $V_i$ to $V_f$, total work is:

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

   Graphically, this work corresponds to the area beneath the process path on a Pressure-Volume ($P$-$V$) diagram.

   Four Fundamental Thermodynamic Processes

   The First Law takes simplified forms depending on which thermodynamic variable remains constant during a quasi-static process of an ideal gas:

   1. Isochoric (Constant Volume)

   • Since the volume remains constant, $dV = 0$, which implies $W = 0$.
   • The first law reduces to: $\Delta U = Q_V$.
   • All heat transferred directly changes the system's internal energy and temperature.

   2. Isobaric (Constant Pressure)

   • Pressure $P$ is held constant, so work simplifies to: $W = P(V_f - V_i) = P\Delta V$.
   • The first law becomes: $\Delta U = Q_P - P\Delta V$.
   • Heat supplied is divided between increasing internal energy and doing mechanical boundary work.

   3. Isothermal (Constant Temperature)

   • For an ideal gas, internal energy depends only on temperature. Since $T$ is constant, $\Delta U = 0$.
   • The first law yields: $Q = W$.
   • All absorbed heat is converted entirely into work during an isothermal expansion: $W = nRT \ln(V_f / V_i)$.

   4. Adiabatic (No Heat Exchange)

   • The system is thermally insulated from its surroundings, so $Q = 0$.
   • The first law yields: $\Delta U = -W$.
   • If the gas expands adiabatically ($W > 0$), its internal energy decreases ($\Delta U < 0$), causing the temperature to drop.

   Thermodynamic Cycles

   A thermodynamic cycle consists of a sequence of processes that eventually returns the working substance to its initial state.

   • Because internal energy is a state function and the initial and final states are identical: $\Delta U_{\text{cycle}} = 0$.
   • Applying the first law to an entire cycle gives: $$Q_{\text{net}} = W_{\text{net}}$$
   • On a $P$-$V$ diagram, the net work done during a complete cycle equals the enclosed area of the loop. If the cycle operates clockwise, net work is positive (heat engine); if counterclockwise, net work is negative (refrigerator or heat pump).

   Summary of Key Takeaways

   1. Energy is conserved across all thermal and mechanical interactions.
   2. Heat and work depend on the path, but their difference $Q - W$ depends solely on the initial and final states.
   3. For an isolated system where $Q = 0$ and $W = 0$, internal energy remains strictly constant: $\Delta U = 0$.