First Law of Thermodynamics

The First Law of Thermodynamics is the cornerstone of energy analysis in every branch of engineering, from designing more efficient jet engines to optimizing power plant cycles. It provides the rigorous accounting framework that ensures you can track where energy comes from, where it goes, and how it transforms. Without this fundamental principle, the systematic design and improvement of any thermal system—from a refrigerator to a spacecraft—would be impossible.

The Conservation of Energy Principle

At its heart, the First Law of Thermodynamics is a statement of the conservation of energy principle. It posits that energy can be neither created nor destroyed; it can only change forms or be transferred between a system and its surroundings. A system is the specific region of space or quantity of matter you define for analysis, while everything external to it is the surroundings. The boundary separating them can be fixed or movable, and it may or may not allow the transfer of mass.

This law elevates a familiar physical concept into a precise, quantitative tool. When you apply it, you are essentially performing an energy balance: the net change in energy within your system must be exactly accounted for by the energy that has crossed its boundary. This rejects the possibility of a perpetual motion machine of the first kind, a device that would produce work without any input of energy. The First Law tells you such a machine is impossible because it would require creating energy from nothing.

Defining System Types and Energy Transfers

To apply the First Law correctly, you must first classify your system. A closed system (or control mass) is one where no mass crosses the boundary, but energy in the form of heat and work can be transferred. Imagine a sealed piston-cylinder assembly containing a gas. An open system (or control volume) allows both mass and energy to cross its boundary, like a turbine or a compressor in a steady flow. For this foundational discussion, we focus on the closed system, which leads to the most direct statement of the law.

Energy crosses the boundary of a closed system via two mechanisms: heat and work. Heat ($Q$) is energy transfer due solely to a temperature difference between the system and its surroundings. By convention, heat added to the system is positive. Work ($W$) is energy transfer associated with a force acting through a distance. Examples include the boundary work from an expanding gas or electrical work. Work done by the system on the surroundings is positive.

The energy stored within the system is its internal energy ($U$). This encompasses the microscopic kinetic and potential energies of the molecules—their translational, rotational, and vibrational motions, as well as intermolecular forces. For most engineering applications with simple compressible substances (like gases and liquids), changes in internal energy are of primary interest.

The First Law Equation for a Closed System

The verbal principle translates into a powerful mathematical equation for a closed system undergoing a process from an initial state (1) to a final state (2):

$$\Delta U=Q-W$$

or, in its more explicit form:

$$U_2-U_1=Q_{net,in}-W_{net,out}$$

This equation reads: The change in the internal energy of a closed system is equal to the net heat transferred into the system minus the net work done by the system on its surroundings.

Let's break down the logic. If you add heat to the system ($Q>0$), you expect its stored energy to increase. If the system does work on the surroundings ($W>0$), it expends some of its stored energy, so its internal energy decreases. The minus sign before work is crucial and stems from the sign convention. It's helpful to think of $Q$ as energy in and $W$ as energy out; the net of these flows is what gets stored ($\Delta U$).

This is a balance equation: Energy In - Energy Out = Change in Energy Stored. For a closed system, the only stored energy change we typically consider is in internal energy ($\Delta U$). The equation is valid for any process, regardless of whether it is slow, fast, reversible, or irreversible.

Applying the Equation: A Worked Example

Consider a rigid, sealed tank containing 2 kg of air. An electrical resistor inside the tank receives 50 kJ of work from an external battery (electrical work into the system). During this process, 10 kJ of heat is lost from the tank to the cooler surroundings. Determine the change in internal energy of the air.

1. Define the system: The closed system is the air inside the rigid tank.
2. Identify energy transfers:

• Electrical work is done on the system. Therefore, by our sign convention, this is negative work done by the system: $W=-50\text{ kJ}$.
• Heat is lost from the system: $Q=-10\text{ kJ}$.

1. Apply the First Law:

$$\Delta U=Q-W=(-10\text{ kJ})-(-50\text{ kJ})=40\text{ kJ}$$

1. Interpretation: The internal energy of the air increases by 40 kJ. The energy input from the electrical work (50 kJ) exceeds the energy output via heat loss (10 kJ), resulting in a net storage of energy within the system.

Governing Energy Balances in Engineering Processes

The First Law is the governing equation for analyzing virtually every engineering device. For a closed system, like the compression stroke in an engine cylinder, you use $\Delta U=Q-W$ directly. You can analyze cycles—series of processes that return the system to its initial state—by applying the law to each process. Since internal energy is a property (its value depends only on the state, not the process path), the net change in internal energy over a complete cycle is zero. This leads to the critical cycle conclusion: For a closed system undergoing a cycle, the net heat transfer equals the net work transfer ($\oint \delta Q=\oint \delta W$).

For an open system (control volume) operating at steady state (conditions don't change with time), the First Law takes a different form, focusing on flow rates. The steady-flow energy equation accounts for enthalpy ($h=u+Pv$) associated with flowing mass, as well as kinetic and potential energy changes. This form is essential for analyzing turbines, compressors, heat exchangers, and nozzles. The core principle, however, remains the same: energy is conserved.

Common Pitfalls

1. Incorrect Sign Convention for Heat and Work: The most frequent error is misassigning signs. Remember the system's perspective: Qin is positive, Wout is positive. If a problem states "work is done on the system," that means $W_{bysystem}$ is negative. If it states "heat is rejected," that means $Q_{in}$ is negative. Always translate the problem statement into the signs the equation requires.

1. Confusing Internal Energy with Heat: Internal energy ($U$) is a stored property. Heat ($Q$) is a path function—it describes energy in transit, not a property of the system. You cannot say "a system contains heat." You can say it contains internal energy, and heat is one mechanism to change that energy. Similarly, work is a path function, not a property.

1. Applying the Closed-System Form to Open Systems: The equation $\Delta U=Q-W$ is strictly for closed systems where mass is constant. Applying it to a device like a steadily operating water heater, where water flows in and out, will lead to an incorrect analysis. You must use the appropriate control volume formulation that includes terms for mass flow carrying energy across the boundary.

1. Ignoring Energy Forms: In the basic closed-system equation, $\Delta U$ represents the change in all stored energy for that system type. In some problems, you might need to account for changes in macroscopic kinetic and potential energy of the system itself (e.g., a falling piston). The full form is $\Delta E=\Delta U+\Delta KE+\Delta PE=Q-W$. For stationary systems, $\Delta KE$ and $\Delta PE$ are zero, simplifying to $\Delta U=Q-W$.

Summary

• The First Law of Thermodynamics is the conservation of energy principle applied to thermodynamic systems: Energy cannot be created or destroyed, only converted or transferred.
• For a closed system, the law is expressed mathematically as $\Delta U=Q-W$, where a positive $Q$ represents heat added to the system and a positive $W$ represents work done by the system.
• Internal energy ($U$) is a property representing microscopic energy storage, while heat ($Q$) and work ($W$) are path-dependent energy transfers across the system boundary.
• This law provides the essential energy accounting framework for analyzing all engineering processes, from simple sealed containers to complex cycles and steady-flow devices like turbines and compressors.
• Correct application requires meticulous attention to the sign convention for heat and work and selecting the correct system boundary (closed vs. open) for the device being analyzed.