Mathematical Formulation of the First Law of Thermodynamics

Before delving into the mathematical rigor of the First Law, it is essential to establish a clear vocabulary. Thermodynamics relies on the distinction between the system under study and its surroundings.

• The System: The specific quantity of matter or region in space chosen for analysis. Systems are categorized into three types:
  • Closed System (Control Mass): A system where mass remains constant, but energy (in the form of heat or work) can cross the boundary.
  • Open System (Control Volume): A system where both mass and energy can flow across the boundaries (e.g., a turbine or a nozzle).
  • Isolated System: A system that exchanges neither mass nor energy with its surroundings.
• State Variables: These are properties that describe the current condition of a system and are independent of how the system reached that state. Examples include Internal Energy ($U$), Pressure ($p$), Volume ($V$), and Temperature ($T$). Because they depend only on the state, they are referred to as state functions.
• Process Variables: Unlike state variables, these quantities depend on the specific path taken during a change. Heat ($Q$) and Work ($W$) are the primary process variables. They are not properties of the system but rather energy in transit.
• Internal Energy ($U$): This represents the sum of all microscopic forms of energy within the system, including the kinetic energy of molecular translation/rotation and the potential energy of molecular interactions.

Mathematical Formulations

The First Law of Thermodynamics is essentially the Law of Conservation of Energy applied to thermal systems: energy can neither be created nor destroyed, only transformed from one form to another.

2.1 The General Expression for Closed Systems

For a closed system undergoing a change from state 1 to state 2, the change in internal energy is equal to the net heat added to the system minus the net work done by the system:

$$\Delta U = Q - W$$

Where:

• $\Delta U = U_2 - U_1$ is the change in internal energy.
• $Q$ is the net heat transferred to the system (by convention, $Q > 0$ if heat is absorbed).
• $W$ is the net work done by the system (by convention, $W > 0$ if work is performed by the system on the surroundings).

Note: Some engineering texts use the convention $\Delta U = Q + W$, where $W$ is defined as work done on the system. Consistency in sign convention is critical for accurate modeling.

2.2 The Differential Form and Quasi-Static Processes

To analyze infinitesimal changes, we express the law in its differential form:

$$\mathrm{d}U = \delta Q - \delta W$$

Here, we use $\delta$ instead of $\mathrm{d}$ to emphasize that $Q$ and $W$ are inexact differentials; they cannot be integrated to find a change in a property, but only to find the total energy transferred during a specific path.

In a quasi-static (or reversible) process involving a moving boundary, the work done is typically pressure-volume ($p$-$V$) work:

$$\delta W = p,\mathrm{d}V$$

Substituting this into the differential equation gives:

$$\mathrm{d}U = \delta Q - p,\mathrm{d}V$$

For an ideal gas, where internal energy is a function of temperature only ($U = f(T)$), the relationship can be expanded using the specific heat at constant volume ($C_V$):

$$\delta Q = nC_{V},\mathrm{d}T + p,\mathrm{d}V$$

2.3 Extension to Open Systems (Control Volumes)

In many engineering applications, such as jet engines or compressors, we analyze a control volume where mass flows in and out. The energy balance must account for the enthalpy of the flowing fluid, as well as its kinetic and potential energies. The steady-flow energy equation is expressed as:

$$\dot{Q} - \dot{W}{\text{shaft}} + \sum{i}\dot{m}{i}\left(h{i} + \frac{V_{i}^{2}}{2} + gz_{i}\right) = \sum_{e}\dot{m}{e}\left(h{e} + \frac{V_{e}^{2}}{2} + gz_{e}\right)$$

Where:

• $\dot{m}$ is the mass flow rate.
• $h$ is the specific enthalpy ($h = u + pv$), which accounts for both internal energy and the flow work required to move the fluid.
• $\frac{V^2}{2}$ and $gz$ represent the kinetic and potential energy per unit mass, respectively.

Application to Common Thermodynamic Processes

The behavior of the First Law simplifies significantly under specific constraints:

Illustrative Examples

Example 1: Isobaric Heating of an Ideal Gas

Consider 1 mole of an ideal gas at a constant pressure of $101.3\ \text{kPa}$. The gas is heated from $300\ \text{K}$ to $400\ \text{K}$. Calculate the heat absorbed ($Q$) and the work done ($W$).
(Assume $C_V = \frac{5}{2}R$ for a diatomic gas).

1. Calculate Work ($W$):
   Using the ideal gas law $pV = nRT$, the work done during an isobaric process is:
   $$W = p\Delta V = nR\Delta T$$
   $$W = (1\ \text{mol})(8.314\ \text{J/mol}\cdot\text{K})(100\ \text{K}) = 831.4\ \text{J}$$

2. Calculate Internal Energy Change ($\Delta U$):
   $$\Delta U = nC_V\Delta T = (1)(2.5)(8.314)(100) = 2078.5\ \text{J}$$

3. Calculate Heat ($Q$):
   $$Q = \Delta U + W = 2078.5 + 831.4 = 2909.9\ \text{J} \approx 2.91\ \text{kJ}$$

Example 2: Adiabatic Expansion

In an adiabatic expansion of a monatomic ideal gas ($\gamma = 5/3$), no heat enters or leaves the system ($Q=0$). Therefore, the work performed by the gas during expansion comes entirely from the reduction of its internal energy:
$$W = -\Delta U$$
This explains why a gas cools down during rapid expansion in a vacuum.

Critical Engineering Considerations

To avoid errors in thermodynamic modeling, practitioners must remain mindful of the following:

• Path Dependency: Never attempt to calculate $Q$ or $W$ by simply subtracting two state values. These are not properties. You must know the specific path (e.g., isothermal vs. adiabatic) to determine their values.
• Sign Convention Rigor: Always define your sign convention at the start of a calculation. A mistake in the sign of $W$ or $Q$ will lead to a fundamental violation of the energy balance.
• The "Adiabatic" Approximation: In real-world engineering, a process is rarely perfectly adiabatic. Insulation reduces heat transfer, but it does not eliminate it. In high-precision thermal analysis, heat leakage must be treated as a non-zero term.
• Control Volume vs. Control Mass: When dealing with flowing fluids, failing to include the enthalpy ($h$) term—which incorporates flow work—is a common error that leads to incorrect energy balances.