2.1 Energy conservation and the First Law

Energy Conservation and the First Law

The First Law of Thermodynamics is the energy accounting rule for every process in the universe: energy can't be created or destroyed, only converted from one form to another. This principle gives you a single equation that connects heat, work, and energy changes in any system, making it the foundation for analyzing everything from car engines to power plants.

First Law of Thermodynamics

The First Law is really just the law of conservation of energy applied to thermodynamic systems. It establishes a precise relationship between three quantities: the heat added to a system, the work done by that system, and the resulting change in the system's total energy.

The mathematical statement is:

$\Delta E = Q - W$

where $\Delta E$ is the change in the system's total energy, $Q$ is the heat added to the system, and $W$ is the work done by the system.

Pay close attention to the sign convention here, because it's a common source of errors:

• $Q > 0$ means heat flows into the system
• $Q < 0$ means heat flows out of the system
• $W > 0$ means the system does work on its surroundings (e.g., a gas expanding)
• $W < 0$ means the surroundings do work on the system (e.g., a gas being compressed)

Some textbooks use $\Delta E = Q + W$, where $W$ represents work done on the system. Both are correct, but you need to know which convention your course uses and stick with it consistently.

Forms of Energy in Systems

A thermodynamic system can hold energy in several forms. The First Law accounts for all of them, so it helps to know what you're tracking.

Kinetic energy (KE) is energy of motion:

$KE = \frac{1}{2}mv^2$

where $m$ is mass and $v$ is velocity. A 1,000 kg car traveling at 30 m/s has $KE = \frac{1}{2}(1000)(30)^2 = 450{,}000 \text{ J}$.

Potential energy (PE) is energy stored due to position or configuration:

• Gravitational PE: $PE_g = mgh$, where $h$ is height above a reference point. A 5 kg book on a 2 m shelf has $PE_g = (5)(9.8)(2) = 98 \text{ J}$.
• Elastic PE: $PE_e = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is displacement from equilibrium.

Internal energy (U) is the sum of all the microscopic kinetic and potential energies of the particles within a system. You can't measure $U$ directly, but you can measure changes in $U$. For an ideal gas, internal energy depends only on temperature: if the temperature doesn't change, $\Delta U = 0$.

Other forms include chemical energy (stored in molecular bonds, like in fuel or batteries), electrical energy (associated with charge separation and electric fields), and thermal energy (the portion of internal energy associated with random particle motion). During thermodynamic processes, energy converts between these forms. A heat engine, for example, converts thermal energy into mechanical work.

First Law in Thermodynamic Processes

How you apply the First Law depends on the type of system and the type of process.

Closed vs. Open Systems

For a closed system (no mass crosses the boundary), the only energy changes come from heat and work. The total energy change reduces to the change in internal energy:

$\Delta U = Q - W$

For an open system (mass flows in or out, like a turbine or nozzle), you also need to account for the energy carried by that mass:

$\Delta E = Q - W + \Delta E_{mass}$

Most introductory First Law problems deal with closed systems, so $\Delta U = Q - W$ is the equation you'll use most often.

Four Key Thermodynamic Processes

Each special process constrains one variable, which simplifies the First Law:

• Isothermal (constant temperature): For an ideal gas, $\Delta U = 0$ because internal energy depends only on temperature. So $Q = W$, and all heat added is converted to work. The work done during isothermal expansion is:

$W = nRT\ln\frac{V_2}{V_1}$

• Adiabatic (no heat transfer, $Q = 0$): The system is perfectly insulated. The First Law becomes $\Delta U = -W$. If the gas expands and does positive work, its internal energy drops and it cools. For an ideal gas, the relation $PV^\gamma = \text{constant}$ holds, where $\gamma$ is the specific heat ratio $C_p/C_v$. Real-world examples include the compression stroke in diesel engines and rapid gas expansion in turbines.
• Isobaric (constant pressure): At constant pressure, the heat transferred equals the change in enthalpy: $Q = \Delta H$. Enthalpy ($H = U + PV$) is a convenient quantity here because it automatically accounts for the $PV$ work the system does as it expands or contracts. Heating water in an open pot is an everyday isobaric process.
• Isochoric / Isovolumetric (constant volume): No expansion or compression means $W = 0$. The First Law simplifies to $\Delta U = Q$. All heat added goes directly into changing the internal energy. Heating a gas in a rigid, sealed container is a classic example.

Problem-Solving with the First Law

When you face a First Law problem, a systematic approach keeps you from making sign errors or using the wrong equation.

1. Define the system and boundaries. What's inside your system? What counts as surroundings? Is mass crossing the boundary (open) or not (closed)?

2. Identify the process type. Is temperature, pressure, volume, or heat transfer held constant? This tells you which simplification to use.

3. Write the appropriate First Law equation.

   • Closed system: $\Delta U = Q - W$
   • Open system: $\Delta E = Q - W + \Delta E_{mass}$

4. Apply the process constraint to eliminate or relate variables:

   • Isothermal ideal gas: $\Delta U = 0$, so $Q = W$
   • Adiabatic: $Q = 0$, so $\Delta U = -W$
   • Isobaric: $Q = \Delta H$
   • Isochoric: $W = 0$, so $\Delta U = Q$

5. Calculate work and/or heat using the relevant formulas. For example, work in an isothermal expansion: $W = nRT\ln\frac{V_2}{V_1}$.

6. Solve for the unknown using algebra.

7. Check your signs and units. Does the direction of energy flow make physical sense? If a gas expanded, $W$ should be positive (work done by the system). If heat was removed, $Q$ should be negative.

The most common mistakes on exams are sign errors (mixing up work done by vs. on the system) and applying the wrong process simplification. Always write out which convention you're using before plugging in numbers.