4.1 First law of thermodynamics

The first law of thermodynamics is the principle of energy conservation applied to thermodynamic systems. It gives you the mathematical relationship between heat, work, and internal energy, which is the starting point for every energy balance you'll do in chemical engineering.

First Law of Thermodynamics

Energy Conservation and the First Law

Energy cannot be created or destroyed. It can only be converted from one form to another (kinetic, potential, thermal) or transferred between a system and its surroundings. That's the first law in a nutshell.

For an isolated system (no heat or mass crossing the boundary), total energy stays constant. Nothing gets in, nothing gets out, so the energy you start with is the energy you end with.

In a cyclic process, where the system returns to its exact initial state, the net change in internal energy is zero. That means the net heat added to the system equals the net work done by the system over the full cycle. This is the principle behind heat engines and refrigeration cycles.

Closed Systems and Internal Energy Changes

For a closed system (fixed mass, no mass flow across boundaries), the first law is written as:

$\Delta U = Q - W$

• $\Delta U$ = change in internal energy of the system
• $Q$ = heat transferred to the system
• $W$ = work done by the system on its surroundings

The sign convention here matters. $Q > 0$ means the system absorbs heat from the surroundings. $W > 0$ means the system does work on the surroundings. If you mix up the signs, your entire energy balance will be wrong, so nail this down early.

This single equation is the framework for analyzing all the major thermodynamic process types: isothermal (constant temperature), isobaric (constant pressure), isochoric (constant volume), and adiabatic (no heat transfer).

Heat, Work, and Internal Energy

Defining Heat, Work, and Internal Energy

Heat ($Q$) is energy transferred between a system and its surroundings because of a temperature difference. No temperature difference, no heat transfer. It's measured in joules ($J$) or sometimes calories ($cal$).

Work ($W$) is energy transferred when a force acts through a distance. In thermodynamics, the most common form is boundary work (also called $PV$ work), where a gas expands or compresses against a piston. Also measured in joules.

Internal energy ($U$) is the total microscopic kinetic and potential energy of all the molecules within the system. You can't measure $U$ directly, but you can measure changes in it ($\Delta U$), which is what the first law lets you do.

Relating Heat, Work, and Internal Energy

The equation $\Delta U = Q - W$ tells you that any energy entering the system as heat, minus whatever leaves as work, accounts for the change in the system's internal energy. Think of it like a bank account: deposits (heat in) minus withdrawals (work out) equals the change in your balance (internal energy).

A useful example: during an isothermal expansion of an ideal gas, temperature stays constant. For an ideal gas, internal energy depends only on temperature, so $\Delta U = 0$. The first law then simplifies to $Q = W$. All the heat you add goes directly into doing work on the surroundings.

Applying the First Law

Problem-Solving Steps

When you're working an energy balance problem, follow this sequence:

1. Define the system and boundaries. Decide what's inside your system and what's outside. Is it a closed system (fixed mass) or an open system (mass flows in or out)?
2. Identify the initial and final states. Note temperature, pressure, volume, phase, and any other properties given or implied.
3. Identify the process type. Is it isothermal, isobaric, isochoric, or adiabatic? Each type gives you a simplifying constraint.
4. Write the first law equation and substitute what you know. Solve for the unknown ($\Delta U$, $Q$, or $W$).
5. Check your signs. Make sure your sign convention is consistent throughout the problem.

Additional Considerations

You'll often need supplementary relationships alongside the first law:

• The ideal gas law ($PV = nRT$) connects pressure, volume, temperature, and moles for ideal gases.
• For an isobaric process (constant pressure), boundary work simplifies to $W = P\Delta V$, where $P$ is the constant pressure and $\Delta V$ is the change in volume.
• For an isochoric process (constant volume), $\Delta V = 0$, so boundary work is zero and $\Delta U = Q$.
• For an adiabatic process, $Q = 0$, so $\Delta U = -W$. All the work comes at the expense of internal energy.

These simplifications are what make identifying the process type so important. They turn a general equation into something you can actually solve.

State Functions vs. Path Functions

Defining State and Path Functions

A state function depends only on the current condition of the system, not on how it got there. Internal energy ($U$), enthalpy ($H$), entropy ($S$), and Gibbs free energy ($G$) are all state functions. If you know the initial and final states, you can calculate the change in any state function regardless of the process path.

A path function depends on the specific route the system takes between two states. Heat ($Q$) and work ($W$) are path functions. Two different processes connecting the same initial and final states can involve very different amounts of heat and work.

Implications for Thermodynamic Processes

This distinction has practical consequences for how you solve problems:

• Changes in state functions ($\Delta U$, $\Delta H$) can be calculated from initial and final state properties alone. You can pick any convenient path between those states to do the calculation, even if it's not the actual path the system followed.
• Path functions ($Q$ and $W$) require you to know the actual process. You can't just look at the endpoints.
• In a cyclic process, every state function returns to its starting value, so $\Delta U = 0$, $\Delta H = 0$, etc. But the net $Q$ and net $W$ over the cycle are generally not zero. That's exactly how a heat engine produces net work over repeated cycles.