15.1 The First Law of Thermodynamics

The First Law of Thermodynamics

The first law of thermodynamics is the energy conservation law applied to thermal systems. It tells you exactly how a system's internal energy changes when heat flows in or out and when work is done. If you can track heat and work, you can predict what happens to the energy inside any system.

The Law and Its Equation

The first law states that the change in a system's internal energy ($\Delta U$) equals the heat added to the system ($Q$) minus the work done by the system ($W$):

$\Delta U = Q - W$

This is really just a specific version of energy conservation: energy can't be created or destroyed, only transferred or converted between forms (kinetic, potential, thermal, chemical, etc.). The total energy of an isolated system always stays constant.

Heat and work are the two ways energy moves into or out of a system:

• Heat ($Q$) transfers energy because of a temperature difference (through conduction, convection, or radiation)
• Work ($W$) transfers energy through a force acting over a distance (mechanical compression, electrical work, etc.)

Sign Conventions

Getting the signs right is critical for solving problems. For the equation $\Delta U = Q - W$:

• $Q$ is positive when heat is added to the system; negative when heat is removed
• $W$ is positive when work is done by the system (energy leaves); negative when work is done on the system (energy enters)

Calculating Internal Energy Changes

Example 1: 100 J of heat is added to a system, and the system does 50 J of work.

$\Delta U = Q - W = 100 \text{ J} - 50 \text{ J} = 50 \text{ J}$

The system's internal energy increases by 50 J. Half the heat went into doing work; the other half stayed in the system.

Work done by a gas at constant pressure is calculated using:

$W = P\Delta V$

where $P$ is pressure and $\Delta V$ is the change in volume.

• When a gas expands ($\Delta V > 0$), it does positive work on its surroundings, so energy leaves the system.
• When a gas is compressed ($\Delta V < 0$), work is done on the gas, so energy enters the system.

Example 2: A gas expands from 2 L to 4 L against a constant pressure of 1 atm.

1. Convert units: $1 \text{ atm} = 101{,}325 \text{ Pa}$, and $\Delta V = 4 \text{ L} - 2 \text{ L} = 2 \text{ L} = 0.002 \text{ m}^3$

2. Calculate work: $W = P\Delta V = (101{,}325 \text{ Pa})(0.002 \text{ m}^3) = 202.65 \text{ J}$

3. The positive value means the gas did about 203 J of work on its surroundings as it expanded.

Note the unit conversion: you need volume in $\text{m}^3$ and pressure in Pa to get work in joules. The original guide's answer of 202.65 kJ was off by a factor of 1000 because the liter-to-cubic-meter conversion was missed.

Applications in Everyday Situations

Closed systems exchange heat and work but not matter with their surroundings:

• Compressing a gas in a piston does work on the gas, increasing its internal energy and temperature.
• Car engines convert heat from burning fuel into mechanical work through repeated expansion and compression of gases.

Open systems also exchange matter with their surroundings:

• Living organisms take in food (chemical energy) and convert it into heat and work (muscle movement, brain function).
• Refrigerators transfer heat from a cold space to a warm space, which requires a work input (electricity). This doesn't violate the first law because the electrical energy accounts for the difference.

Biological energy conversions follow the first law too:

• Photosynthesis converts light energy into chemical energy stored in glucose.
• Cellular respiration breaks down glucose to produce ATP (the cell's usable energy currency), releasing heat in the process.

Key Thermodynamic Concepts

• Thermodynamics is the study of heat, work, and energy transformations in physical systems.
• State functions are properties that depend only on the system's current state, not on how it got there. Internal energy is a state function. Heat and work are not state functions because they depend on the process path.
• Thermal equilibrium is the condition where two systems in contact reach the same temperature, so no net heat flows between them.
• Adiabatic process is any change where no heat is exchanged with the surroundings ($Q = 0$). In an adiabatic process, $\Delta U = -W$, meaning internal energy changes come entirely from work. Rapid compressions and expansions are often approximately adiabatic because there isn't enough time for heat to transfer.