KAHIBARO
Table of Contents
Energy bookkeeping in thermodynamics
The first law of thermodynamics is the statement that energy is conserved when a system exchanges heat and work with its surroundings. It tells us how the internal energy of a system changes.
If a system receives heat, its internal energy can increase. If the system does work on its surroundings, its internal energy can decrease. The first law puts these ideas into one equation.
The first law of thermodynamics is
$$
\Delta U = Q - W
$$
where $\Delta U$ is the change in internal energy of the system, $Q$ is the heat added to the system, and $W$ is the work done by the system.
This equation is a bookkeeping rule for energy. It does not tell us how fast a process happens or whether it happens naturally. It only tracks where energy goes.
Meaning of the quantities
Internal energy, written as $U$, is the total microscopic energy inside the system. It includes the random kinetic energy of particles and the potential energy associated with their interactions. For an ideal gas, internal energy depends only on temperature.
Heat, written as $Q$, is energy transferred because of a temperature difference between the system and its surroundings. Work, written as $W$, is energy transferred when a force causes displacement or, in thermodynamics, when the system changes volume against an external pressure.
A very important point is that heat and work are not properties stored inside a system. They are methods of energy transfer. Internal energy is a property of the system, but heat and work are not.
Internal energy is a state function, but heat and work are not state functions.
This means $U$ depends only on the state of the system, while $Q$ and $W$ depend on the path taken between states.
Sign convention
In this course, we use the convention
$$
\Delta U = Q - W
$$
This means positive $Q$ adds energy to the system, and positive $W$ means the system gives energy to the surroundings by doing work.
The sign convention is often the hardest part for beginners, so it helps to keep a simple picture in mind. Heat entering the system is positive. Work leaving the system is positive.
| Quantity | Positive when | Negative when |
|---|---|---|
| $Q$ | heat enters the system | heat leaves the system |
| $W$ | system does work | surroundings do work on system |
| $\Delta U$ | internal energy increases | internal energy decreases |
For the convention used here,
$$
\Delta U = Q - W
$$
Do not switch signs in the middle of a problem.
A simple physical picture
Imagine a gas in a cylinder with a movable piston. If you heat the gas, energy enters as heat. The gas may warm up, so its internal energy increases. It may also expand and push the piston upward, so it does work on the surroundings.
If the gas gets $100 \ \text{J}$ of heat and does $40 \ \text{J}$ of work, then
$$
\Delta U = 100 - 40 = 60 \ \text{J}
$$
So the internal energy increases by $60 \ \text{J}$.
If instead the gas is compressed by the surroundings, then work is done on the gas. In our sign convention, the work done by the system is negative. For example, if no heat enters and the surroundings compress the gas with $30 \ \text{J}$ of work, then $W = -30 \ \text{J}$ and
$$
\Delta U = 0 - (-30) = 30 \ \text{J}
$$
The gas gains internal energy.
Work in volume changes
A common kind of thermodynamic work is pressure-volume work. When a gas expands, it can push back the surroundings. For a small change in volume, the work done by the system is
$$
dW = P \, dV
$$
for a process carried out against pressure $P$. Over a finite change,
$$
W = \int P \, dV
$$
If the pressure is constant, this becomes
$$
W = P \Delta V
$$
where $\Delta V = V_f - V_i$.
When the volume increases, $\Delta V > 0$, so the system does positive work. When the volume decreases, $\Delta V < 0$, so the work done by the system is negative.
For constant pressure volume work,
$$
W = P \Delta V
$$
Expansion gives $W > 0$, compression gives $W < 0$.
Rearranging the first law
The first law can be rearranged depending on what quantity you want to find.
If you want heat added to the system,
$$
Q = \Delta U + W
$$
If you want the work done by the system,
$$
W = Q - \Delta U
$$
These forms are all the same law. They just emphasize different energy transfers.
Examples
Consider a gas that absorbs $250 \ \text{J}$ of heat and does $100 \ \text{J}$ of work. Then
$$
\Delta U = 250 - 100 = 150 \ \text{J}
$$
The internal energy increases.
Now consider a gas that releases $80 \ \text{J}$ of heat, so $Q = -80 \ \text{J}$, and is compressed so that the work done by the system is $W = -50 \ \text{J}$. Then
$$
\Delta U = -80 - (-50) = -30 \ \text{J}
$$
The internal energy decreases by $30 \ \text{J}$.
A third example is an insulated compression, where no heat enters or leaves, so $Q = 0$. If the surroundings do $60 \ \text{J}$ of work on the gas, then $W = -60 \ \text{J}$ and
$$
\Delta U = 0 - (-60) = 60 \ \text{J}
$$
The internal energy rises.
Cyclic processes
Sometimes a system goes through a cycle and returns to its original state. Because internal energy is a state function, its total change over a complete cycle is zero.
$$
\Delta U_{\text{cycle}} = 0
$$
So the first law becomes
$$
Q_{\text{cycle}} = W_{\text{cycle}}
$$
This means the net heat added over the cycle equals the net work done by the system over the cycle.
For any complete cycle,
$$
\Delta U = 0
$$
Therefore,
$$
Q = W
$$
Differential form
For very small changes, the first law is often written as
$$
dU = \delta Q - \delta W
$$
The symbol $dU$ is used because internal energy is an exact differential, since $U$ is a state function. The symbols $\delta Q$ and $\delta W$ remind us that heat and work depend on the process, not just the initial and final states.
This notation is useful later when studying specific thermodynamic processes.
Special importance for ideal gases
For an ideal gas, the internal energy depends only on temperature. So if the temperature increases, the internal energy increases. If the temperature stays constant, the internal energy does not change.
Thus, for an ideal gas with no temperature change,
$$
\Delta U = 0
$$
and the first law becomes
$$
Q = W
$$
This result is especially important in constant-temperature processes, which are studied separately.
Common mistakes
A common mistake is to say that a system contains heat. This is not correct. A system contains internal energy, not heat. Heat is energy in transfer.
Another common mistake is to forget the sign of work. If the gas expands, the system does work, so $W$ is positive in this chapter's convention. If the gas is compressed, $W$ is negative.
Students also sometimes confuse $\Delta U$ with $Q$. Adding heat to a system does not always mean all that energy becomes internal energy. Some of it may leave immediately as work.
Summary
The first law of thermodynamics is the energy conservation law for thermodynamic systems. It relates internal energy, heat, and work through
$$
\Delta U = Q - W
$$
Heat added to the system increases internal energy, while work done by the system decreases it. Internal energy is a property of state, but heat and work depend on the process. In any complete cycle, the net change in internal energy is zero. The first law is the basic tool for analyzing energy transfer in thermodynamic processes.
Core statement of this chapter:
$$
\Delta U = Q - W
$$
Interpretation, energy added by heat minus energy lost by doing work equals the change in internal energy.