Thermodynamics is concerned with how energy flows into and out of systems. A system is the portion of the universe that we choose to study, while everything outside the system is called the surroundings. The combination of the system and surroundings constitutes the universe.
One of the most important quantities in thermodynamics is the internal energy, \(U\). Internal energy is a measure of a system's capacity to do work on its surroundings. Any process that changes the internal energy of a system must involve the transfer of energy across the system boundary.
There are two primary ways energy can be transferred:
Examples of work include expanding a gas against an external pressure, stretching a spring, lifting a weight against gravity, or moving charge through an electrical circuit.
The First Law of Thermodynamics states that a system's capacity to do work is increased by heating the system or by doing work on the system. Mathematically, the First Law is expressed as
\[ \Delta U = q + w \]
or, in differential form,
\[ dU = dq + dw \]
This equation is simply a statement of conservation of energy. Any increase in the internal energy of a system must come from heat flowing into the system, work being done on the system, or both.
Likewise, if the system loses energy by releasing heat or doing work on the surroundings, the internal energy decreases.
In this course, we will use the convention that energy entering the system is positive and energy leaving the system is negative.
| Process | Sign |
|---|---|
| Heat flows into the system | \(q > 0\) |
| Heat flows out of the system | \(q < 0\) |
| Work is done on the system | \(w > 0\) |
| System does work on the surroundings | \(w < 0\) |
| Internal energy increases | \(\Delta U > 0\) |
| Internal energy decreases | \(\Delta U < 0\) |
For example, when a gas expands against an external pressure, the gas is doing work on the surroundings. As a result, the work term is negative:
\[ w < 0 \]
Conversely, when a gas is compressed, the surroundings are doing work on the gas and the work term is positive:
\[ w > 0 \]
Internal energy is an example of a state function. State functions depend only on the current condition of the system and not on how the system arrived at that condition.
Heat and work are different. They are path functions, meaning that their values depend on the specific pathway followed by the process.
For example, two different pathways may connect the same initial and final states. The amount of heat transferred and the amount of work performed may differ for the two pathways, but the change in internal energy will be the same.
This distinction is one of the most important ideas in thermodynamics:
Big picture: The First Law of Thermodynamics tracks the flow of energy through a system. Heat and work describe how energy is transferred, while internal energy describes how much energy is available within the system. Although heat and work depend on the pathway followed, the change in internal energy depends only on the initial and final states of the system.
A system absorbs \(3.0\ \mathrm{J}\) of energy in the form of heat while expending \(21.0\ \mathrm{J}\) of energy by doing work on the surroundings. Calculate the change in internal energy of the system.
The First Law of Thermodynamics states
\[ \Delta U = q + w \]
The system absorbs heat, so
\[ q = +3.0\ \mathrm{J} \]
The system does work on the surroundings, so energy leaves the system and the work term is negative:
\[ w = -21.0\ \mathrm{J} \]
Substituting these values into the First Law gives
\[ \Delta U = (+3.0\ \mathrm{J}) + (-21.0\ \mathrm{J}) \]
\[ \Delta U = -18.0\ \mathrm{J} \]
Therefore,
\[ \Delta U = -18.0\ \mathrm{J} \]
Physical interpretation: Although the system gains \(3.0\ \mathrm{J}\) of energy through heating, it loses \(21.0\ \mathrm{J}\) by doing work on the surroundings. The work loss is larger than the heat gain, so the internal energy of the system decreases by \(18.0\ \mathrm{J}\).
Use \(\Delta U = q + w\) to solve for the missing quantity. Enter only the numerical value.
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Big picture: The First Law of Thermodynamics tracks the flow of energy through a system. Heat and work transfer energy across the system boundary, and the resulting balance determines whether the internal energy of the system increases or decreases.