The First Law of Thermodynamics
You have now mastered the two ways energy can be transferred between a system and its surroundings: work (w), the transfer of energy via organized motion, and heat (q), the transfer of energy via random molecular motion. You also know that a system’s total energy content is its internal energy (U).
We are now ready to connect these three concepts with one of the most fundamental principles in all of science.
The First Law of Thermodynamics: Energy Conservation
The First Law of Thermodynamics is a formal statement of the Law of Conservation of Energy. It provides a simple, powerful equation for our energy accounting:
The change in the internal energy of a system (ΔU) is equal to the sum of the heat transferred (q) and the work done (w).
\[ \Delta U = q + w \]
This equation is the central pillar of thermochemistry. It states that the change in the system’s “energy bank account” (ΔU) must be precisely equal to the sum of the two energy “transactions” of heat transfer and work performed.
Because ΔU depends only on the initial and final states of the system, it is a state function. However, as we saw with reversible and irreversible pathways, the specific amounts of heat and work depend on the path taken between those states. This makes q and w path functions.
The First Law tells us that while q and w can vary individually depending on the path, their sum (q + w) is constant for any process that connects the same initial and final states.
NoteState vs. Path Functions: A Mountain Climbing Analogy
Understanding the difference between state and path functions is essential for thermochemistry. Imagine you are climbing a mountain. Your starting point is the Trailhead (the initial state) at an elevation of 1,000 meters, and your goal is the Summit (the final state) at an elevation of 3,000 meters.
The State Function: Change in Elevation
Your net change in elevation (ΔElevation) is a fixed value. It depends only on your starting elevation and your final elevation: \[ \begin{align*} \Delta\text{Elevation} &= \text{Elevation}_{\text{final}} - \text{Elevation}_{\text{initial}} \\[1.5ex] &= 3000~\text{m} - 1000~\text{m} \\[1.5ex] &= 2000~\text{m} \end{align*} \] This value is constant regardless of the path you take. You are 2,000 meters higher than when you started, period.
This is exactly how internal energy (ΔU) works. It is a state function because its change depends only on the initial and final conditions of the chemical system.
The Path Functions: Distance and Energy
Now, consider two different routes to the summit:
- Path A (The Gentle Trail): A long, winding trail with many switchbacks. It is 10 kilometers long.
- Path B (The Steep Climb): A very direct, steep, and physically demanding scramble up the mountain face. It is only 3 kilometers long.
Two quantities are now completely different depending on the route:
The Distance Traveled: This is a path function. You travel a much greater distance on the gentle trail (10 km) than on the steep climb (3 km). This is analogous to work (w), as the amount of work performed can differ greatly depending on the process.
The Energy Expended: This is also a path function. The total energy you burn (in calories) depends on the path’s length and difficulty. This is analogous to heat (q), the other mode of energy transfer.
Even though the distance traveled and the energy expended were completely different for Path A and Path B, the final change in elevation was exactly the same for both.
This is the essence of the First Law: The change in a state function like internal energy (ΔU) is independent of the journey, while the values of path functions like work (w) and heat (q) depend entirely on the specific route taken.
The table below highlights some state and path functions.
A Unified View of Sign Conventions
The First Law equation only works if we are rigorously consistent with the sign conventions, which are always defined from the system’s point of view. The table below summarizes the conventions you learned in the previous sections.
Example: A system undergoes a two-step process:
- In the first step, the system absorbs 315 J of heat while expanding and doing 127 J of work on its surroundings.
- In the second step, 89 J of work is performed on the system while it releases 44 J of heat.
What is the total change in internal energy for the complete process?
Solution:
The First Law states that ΔU = q + w. We need to apply this to each step, carefully tracking signs, then add the results.
Step 1: - Heat absorbed by system: q1 = +315 J (positive because energy enters) - Work done by system: w1 = −127 J (negative because system does work on surroundings)
\[ \begin{align*} \Delta U_1 &= q_1 + w_1 \\[1.5ex] &= (+31\bar{5}~\mathrm{J}) + (-12\bar{7}~\mathrm{J}) \\[1.5ex] &= +18\bar{8}~\mathrm{J} \end{align*} \]
Step 2: - Work done on system: w2 = +89 J (positive because energy enters) - Heat released by system: q2 = −44 J (negative because energy leaves)
\[ \begin{align*} \Delta U_2 &= q_2 + w_2 \\[1.5ex] &= (-4\bar{4}~\mathrm{J}) + (+8\bar{9}~\mathrm{J}) \\[1.5ex] &= +4\bar{5}~\mathrm{J} \end{align*} \]
Total change:
Since internal energy is a state function, changes are additive:
\[ \begin{align*} \Delta U_{\text{total}} &= \Delta U_1 + \Delta U_2 \\[1.5ex] &= (+18\bar{8}~\mathrm{J}) + (+4\bar{5}~\mathrm{J}) \\[1.5ex] &= +23\bar{3}~\mathrm{J} \\[1.5ex] &= +233~\mathrm{J} \end{align*} \]
The system’s internal energy increased by 233 J over the complete two-step process.
Notice: While the individual values of q and w differed between steps (path-dependent), the total ΔU depends only on the initial and final states (state function property).
Applying the First Law
By measuring the heat transfer and work done in a process, we can determine the exact change in the system’s internal energy, a quantity that is often impossible to measure directly.
Practice
A cylinder containing a gas is compressed by a piston, and the surroundings perform 462 J of work on the system. During this process, the system releases 128 J of heat to the surroundings.
What is the change in the internal energy (in J) of the system?
Solution
This problem requires us to apply the First Law of Thermodynamics, ΔU = q + w, by first assigning the correct signs to the values of heat and work.
Determine the sign of work (w): The problem states that work is performed on the system. By convention, this means energy is being “deposited” into the system, so w is positive. \[w = +46\bar{2}~\mathrm{J}\]
Determine the sign of heat (q): The problem states that the system releases heat. By convention, this means energy is being “withdrawn” from the system, so q is negative. \[q = -12\bar{8}~\mathrm{J}\]
Calculate ΔU: Now substitute these signed values into the First Law equation. \[ \begin{align*} \Delta U &= q + w \\[1.5ex] &= (-12\bar{8}~\mathrm{J}) + (+46\bar{2}~\mathrm{J}) \\[1.5ex] &= +33\bar{4}~\mathrm{J} \end{align*} \] The internal energy of the system increased by 334 J.
Special Cases Under Constraints
The First Law of Thermodynamics, ΔU = q + w, provides a complete energy accounting for any process. However, just as we discovered that specific constraints make certain relationships particularly useful for laboratory measurements, we find that controlled experimental conditions provide direct access to thermodynamic state functions.
In many laboratory and industrial settings, systems are deliberately constrained to simplify the mathematics while making specific quantities experimentally accessible. Two of the most important constraints are:
- Constant volume conditions - where the system cannot change its volume
- Constant pressure conditions - where the system experiences constant external pressure
Understanding how these constraints simplify the First Law provides both theoretical insight and practical laboratory advantages. As we’ll discover, each constraint makes a different state function directly measurable through heat transfer.
Internal Energy at Constant Volume
Many chemical processes are studied in rigid containers where the volume cannot change. In bomb calorimetry, for example, reactions occur in sealed steel vessels designed to withstand high pressures without expanding. Under these constant-volume conditions, the First Law simplifies dramatically.
Starting with the First Law:
\[ \Delta U = q + w \]
For pressure-volume work, the work term is:
\[ w = -P_{\mathrm{ext}} \Delta V \]
Under constant volume conditions:
\[ \Delta V = 0 \]
Therefore:
\[ w = -P_{\mathrm{ext}}(0) = 0 \]
Substituting back into the First Law:
\[ \Delta U = q + 0 = q \]
At constant volume, the heat transferred equals the change in internal energy:
\[ \Delta U = q_{\mathrm{V}} \]
where qV is the heat transferred under constant-volume conditions.
Conceptual Understanding: ΔU = qV
This result reveals a crucial conceptual point:
- Internal Energy (U) is a state function that exists for all conditions regardless of volume constraints
- Heat at constant volume (qV) is a path function that depends on the specific process pathway taken
- The equality ΔU = qV holds only under the specific condition of constant volume
This is not a conceptual identity but rather a mathematical equality that occurs when volume constraints make the path-dependent heat transfer numerically identical to the path-independent internal energy change.
Why Constant Volume is Useful
The relationship ΔU = qV is experimentally valuable because it allows us to:
- Directly measure internal energy changes using calorimetry
- Study reactions in sealed systems where pressure changes would complicate measurements
- Access fundamental energy changes without the complication of work terms
This is the principle behind bomb calorimetry, where reactions occur in rigid, sealed steel vessels. The constant-volume constraint eliminates PV work, making ΔU directly measurable from the heat released or absorbed.
Looking Ahead: Constant Pressure Conditions
While constant-volume conditions simplify the First Law beautifully, most laboratory chemistry occurs at constant pressure rather than constant volume. When you perform a reaction in an open beaker or flask, the system can expand or contract against atmospheric pressure, which means work is being done.
Under constant-pressure conditions, a different state function becomes directly measurable through heat transfer. This function, called enthalpy, is the subject of the next section. There you’ll discover that constant-pressure constraints lead to an equally elegant simplification of the First Law, one that proves even more practical for everyday chemistry.
Summary
The First Law of Thermodynamics
The First Law of Thermodynamics expresses the conservation of energy for a closed system:
\[ \Delta U = q + w \]
where ΔU is the change in internal energy, q is heat transferred to the system (positive when absorbed), and w is work done on the system (positive when compressed).
While q and w individually depend on the path taken (path functions), their sum ΔU depends only on the initial and final states (state function).
At Constant Volume
When volume is held constant (V = constant), no PV work is possible, so:
\[ \Delta U = q_{\mathrm{V}} \]
This relationship is the basis of bomb calorimetry, allowing direct measurement of internal energy changes in sealed containers.