Internal energy models

Internal energy is the energy associated with microscopic motion, interactions, and internal structure inside a chosen thermodynamic system. It excludes the kinetic energy of the system moving as a whole and the potential energy assigned to the system as a whole in an external field. Its precise list of contributions depends on the model and boundary.

Microscopic content

For a classical particle model, a schematic internal-energy expression is

\[U=K_{\mathrm{random}}+V_{\mathrm{interactions}}+E_{\mathrm{rotation}}+E_{\mathrm{vibration}}+E_{\mathrm{electronic}}+E_{\mathrm{chemical}}+\cdots\]

Not every model contains every term. A monatomic ideal gas neglects intermolecular potential energy except during brief elastic collisions and has no molecular rotation or vibration. A liquid or solid has substantial interaction energy. Molecules can store rotational and vibrational energy, while atoms and electrons can occupy different internal states.

The ellipsis is a warning to specify scope, not permission to hide discrepancies. A coarse model may omit chemical reactions or radiation. Its output should name these as unmodeled reservoirs.

Separate bulk motion

For particles with masses \(m_i\), positions \(\mathbf r_i\), and velocities \(\mathbf v_i\), the center-of-mass velocity is

\[\mathbf V_{\mathrm{cm}}=\frac{\sum_i m_i\mathbf v_i}{\sum_i m_i}\]

The total translational kinetic energy separates as

\[\sum_i\frac12m_i|\mathbf v_i|^2=\frac12M|\mathbf V_{\mathrm{cm}}|^2+\sum_i\frac12m_i|\mathbf v_i-\mathbf V_{\mathrm{cm}}|^2\]

where \(M=\sum_i m_i\). The first term is bulk center-of-mass kinetic energy; the second can contribute to internal energy. Adding one constant velocity to every particle changes the bulk term but leaves the relative term unchanged.

If the entire body also rotates coherently, thermodynamic bookkeeping may separate that macroscopic rotational energy from random microscopic energy. The split must match the question and remain consistent throughout a calculation.

Interaction energy without double counting

For pairwise interactions described by \(V(r_{ij})\), the potential part is

\[U_{\mathrm{pair}}=\sum_{i<j}V(r_{ij})\]

Summing over every ordered pair \(i\ne j\) counts each interaction twice unless a factor of one half is included. Also state cutoff, boundary, and long-range correction policies; they change the modeled energy.

import numpy as np

positions = np.array([0.0, 1.1, 2.0])
velocities = np.array([0.2, -0.1, 0.1])
masses = np.ones(3)
centre_velocity = np.dot(masses, velocities) / masses.sum()
random_kinetic = 0.5 * np.sum(masses * (velocities - centre_velocity) ** 2)
pair_energy = sum(
    0.5 * 10 * (positions[j] - positions[i] - 1) ** 2 for i, j in [(0, 1), (1, 2)]
)
print(
    "Internal energy of the illustrative spring chain (J):",
    random_kinetic + pair_energy,
)

Potential energy has an arbitrary additive zero. Changes in internal energy are physically meaningful even when its absolute numerical zero is conventional. Never compare totals from two potential models with different zeros without aligning them.

Internal energy is a state function

At equilibrium, internal energy is a state function: its value depends on the present thermodynamic state, not on the path used to reach it. Heat and work are different. They describe energy crossing the boundary during a process and are path-dependent.

Using the convention that heat entering the system is positive and work done by the system is positive, the first law is

\[\Delta U=Q-W\]

The same initial and final equilibrium states have the same \(\Delta U\) even if different paths have different heat \(Q\) and work \(W\). Heat and work are not parts stored inside \(U\).

Some engineering sources define work done on the system as positive, giving \(\Delta U=Q+W_{\mathrm{on}}\). These conventions agree because \(W_{\mathrm{on}}=-W\).

Ideal-gas internal energy

For a classical monatomic ideal gas,

\[U=\frac32Nk_BT=\frac32nRT\]

where \(N\), \(n\), \(k_B\), and \(R\) have their usual meanings. Hence

\[\Delta U=\frac32Nk_B\Delta T=\frac32nR\Delta T\]

For an ideal gas with a constant number \(f\) of active quadratic degrees of freedom,

\[U=\frac{f}{2}Nk_BT\]

This is not a universal formula for matter. Quantum freeze-out can make \(f\) temperature-dependent, and real gases have interaction energy that can make internal energy depend on density or volume as well as temperature.

Heat capacity connects state changes

At constant volume, no pressure–volume boundary work occurs for a closed rigid container, so added heat changes internal energy. Molar heat capacity at constant volume is defined by

\[C_V=\frac{1}{n}\left(\frac{\partial U}{\partial T}\right)_V\]

For a monatomic ideal gas, \(C_V=3R/2\). More generally, heat capacity can vary with temperature and phase. During a first-order phase change, energy can change substantially while temperature remains nearly constant, so a single constant heat capacity is inadequate.

Tabulated heat capacities also depend on whether pressure or volume is held fixed. Using \(C_P\) in a constant-volume calculation silently includes the wrong work conditions.

Boundary choice changes classification

Imagine a block sliding across a rough surface and losing \(5\,\mathrm J\). If the system is the block alone, energy transferred through the contact can cross the boundary, while some deformation may increase the block's own internal energy. If the system includes block and surface, frictional forces between them are internal and the lost bulk kinetic energy can appear as internal energy of the combined system plus sound or other explicitly tracked forms.

Neither choice is automatically wrong. Switching boundaries halfway through a ledger is wrong.

initial_energy, final_energy, heat_in, work_by = 100.0, 107.0, 10.0, 3.0
residual = final_energy - initial_energy - heat_in + work_by
print("First-law residual (J):", residual)
assert residual == 0

The residual should not be renamed “internal energy” merely to force conservation. First look for time-integration error, omitted external work, inconsistent potential zeros, energy carried through open boundaries, or an unmodeled reservoir.

Local equilibrium and internal-energy fields

A large system with gradients may use internal-energy density \(u(\mathbf r,t)\). This requires coarse-graining: each cell must be small compared with the macroscopic variation but large enough to contain meaningful statistics. The local flow velocity must be removed before random kinetic energy is assigned.

Far from equilibrium, the same total internal energy can coexist with very different velocity distributions, stresses, or chemical populations. One scalar \(U\) does not fully specify such a state.

Verification

Add a constant velocity to every particle and confirm that total laboratory kinetic energy changes while internal random kinetic energy does not. Duplicate a noninteracting equilibrium sample: total internal energy should double, while temperature remains the same. For pair potentials, compare a two-particle analytic value and verify each pair is counted once.

Take two different thermodynamic paths between the same ideal-gas endpoint states. Compute heat and work separately; their values may differ, but \(Q-W\) must give the same internal-energy change. Refine numerical time steps and require any closed-system residual to shrink.

Real internal energy may include phase, chemical, nuclear, electronic, magnetic, defect, surface, and radiative contributions. State the temperature, pressure, composition, and model range over which a simplified relation is intended.

A worked Python notebook

Model one mole of monatomic ideal gas with translational internal energy only. The question for this calculation is precise: how does internal energy of one mole change when we change temperature, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Interparticle potential energy and internal molecular excitation are omitted. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 100 to 600 K. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for temperature, expressed in K. The output is internal energy of one mole, expressed in J. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 12.471693927 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 300 K, the expression gives 3741.51 J. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("3*8.314462618*q/2", locals={"q": q})
reference_input = 300.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "K")
print("Predicted internal energy of one mole:", reference_output, "J")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 1247.17 J. The midpoint, at 350 K, gives 4365.09 J. The upper endpoint gives 7483.02 J. Read these together with the assumptions: they describe different choices of temperature within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([100.0, 350.0, 600.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} K -> {output_value:.6g} J")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows temperature; the vertical axis shows internal energy of one mole. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(100.0, 600.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Temperature (K)")
ax.set_ylabel("Internal energy of one mole (J)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does internal energy of one mole change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 300 K is 12.4717 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 12.471693927\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 302.5
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 302.5 K. The tangent estimate is 3772.69 J, while the full expression gives 3772.69 J. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 500.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of temperature inside the stated interval, predict whether internal energy of one mole will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Interparticle potential energy and internal molecular excitation are omitted. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of J: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(600.0) - evaluate(100.0))
for points in [101, 201, 401]:
    grid = np.linspace(100.0, 600.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

• OpenStax, University Physics Volume 2, Section 3.2, “Work, Heat, and Internal Energy”.
• OpenStax, University Physics Volume 2, Section 3.3, “First Law of Thermodynamics”.
• MIT OpenCourseWare 8.01SC, Chapter 29, “Kinetic Theory of Gases”.
• Benjamin Crowell, Light and Matter, pressure, temperature, and microscopic energy.
• The Physics Hypertextbook, “Heat and Work”.