Zeroth Law Transitivity Checker

Physics Thermodynamics • Temperature and Zeroth Law

Written by STEM Calculators Team Published January 2, 2026 Updated February 24, 2026

8. Zeroth Law Transitivity Checker

Verify thermal-equilibrium transitivity (if \(A\) is in equilibrium with \(B\), and \(B\) with \(C\), then \(A\) with \(C\)), and simulate pairwise contacts (mixing) to see temperatures converge to a common equilibrium.

Inputs support: pi, e, sqrt( ), sin, cos, exp, log. Use *. Temperatures are in °C, mass in kg, and \(c\) in J/(kg·K).

Equilibrium tolerance \(\varepsilon\) (°C): Mixing strategy:

Heat loss option (optional)

Loss fraction per contact step: Ambient \(T_{\text{amb}}\) (°C):

If loss fraction is \(f\), after each mixing step the tool applies: \(T \leftarrow T + f\,(T_{\text{amb}}-T)\). Set \(f=0\) for an ideal insulated contact.

Bodies

#	Label	Mass \(m\) (kg)	Specific heat \(c\) (J/kg·K)	Temp \(T\) (°C)

Zeroth-law equilibrium test uses only temperatures (within \(\varepsilon\)). Mixing uses \(m\) and \(c\).

Transitivity check (choose a triple)

A: B: C:

The checker verifies: if \(A\sim B\) and \(B\sim C\) (both within \(\varepsilon\)), then \(A\sim C\).

Animation settings

Progress \(\tau\in[0,1]\): Play speed:

Graphs interpolate from “start” (\(\tau=0\)) to “end” (\(\tau=1\)).

Ready

Results & Steps

Add bodies and click Run.

Equilibrium / contact diagram

Equilibrium relations & mixing sequence

Solid lines: equilibrium (within \(\varepsilon\)). Dashed: contact/mixing steps.

Temperature vs step

Drag to pan, wheel to zoom (step index on x-axis)

Hover to probe \((\text{step},T)\).

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Theory — Zeroth Law Transitivity

The zeroth law of thermodynamics says that thermal equilibrium is a transitive relation: if \(A\) is in thermal equilibrium with \(B\), and \(B\) is in thermal equilibrium with \(C\), then \(A\) is in thermal equilibrium with \(C\). This is what makes the concept of temperature meaningful and measurable.

1) Equilibrium as a relation

In an idealized setting, “\(A\) is in equilibrium with \(B\)” means that if you put \(A\) and \(B\) in thermal contact, there is no net heat flow and their temperatures are equal.

\[ A\sim B \iff T_A = T_B \]

In experiments and simulations we often use a tolerance \(\varepsilon\) to account for measurement limits:

\[ A\sim B \iff |T_A - T_B|\le \varepsilon \]

2) Transitivity (the zeroth law)

Transitivity is the logical statement:

\[ (A\sim B)\wedge (B\sim C)\ \Rightarrow\ (A\sim C) \]

This allows a thermometer (system \(B\)) to compare temperatures of different bodies \(A\) and \(C\) consistently.

3) Mixing/contact simulation (energy balance)

When bodies are placed in contact and allowed to reach a common equilibrium temperature \(T_f\), and we ignore phase changes and heat loss, conservation of energy gives:

\[ \sum_i Q_i = 0,\qquad Q_i = m_i c_i (T_f - T_i) \] \[ \Rightarrow\quad T_f=\frac{\sum_i m_i c_i T_i}{\sum_i m_i c_i} \]

This equilibrium formula is associative in the ideal case: mixing sequentially or “all at once” gives the same final \(T_f\). If heat loss is present during each contact step, different contact sequences can yield slightly different outcomes.

4) What this calculator shows

Equilibrium matrix: which pairs satisfy \( |T_i-T_j|\le \varepsilon \).
Triple check: evaluates the transitivity implication for a chosen \((A,B,C)\).
Mixing plot: temperature evolution vs step index (not time) to keep interpretation physical and generic.
Diagram: solid edges show equilibrium relations; dashed edges show the chosen contact/mixing sequence.

Note: Real systems may have temperature-dependent \(c\), heat exchange with the environment, and phase changes. Those effects belong to more advanced models; here we focus on the zeroth-law logic and basic calorimetry mixing.