# The Zeroth Law and Thermodynamic Coefficients

Physics II · Thermodynamics · https://tryals.app/learn/physics-ii/the-zeroth-law-and-thermodynamic-coefficients

## What Makes a Thermometer Possible

The **zeroth law** looks trivial and is not: if A is in thermal equilibrium with C, and B is in thermal equilibrium with C, then A and B are in thermal equilibrium with each other. Thermal equilibrium is **transitive**.

That transitivity is exactly what licenses temperature. It means all mutually equilibrated systems share one common property, so a single number can label the equivalence class, and it means a thermometer works, since C can be a small instrument brought to each body in turn.

Empirical scales rely on some property that varies with heat: the length of a mercury column, a resistance, a gas pressure. These agree at their calibration points and drift between them, because different substances vary differently. The **ideal gas scale** is better behaved: at low density all gases converge on the same reading, giving $T$ from $\lim_{P\to 0} PV$. The **thermodynamic scale**, defined later from Carnot efficiencies, turns out to coincide with it while depending on no substance whatever.

A substance's response to changes is captured by three coefficients:

$$\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P, \quad \kappa_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T, \quad \beta = \frac{1}{P}\left(\frac{\partial P}{\partial T}\right)_V$$

the **thermal expansivity**, the **isothermal compressibility**, and the pressure coefficient. The minus sign in $\kappa_T$ makes it positive, since raising the pressure always reduces the volume, a substance with negative $\kappa_T$ would be mechanically unstable and would collapse.

The three are not independent. Because any two of $P$, $V$, $T$ fix the third, the chain rule ties them:

$$\alpha = \kappa_T \beta P$$

> **Common pitfall:** treating the zeroth law as an empty statement. Transitivity is a genuine empirical fact about nature, not a logical necessity, and without it "temperature" would not be a well-defined property at all, since two bodies matching a third could still disagree with each other.

## Practice questions

7 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A metal rod of length 2.0 m has a linear expansivity of $1.2 \times 10^{-5}$ K$^{-1}$. Compute its expansion in mm when heated by 50 K, to one decimal place.

**Answer:** 1.2 (within ±0.05)

**Why:** $\Delta L = L\alpha\Delta T = 2.0 \times 1.2\times10^{-5} \times 50 = 1.2 \times 10^{-3}$ m $= 1.2$ mm. Small, but bridges and rails are built with expansion joints for exactly this.

Page: https://tryals.app/practice/physics-ii/the-zeroth-law-and-thermodynamic-coefficients/a-metal-rod-of-length-2-0-m-has-a-linear-expansivity-of-1-2-10

### 2. Two different empirical thermometers calibrated at the same two fixed points will agree at all temperatures between them.

**Answer:** False

**Why:** False, they agree at the calibration points by construction, then drift apart, because mercury, resistance and gas pressure vary differently with temperature. This is exactly why a substance-independent scale was needed.

Page: https://tryals.app/practice/physics-ii/the-zeroth-law-and-thermodynamic-coefficients/two-different-empirical-thermometers-calibrated-at-the-same-two-fixed

### 3. Match each thermodynamic coefficient to what it measures.

**Answer:**

- Thermal expansivity → Fractional volume change per unit temperature at fixed pressure
- Isothermal compressibility → Fractional volume reduction per unit pressure at fixed temperature
- Pressure coefficient → Fractional pressure change per unit temperature at fixed volume
- Heat capacity → Energy needed per unit temperature rise

**Why:** Each of the first three is a fractional response with one variable held fixed, and they are linked by $\alpha = \kappa_T\beta P$ because any two of $P$, $V$, $T$ determine the third. Heat capacity is an energy per degree, not a fractional response.

Page: https://tryals.app/practice/physics-ii/the-zeroth-law-and-thermodynamic-coefficients/match-each-thermodynamic-coefficient-to-what-it-measures

### 4. The zeroth law is an empirical observation of transitivity rather than a purely logical truth. What follows from this distinction when we attempt to assign a single temperature to a system?

A. A thermometer can only ever calibrate empirical substance scales
B. Temperature exists as an objective label for an equivalence class
C. Thermal equilibrium is guaranteed by energy conservation alone
D. Systems in contact must exchange heat until their volumes equalize

**Answer:** B. Temperature exists as an objective label for an equivalence class

**Why:** Without empirical transitivity, thermal equilibrium would not partition states into equivalence classes, making a single scalar temperature mathematically invalid. Transitivity is independent of energy conservation and does not require equal volumes.

Page: https://tryals.app/practice/physics-ii/the-zeroth-law-and-thermodynamic-coefficients/the-zeroth-law-is-an-empirical-observation-of-transitivity-rather

### 5. Which statements about the ideal gas temperature scale are correct?

A. It uses the triple point of water as its fixed point
B. It depends on which gas is chosen for the thermometer
C. All gases converge on the same reading as density tends to zero
D. It agrees with the thermodynamic scale defined from Carnot efficiencies

**Answer:** A. It uses the triple point of water as its fixed point; C. All gases converge on the same reading as density tends to zero; D. It agrees with the thermodynamic scale defined from Carnot efficiencies

**Why:** Low-density convergence is what makes the scale nearly substance-independent, it is anchored at the triple point, and it coincides with the thermodynamic scale. Gas dependence is exactly the flaw it was designed to remove.

Page: https://tryals.app/practice/physics-ii/the-zeroth-law-and-thermodynamic-coefficients/which-statements-about-the-ideal-gas-temperature-scale-are-correct

### 6. Complete the account of what the zeroth law provides.

**Answer:** The zeroth law states that thermal equilibrium is **transitive**, which means a single number can label every mutually equilibrated system. That number is the **temperature**, and the third body used to compare two others is a **thermometer**. Without this law, two bodies each matching a third could still **disagree** with each other.

**Why:** Transitivity is what makes temperature a well-defined property and a thermometer a valid instrument. It is an empirical fact rather than a logical necessity, nature did not have to behave this way.

Page: https://tryals.app/practice/physics-ii/the-zeroth-law-and-thermodynamic-coefficients/complete-the-account-of-what-the-zeroth-law-provides

### 7. A gas at constant volume has a pressure of 120 kPa at 300 K. Compute its pressure in kPa at 400 K.

**Answer:** 160 (within ±2)

**Why:** $P_2 = P_1T_2/T_1 = 120 \times 400/300 = 160$ kPa. This proportionality is the principle of the constant-volume gas thermometer.

Page: https://tryals.app/practice/physics-ii/the-zeroth-law-and-thermodynamic-coefficients/a-gas-at-constant-volume-has-a-pressure-of-120-kpa-at-300-k-compute