18-1 Temperature

18-1 Temperature#

Prompts

  • State the zeroth law of thermodynamics. Why does it allow us to use a thermometer to compare temperatures of two objects without bringing them into contact?

  • What is thermal equilibrium? When you place a thermometer in a cup of coffee, what happens over time?

  • What is the triple point of water? What Kelvin temperature is assigned to it?

  • How does a constant-volume gas thermometer measure temperature? Why is pressure proportional to temperature at fixed volume?

  • Why do we take the limit \(p_3 \to 0\) (or gas density \(\to 0\)) to define the ideal gas temperature?

Lecture Notes#

Overview#

  • Temperature is an SI base quantity that measures “hotness” or “coldness.” It is measured with a thermometer—a device whose measurable property (length, pressure, etc.) changes with hotness.

  • The zeroth law of thermodynamics underlies temperature: if A and B each equilibrate with a third body (the thermometer), then A and B are in thermal equilibrium with each other.

  • The Kelvin scale is defined using the triple point of water (273.16 K) and a constant-volume gas thermometer.

Thermal equilibrium and the zeroth law#

Thermal equilibrium: When two bodies are in contact and no measurable properties change with time, they are in thermal equilibrium—they have the same temperature.

Zeroth law of thermodynamics: If body A and body B are each in thermal equilibrium with a third body T (e.g., a thermometer), then A and B are in thermal equilibrium with each other.

  • Implication: Temperature is a well-defined property. We can compare temperatures of two objects by measuring each with a thermometer, without bringing them into contact.

  • Historical note: Named “zeroth” because it logically precedes the first and second laws—it defines the concept of temperature used in those laws.

The Kelvin scale and triple point#

Temperature is measured on the Kelvin scale (units: kelvins, K). There is a lower limit (absolute zero, 0 K) but no upper limit.

To fix the scale, we use a reproducible reference: the triple point of water—the single pressure and temperature at which ice, liquid water, and water vapor coexist in equilibrium:

(147)#\[ T_3 = 273.16\;\text{K} \quad \text{(triple-point temperature)} \]

Constant-volume gas thermometer#

A constant-volume gas thermometer uses a gas in a bulb at fixed volume. The gas pressure \(p\) changes with temperature. By convention:

(148)#\[ T = (273.16\;\text{K})\,\frac{p}{p_3} \quad \text{(provisional)} \]

where \(p_3\) is the pressure when the bulb is at the triple point.

  • Different gases give slightly different readings. As the gas density is reduced (less gas in the bulb), readings converge.

  • The ideal gas temperature is defined by taking the limit as gas density → 0:

(149)#\[ T = (273.16\;\text{K})\,\lim_{p_3 \to 0}\frac{p}{p_3} \]

This limit removes gas-specific effects and yields a universal temperature scale.

Poll: Zeroth law

Thermometer T reads 20°C when in contact with object A, and 20°C when in contact with object B. What can we conclude if we bring A and B into contact?

  • (A) They will exchange heat until both reach 20°C

  • (B) They are already in thermal equilibrium; no net heat flow

  • (C) Heat will flow from A to B

Why pressure ∝ temperature?

At constant volume, gas pressure increases with temperature (kinetic theory: faster molecules → more collisions per second → higher pressure). The proportionality \(T \propto p\) is the basis of the gas thermometer.

Summary#

  • Zeroth law: If A and B each equilibrate with T, then A and B are in thermal equilibrium with each other.

  • Thermal equilibrium: no net energy flow; same temperature.

  • Triple point of water: \(T_3 = 273.16\) K.

  • Gas thermometer: \(T \propto p\) at constant volume; ideal gas temperature uses the limit of zero gas density.