Thermometer

Thermometry is the branch of physics that deals with the measurement of temperature.

A thermometric property is any physical property of a material that varies in a predictable and measurable way with temperature. These properties are used to construct thermometers and establish temperature scales.

Common Thermometric Properties

1. Expansion of solids, liquids, and gases (e.g., mercury or alcohol expansion in thermometers)
2. Electrical resistance of a conductor (used in resistance thermometers)
3. Electromotive force (e.m.f.) in thermocouples
4. Pressure of a fixed mass of gas at constant volume

These properties must: Respond consistently to temperature changes, Be measurable over a wide temperature range, Be reproducible and agree with other thermometric responses

To define a temperature scale, three key criteria must be satisfied:

1. Selection of a thermometric property
   A property that reliably changes with temperature must be chosen (e.g., volume, resistance, pressure).
2. Establishment of a functional relationship
   A mathematical or empirical relationship must relate temperature to the thermometric property (e.g., linear relationship between resistance and temperature in metals).
3. Use of fixed calibration points
   Defined and reproducible temperatures, such as the ice point (0°C) and steam point (100°C), are assigned numerical values to calibrate the scale.

To quantify temperature, a temperature scale must be established based on a chosen thermometric property. A good temperature scale relies on:

• A sensitive and reproducible thermometric property
• Fixed reference points for calibration
• A well-defined numerical scale

Common Temperature Scales

• Celsius (°C): Based on the ice and steam points (0°C and 100°C)
• Fahrenheit (°F): Uses 32°F as freezing point and 212°F as boiling point of water
• Kelvin (K): Absolute temperature scale starting at absolute zero (0 K = –273.15°C)

A thermometer is an instrument used to measure temperature. It is based on a physical (thermometric) property of a substance that changes predictably with temperature. Examples include:

• Liquid-in-glass thermometers (volume expansion of mercury or alcohol)
• Resistance thermometers (variation of resistance with temperature)
• Thermocouples (variation of voltage generated at junctions of different metals)

To create a temperature scale, we rely on a thermometric property ($X$) that changes uniformly with temperature. This property could be volume, pressure, resistance, etc. Two fixed points are commonly used:

• Ice point: 0°C or 273.15 K
• Steam point: 100°C or 373.15 K

Let:

• $X_0$ = value of the property at ice point
• $X_{100}$ = value at steam point
• $X_\theta$ = value at unknown temperature $\theta$

The equation is:

$\frac{\theta}{100} = \frac{X_\theta - X_0}{X_{100} - X_0}$

\Rightarrow \theta = \left(\frac{X_\theta - X_0}{X_{100} - X_0}\right) \times 100^\circ C \tag{7.3}

This equation shows how equal changes in the property correspond to equal changes in temperature, indicating a linear temperature scale.

• Ice point = 32°F
• Steam point = 212°F
• Fundamental interval = $212 - 32 = 180^\circ F$

Celsius to Fahrenheit Conversion

T_F = \frac{9}{5}T_C + 32 \tag{7.4}

Fahrenheit to Celsius Conversion

$T_C = \frac{5}{9}(T_F - 32)$

The thermodynamic or Kelvin scale is absolute and non-empirical—it does not depend on the properties of any specific material. Instead, it's based on theoretical principles of an ideal gas behavior.

Key Fixed Points

• Absolute zero: 0 K = –273.15°C — the temperature where molecular motion theoretically stops.
• Triple point of water: 273.16 K — the unique condition where ice, liquid water, and vapor coexist in equilibrium.

Let:

• $X_{tr}$ = thermometric property at the triple point
• $X_T$ = property at unknown temperature T

The equation becomes:

T = \left( \frac{X_T}{X_{tr}} \right) \times 273.16 \text{ K} \tag{7.9}

• Independent of specific substances
• Basis for scientific temperature measurements
• Used in laws like Boyle's Law and Charles's Law

Absolute Zero

Absolute zero is the lowest possible temperature, where the pressure of an ideal gas becomes zero. At this point, the random motion of atoms and molecules is minimal.

• Value:

$\boxed{0\, \text{K} = -273.15^\circ\text{C} = -459.67^\circ\text{F}}$

• The Kelvin scale begins at absolute zero and is used in scientific calculations because it avoids negative temperatures in thermodynamic equations.

Triple Point

The triple point of a substance is the unique temperature and pressure at which its solid, liquid, and gas phases coexist in equilibrium.

• For water:
  • Temperature: $\boxed{0.01^\circ \text{C} = 273.16\, \text{K}}$
  • Pressure: $\boxed{610\, \text{Pa}}$ (about)
• Importance: The triple point of water is used as a precise fixed reference to define the Kelvin scale. Even tiny changes in temperature or pressure at this point can shift the phase balance.

Let:

• $X_{tr}$ = value of thermometric property at triple point
• $X_T$ = value at unknown temperature
• $T$ = temperature in Kelvin
• $T_{tr} = 273.16\, \text{K}$

From proportionality:

T = \left( \frac{X_T}{X_{tr}} \right) \times T_{tr} \tag{7.9}

This equation allows temperature $T$ to be calculated using a thermometric property $X$, assuming a linear relationship.

To construct a thermometer, you depend on those materials whose properties change uniformly with temperature. For example, the volume of a gas or a liquid increases uniformly with the increase of its temperature, and the length of a solid changes uniformly with increase of its temperature. Similarly, the electric resistance of a wire increases with increase of its temperature. Thermometers are designed to measure the temperature of a body by assigning a numerical value to any given temperature. It uses some measurable properties of matter that change continuously with temperature to measure the unknown temperature. There are many types of thermometers. Some of the common thermometers are liquid-in glass thermometers, gas thermometers, platinum resistance thermometers, thermoelectric thermometers and radiation thermometers (pyrometers).

a. Liquid-in Glass Thermometer

A liquid-in glass thermometer is the simplest and most commonly employed type of temperature measurement device. It is one of the oldest thermometers available in the industry. It mainly comprises of:

1. A bulb which acts as the container holding the liquid whose volume changes with temperature. The bulb also acts as a sensor or gauge which is inserted in the body whose temperature is to be measured
2. A stem which is a glass tube containing a tiny capillary tube enlarged at the bottom into a bulb that is partially filled with a "working fluid".
3. A temperature scale which is basically preset or imprinted on the stem for displaying temperature readings.
4. Point of reference, i.e., a calibration point which is most commonly the ice point.
5. A working liquid which is generally either mercury or alcohol.
6. An inert gas, mainly argon or nitrogen which is filled inside the thermometer above the working liquid to trim down its volatization.

The liquid-in-glass thermometer utilizes the variation in the length of the liquid column in a glass tube as a thermometric property. As temperature changes, the liquid expands or contracts uniformly within the capillary. By replacing the thermometric property $X$ in Equation (7.3) with the length of the liquid column, the temperature $\theta$ in degrees Celsius is given by:

\theta = \left( \frac{l_\theta - l_0}{l_{100} - l_0} \right) \times 100^\circ \text{C} \tag{7.10}

Where:

• $l_0$ = length of the liquid column at the ice point
• $l_{100}$ = length at the steam point
• $l_\theta$ = length at the unknown temperature $\theta$

i. Constant-Volume Gas Thermometer

Principle: Measures pressure change at constant volume.

$\frac{P}{T} = \text{constant} \Rightarrow \frac{P_1}{T_1} = \frac{P_2}{T_2}$

Example 1:

$\frac{100}{273} = \frac{150}{T_2} \Rightarrow T_2 = \frac{150 \times 273}{100} = 409.5\,\text{K}$

ii. Constant-Pressure Gas Thermometer

Principle: Measures volume change at constant pressure.

$\frac{V}{T} = \text{constant} \Rightarrow \frac{V_1}{T_1} = \frac{V_2}{T_2}$

Example 2:

$\frac{1.0}{300} = \frac{V_2}{600} \Rightarrow V_2 = \frac{600 \times 1.0}{300} = 2.0\,\text{L}$

i. Resistance Thermometers

Principle: Resistance of metals increases with temperature.

$R = R_0(1 + \alpha \Delta T)$

Example 3:

$119.6 = 100(1 + 0.00392 \Delta T) \Rightarrow \Delta T = \frac{0.196}{0.00392} \approx 50^\circ C$

ii. Thermocouple Thermometer

Principle: Two different metals create a voltage when heated.

$V = \beta \Delta T$

Example 4:

$2.5 = \beta \times 100 \Rightarrow \beta = 0.025\, \text{mV/}^\circ\text{C}$

Principle: Measures radiation to estimate temperature without contact.

• Optical Pyrometer – Visible light
• Infrared Pyrometer – Infrared radiation

Applications: Molten metals, kilns, furnaces.