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Thermal Physics

Temperature Measuring Scales and Zeroth Law of Thermodynamics

Temperature scales provide a quantitative metric for thermal states based on reproducible reference thermal equilibrium points. Conversion between different temperature scales is established using linear interpolation between ice and steam fixed points.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Conversion formulas between Celsius, Fahrenheit, Kelvin, and Reaumur scales; determination of faulty thermometer readings; triple point of water.

PGT extension

Thermodynamic temperature scale based on Carnot engines; platinum resistance thermometry; thermoelectric EMF scales; gas thermometry corrections at low pressures.

Prerequisites

Concept of thermal equilibrium and Zeroth Law of Thermodynamics
Thermal expansion of solids and liquids
Ideal gas equation and Charles's Law

Thermal Equilibrium and Zeroth Law

Two systems in thermal equilibrium with a third system are in thermal equilibrium with each other.
This empirical fact justifies the use of thermometers to measure temperature.
Temperature is a property that determines whether two bodies are in thermal equilibrium.

Celsius and Fahrenheit Scales

The Celsius scale defines the ice point (melting point of pure ice at 1 atm) as 0°C and the steam point (boiling point of water at 1 atm) as 100°C.
The Fahrenheit scale defines the ice point as 32°F and the steam point as 212°F, divided into 180 equal intervals.
The linear relation allows direct conversion: (C/100) = (F - 32)/180.

Kelvin Absolute Temperature Scale

The absolute or thermodynamic temperature scale is independent of the working substance.
The zero of the Kelvin scale is absolute zero, the temperature at which the entropy of a system reaches a minimum value.
The triple point of water is assigned an exact value of 273.16 K, defining the kelvin as 1/273.16 of the thermodynamic temperature of the triple point of water.

Constant Volume Gas Thermometer

Relies on the variation of pressure of a fixed mass of gas at constant volume with temperature, following $P = P_0 (1 + \gamma_v t)$.
Provides an ideal gas temperature scale where $T = \lim_{P→0} [273.16 \times (P/P_{tr})]$.
Serves as a standard reference for calibrating secondary thermometers.

Derivations

Derivation of General Temperature Conversion Equation

Conditions: Assuming a linear response of thermometric property X with temperature T across two fixed reference points (lower fixed point LFP and upper fixed point UFP).

Let X be the thermometric property at temperature T.
Assume linear variation: $T = A \times X + B$, where A and B are constants determined by reference points.
Substitute lower fixed point: LF$P = A \times X_{LFP} + B$, and upper fixed point: UF$P = A \times X_{UFP} + B$.
Subtract and solve for A: $A = (UFP - LFP) / (X_{UFP} - X_{LFP})$.
Equate the ratio of measured intervals: (T - LFP) / (UFP - LFP) = (X - X_{LFP}) / (X_{UFP} - X_{LFP}).

Result: frac{T - LFP}{UFP - LFP} = \frac{X - X_{LFP}}{X_{UFP} - X_{LFP}}

Conversion Formula between Celsius and Fahrenheit Scales

Conditions: Standard atmospheric pressure reference points.

Set LF$P_C = 0$, UF$P_C = 100$ for Celsius, and LF$P_F = 32$, UF$P_F = 212$ for Fahrenheit.
Apply the general scale conversion formula: \frac{C - 0}{100 - 0} = \frac{F - 32}{212 - 32}.
Simplify the denominators: \frac{C}{100} = \frac{F - 32}{180}.
Divide denominators by 20 to obtain the standard reduction.

Result: \frac{C}{5} = \frac{F - 32}{9}

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Celsius to Fahrenheit Conversion	$$\frac{C}{5} = \frac{F - 32}{9}$$	C = Celsius temperature, F = Fahrenheit temperature Degree Celsius (°C), Degree Fahrenheit (°F)	Valid for standard linear temperature ranges
Celsius to Kelvin Conversion	$$T = C + 273.15$$	T = Kelvin temperature, C = Celsius temperature Kelvin (K), Degree Celsius (°C)	Exact SI relation
Fahrenheit to Kelvin Conversion	$$T = \frac{5}{9}(F - 32) + 273.15$$	T = Kelvin temperature, F = Fahrenheit temperature Kelvin (K), Degree Fahrenheit (°F)	Standard conversion
General Scale Conversion Formula	$$\frac{X - X_{ice}}{X_{steam} - X_{ice}} = \frac{Y - Y_{ice}}{Y_{steam} - Y_{ice}}$$	X, Y = Thermometric property values on two different scales Arbitrary units	Assumes linear interpolation
Constant Volume Gas Thermometer Equation	$$T = 273.16 \times \left( \frac{P}{P_{tr}} \right)$$	P = Pressure at temperature T, $P_{tr} = Pressure at triple point of water$ Pascal (Pa), Kelvin (K)	Limit of zero gas density (P → 0)
Platinum Resistance Thermometer Equation	$$t = \frac{R_t - R_0}{R_{100} - R_0} \times 100$$	$R_t = Resistance at temperature t$, $R_0 = Resistance at 0°C$, $R_{100} = Resistance at 100°C$ Ohm (Ω), Degree Celsius (°C)	Quadratic temperature coefficient ignored for primary linear approximation
Thermocouple Temperature Relation	$$E = a t + \frac{1}{2} b t^2$$	E = Thermoelectric EMF, t = Temperature, a, b = Constants Volt (V), Degree Celsius (°C)	Cold junction kept at 0°C
Scale Reading Equivalence Temperature	$$C = F implies C = -40$$	C = Celsius reading, F = Fahrenheit reading Degree	At -40 degrees, Celsius and Fahrenheit scales read identical numerical values

Graphs and experiments

Celsius vs. Fahrenheit Calibration Line

Draw: A linear graph with Celsius (C) on the X-axis and Fahrenheit (F) on the Y-axis, showing intercepts at (0, 32) and (-40, -40).

Interpret: The straight line confirms linear proportionality with slope 9/5 and Y-intercept 32.

Exam use: Directly tested in numericals asking for the temperature where two scales match.

Pressure vs. Temperature (Constant Volume Gas Thermometer)

Draw: Straight lines of Pressure (P) versus Celsius temperature (t) for different initial amounts of gas, extrapolated backwards to intersect the temperature axis at -273.15°C.

Interpret: All gases extrapolate to zero pressure at the same temperature, defining absolute zero (-273.15°C).

Exam use: Conceptual questions on absolute zero and independence of gas species at low pressures.

Thermometric Property vs. Temperature Calibration Curve

Draw: Comparison between ideal gas temperature and mercury-in-glass thermometer temperature showing slight divergence between 0°C and 100°C.

Interpret: Different thermometric substances exhibit slight non-linear deviations because their physical properties do not vary in exact unison with thermodynamic temperature.

Exam use: PGT-level questions on calibration discrepancies of real thermometers.

Solved numericals

TGT core

At what temperature do the Celsius and Fahrenheit scales read the same numerical value?

Let the reading on both scales be x. So, C = F = x.
Use the conversion formula: \frac{C}{5} = \frac{F - 32}{9}.
Substitute x for C and F: \frac{x}{5} = \frac{x - 32}{9}.
Cross-multiply: 9$x = 5(x - 32) implies 9x = 5x - 160$.
Solve for x: 4$x = -160 implies x = -40$.

Answer: -40°

TGT core

A faulty thermometer has its lower fixed point marked as 5°C and upper fixed point as 95°C. What is the correct temperature on the Celsius scale if this thermometer reads 41°C?

Apply the general calibration formula: \frac{T - LFP_{faulty}}{UFP_{faulty} - LFP_{faulty}} = \frac{C - LFP_{Celsius}}{UFP_{Celsius} - LFP_{Celsius}}.
Substitute the values: \frac{41 - 5}{95 - 5} = \frac{C - 0}{100 - 0}.
Simplify the fraction: \frac{36}{90} = \frac{C}{100}.
Solve for C: $C = \frac{36}{90} \times 100 = 0.4 \times 100 = 40°C$.

Answer: 40°C

PGT extension

The pressure of a gas in a constant volume gas thermometer is 80 cm of Hg at 0°C, 110 cm of Hg at 100°C, and 96 cm of Hg in a liquid bath. Calculate the temperature of the bath.

Assume linear relation between pressure and temperature: $t = \frac{P_t - P_0}{P_{100} - P_0} \times 100°C$.
Substitute given values: $P_t = 96$, $P_0 = 80$, $P_{100} = 110$.
Calculate numerator: P_t - $P_0 = 96 - 80 = 16$.
Calculate denominator: P_{100} - $P_0 = 110 - 80 = 30$.
Evaluate t: $t = \frac{16}{30} \times 100 = \frac{160}{3} = 53.33°C$.

Answer: 53.33°C

Assumptions and limitations

Linear interpolation assumes thermometric properties vary strictly linearly with temperature, which introduces slight calibration errors in real materials.
Real gases deviate from ideal behavior at high pressures, requiring extrapolation to zero pressure (P → 0) for absolute gas thermometry.

Common misconceptions

Common error: Absolute zero is the temperature where molecular motion completely stops according to classical physics.

Correct: Molecular motion does not stop at absolute zero due to zero-point energy mandated by quantum mechanics.

Heisenberg's Uncertainty Principle prevents zero momentum and position variance simultaneously.

Common error: All thermometers filled with different liquids give identical readings at all intermediate temperatures.

Correct: Different liquids (e.g., mercury and alcohol) expand non-linearly at different rates, causing minor discrepancies between fixed reference points.

Coefficient of cubical expansion is not strictly constant across wide temperature ranges.

Quick revision

Conversion bridge: C/$5 = (F - 32)/9 = (K - 273.15)/5 = (R)/4$.
Celsius to Fahrenheit Conversion: \frac{C}{5} = \frac{F - 32}{9}
Celsius to Kelvin Conversion: $T = C + 273.15$

Must remember

The Zeroth Law of Thermodynamics forms the logical foundation for the concept of temperature and the use of thermometers.
The relation between Celsius and Fahrenheit scales is given by $\frac{C}{5} = \frac{F - 32}{9}$.
At $-40^{\circ}$, the Celsius and Fahrenheit scales show the exact same numerical reading ($C = F = -40$).
The absolute or thermodynamic Kelvin scale defines the triple point of water precisely as $273.16\text{ K}$.
Constant volume gas thermometers define temperature through the limit of zero gas density, $T = \lim_{P\to0} [273.16 \times (P/P_{tr})]$.

Glossary

Zeroth Law of ThermodynamicsStates that if two systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.

Triple Point of WaterThe unique temperature and pressure ($273.16\text{ K}$ and $0.006\text{ atm}$) at which ice, liquid water, and water vapor coexist in equilibrium.

Absolute ZeroThe zero of the Kelvin scale where entropy reaches a minimum value and quantum zero-point motion persists.

Constant Volume Gas ThermometerA primary thermometer measuring temperature via pressure variations of a fixed mass of gas kept at constant volume.

Comparison of Temperature Scales

Scale	Ice Point	Steam Point	Number of Divisions
Celsius	$0^{\circ}\text{C}$	$100^{\circ}\text{C}$	100
Fahrenheit	$32^{\circ}\text{F}$	$212^{\circ}\text{F}$	180
Kelvin	$273.15\text{ K}$	$373.15\text{ K}$	100

Practice Quiz

Choose an option to reveal the explanation.

1. Which law of thermodynamics justifies the use of thermometers to measure temperature?

Explanation: The Zeroth Law states that thermal equilibrium is transitive, which is the foundational principle allowing a thermometer to measure temperature.

2. At what temperature do the Celsius and Fahrenheit temperature scales read the same numerical value?

Explanation: Setting $C = F$ in the relation $\frac{C}{5} = \frac{C - 32}{9}$ yields $9C = 5C - 160$, resulting in $C = -40$.

3. What is the exact assigned temperature of the triple point of water on the Kelvin scale?

Explanation: The triple point of water is assigned an exact value of $273.16\text{ K}$ in the definition of the kelvin.

4. Which of the following equations correctly relates Celsius ($C$) and Fahrenheit ($F$) temperatures?

Explanation: The standard linear conversion formula is $\frac{C}{100} = \frac{F - 32}{180}$, which simplifies to $\frac{C}{5} = \frac{F - 32}{9}$.

5. In a constant volume gas thermometer, the temperature is determined by measuring the variation of which property?

Explanation: A constant volume gas thermometer relies on the variation of pressure of a fixed mass of gas at constant volume.

6. Why do different liquid-in-glass thermometers (e.g., mercury and alcohol) show slight disagreements at intermediate temperatures?

Explanation: Different liquids expand non-linearly at different rates, causing minor discrepancies between fixed reference points.

7. What is the value of absolute zero on the Celsius scale approximately?

Explanation: Absolute zero corresponds to $0\text{ K}$, which is approximately $-273.15^{\circ}\text{C}$.

8. What happens to molecular motion at absolute zero according to quantum mechanics?

Explanation: Due to Heisenberg's Uncertainty Principle, zero-point energy prevents molecular motion from completely stopping.

9. Which temperature scale is completely independent of the working substance used in the thermometer?

Explanation: The absolute or thermodynamic temperature scale is independent of the working substance.

10. What is the formula for a platinum resistance thermometer to find temperature $t$?

Explanation: The linear approximation for a platinum resistance thermometer is $t = \frac{R_t - R_0}{R_{100} - R_0} \times 100$.

11. In thermocouple thermometry, what is the standard temperature usually maintained at the cold junction?

Explanation: Thermocouple temperature relations typically assume the cold junction is kept at $0^{\circ}\text{C}$.

12. How is the ideal gas temperature scale defined using a constant volume gas thermometer at very low pressures?

Explanation: The ideal gas temperature scale is defined by taking the limit as pressure approaches zero: $T = \lim_{P\to0} [273.16 \times (P/P_{tr})]$.

13. What does the symbol $P_{tr}$ represent in gas thermometry equations?

Explanation: $P_{tr}$ denotes the pressure of the gas measured at the triple point of water.

14. If a faulty thermometer reads ice point as $5^{\circ}\text{C}$ and steam point as $95^{\circ}\text{C}$, what is the correct Celsius temperature when it reads $41^{\circ}\text{C}$?

Explanation: Using linear interpolation $\frac{X - X_{ice}}{X_{steam} - X_{ice}} = \frac{C - 0}{100 - 0}$, we get $\frac{41 - 5}{95 - 5} = \frac{C}{100} \implies \frac{36}{90} = \frac{C}{100} \implies C = 40^{\circ}\text{C}$.

15. What is the primary condition for two bodies to be in thermal equilibrium according to the Zeroth Law?

Explanation: Temperature is precisely the property that determines whether two bodies are in thermal equilibrium.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The reading of a centigrade (Celsius) thermometer coincides with that of a Fahrenheit thermometer at:

Explanation: Solving $\frac{C}{5} = \frac{C - 32}{9}$ yields $C = -40$.

2. The triple point of water is a standard fixed point in thermometry and has a numerical value of:

Explanation: By definition, the triple point of water is assigned an exact value of $273.16\text{ K}$.

3. Two systems $A$ and $B$ are in thermal equilibrium with a third system $C$. By which law are $A$ and $B$ in thermal equilibrium with each other?

Explanation: This statement is the exact definition of the Zeroth Law of Thermodynamics.

4. On a new scale of temperature (say $X$), the ice point is $0^{\circ}X$ and the steam point is $150^{\circ}X$. What is the temperature on the Celsius scale corresponding to $60^{\circ}X$?

Explanation: Using linear interpolation $\frac{X - 0}{150 - 0} = \frac{C - 0}{100 - 0} \implies \frac{60}{150} = \frac{C}{100} \implies C = \frac{2}{3} \times 100 = 40^{\circ}\text{C}$.

5. In a constant volume gas thermometer, the pressure of the gas at the ice point is $1.5 \times 10^4\text{ Pa}$ and at the steam point is $2.0 \times 10^4\text{ Pa}$. What is the temperature when the pressure is $1.8 \times 10^4\text{ Pa}$?

Explanation: Using linear interpolation: $t = \frac{P_t - P_0}{P_{100} - P_0} \times 100 = \frac{1.8 \times 10^4 - 1.5 \times 10^4}{2.0 \times 10^4 - 1.5 \times 10^4} \times 100 = \frac{0.3}{0.5} \times 100 = 60^{\circ}\text{C}$.

6. Which of the following statements about absolute zero ($0\text{ K}$) is correct?

Explanation: Absolute zero is defined as the temperature at which the entropy of a system reaches a minimum value.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. If temperature is increased by $20^{\circ}\text{C}$, what is the corresponding increase on the Fahrenheit scale?

Explanation: Since $100^{\circ}\text{C} = 180^{\circ}\text{F}$, a $1^{\circ}\text{C}$ change equals a $\frac{9}{5}^{\circ}\text{F}$ change. Therefore, a $20^{\circ}\text{C}$ increase equals $20 \times \frac{9}{5} = 36^{\circ}\text{F}$.

2. A platinum resistance thermometer has a resistance of $2\ \Omega$ at $0^{\circ}\text{C}$ and $6\ \Omega$ at $100^{\circ}\text{C}$. When immersed in a liquid, its resistance is $5\ \Omega$. What is the temperature of the liquid?

Explanation: Using $t = \frac{R_t - R_0}{R_{100} - R_0} \times 100 = \frac{5 - 2}{6 - 2} \times 100 = \frac{3}{4} \times 100 = 75^{\circ}\text{C}$.

3. What is the physical significance of the Zeroth Law of Thermodynamics?

Explanation: The Zeroth Law establishes the existence of temperature as a property determining thermal equilibrium, justifying thermometry.

4. In a thermocouple, the electromotive force (EMF) is given by $E = at + \frac{1}{2}bt^2$. If $a = 10\ \mu\text{V}/^{\circ}\text{C}$ and $b = -0.1\ \mu\text{V}/^{\circ}\text{C}^2$, what is the EMF at $t = 100^{\circ}\text{C}$?

Explanation: $E = (10)(100) + \frac{1}{2}(-0.1)(100)^2 = 1000 - 0.05(10000) = 1000 - 500 = 500\ \mu\text{V}$.

5. Which of the following thermometric properties is utilized in a constant volume gas thermometer?

Explanation: A constant volume gas thermometer relies specifically on the variation of pressure of a fixed mass of gas at constant volume.

6. If a faulty thermometer reads the lower fixed point as $-10^{\circ}\text{C}$ and upper fixed point as $110^{\circ}\text{C}$, what is the true temperature when this thermometer reads $60^{\circ}\text{C}$?

Explanation: Using linear interpolation: $\frac{60 - (-10)}{110 - (-10)} = \frac{C}{100} \implies \frac{70}{120} = \frac{C}{100} \implies C = \frac{7}{12} \times 100 = 58.33^{\circ}\text{C}$.

7. Why is the constant volume gas thermometer considered a standard reference for calibrating other thermometers?

Explanation: In the limit of zero density ($P \to 0$), real gases behave ideally, making the constant volume gas thermometer an independent standard reference.

8. What is the conversion relation between Fahrenheit ($F$) and Kelvin ($T$) temperature?

Explanation: Combining Celsius to Fahrenheit and Celsius to Kelvin yields $T = \frac{5}{9}(F - 32) + 273.15$.

9. Which scale assigns $0^{\circ}$ to the ice point and $80^{\circ}$ to the steam point?

Explanation: The Reaumur scale defines the ice point as $0^{\circ}\text{R}$ and the steam point as $80^{\circ}\text{R}$.

10. What is the primary reason why Heisenberg's Uncertainty Principle prevents molecular motion from completely halting at absolute zero?

Explanation: Heisenberg's Uncertainty Principle ($\Delta x \Delta p \ge \hbar / 2$) prevents both position and momentum from simultaneously having zero uncertainty, meaning zero-point kinetic energy must remain.