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Thermal Physics
Concept of Temperature and Zeroth Law of Thermodynamics
Temperature is a macroscopic scalar physical property that quantitatively determines the thermal state of a system and dictates the direction of spontaneous net heat transfer. It originates microscopically from the average translational kinetic energy of the constituent particles.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of temperature, thermal equilibrium, empirical temperature scales (Celsius, Fahrenheit, Kelvin), principle of thermometry, and simple conversion relations.
PGT extension
Zeroth Law of Thermodynamics, kinetic interpretation of temperature using degrees of freedom, ideal gas temperature scale, absolute zero, and thermodynamic temperature scale based on Carnot engines.
Prerequisites
Kinetic theory of gases and microscopic random motion
Degrees of freedom and the law of equipartition of energy
Ideal gas equation of state
Thermal Equilibrium and the Zeroth Law
Two systems are in thermal equilibrium if, when placed in thermal contact through a diathermal wall, their macroscopic properties do not change with time.
The Zeroth Law of Thermodynamics states that if two systems A and B are each in thermal equilibrium with a third system C, then they are also in thermal equilibrium with each other.
This law provides the logical foundation for the existence of temperature as a universal property shared by systems in thermal equilibrium.
Microscopic Definition of Temperature
According to the kinetic theory of gases, the absolute temperature T of an ideal gas is directly proportional to the average translational kinetic energy per molecule.
Expressed via the equipartition theorem, each translational degree of freedom contributes (1/2)kBT to the internal energy per molecule.
Temperature measures the intensity of thermal agitation rather than the total thermal energy contained within the system.
Empirical Temperature Scales
Temperature scales are constructed by choosing two reproducible fixed points, such as the ice point and the steam point of pure water at standard atmospheric pressure.
The Celsius scale defines the ice point as 0 degrees C and the steam point as 100 degrees C, divided into 100 equal intervals.
The Kelvin scale sets its zero at absolute zero (-273.15 degrees C) and uses the triple point of water (273.16 K) as its single defining fixed point.
Thermodynamic Temperature Scale
The thermodynamic (Kelvin) scale is independent of the specific thermometric substance or property used to measure it.
It is defined using the efficiencies of reversible heat engines operating between two thermal reservoirs according to Kelvin's formulation.
Absolute zero (0 K) represents the theoretical limit at which the entropy of a pure crystalline substance vanishes, per the Third Law of Thermodynamics.
Derivations
Kinetic Interpretation of Temperature
Conditions: Ideal gas containing N molecules in volume V at absolute temperature T
Start with the kinetic theory pressure relation for an ideal gas: $P = (1/3)(Nm/V)(v_rms)^2$
Substitute the ideal gas equation of state: P$V = N k_B T$, yielding (1/3)m($v_rms)^2 = k_B T$
Multiply both sides by (3/2) to isolate the average translational kinetic energy per molecule: (1/2)m($v_rms)^2 = (3/2)k_B T$
Result: $E_k = (3/2)k_B T$, proving that temperature is a direct measure of average molecular kinetic energy.
Conversion Relation Between Celsius and Fahrenheit Scales
Conditions: Standard linear thermometric scaling between ice point and steam point
Establish the linear mapping equation: (C - C_ice)/(C_steam - $C_ice) = (F - F_ice)/(F_steam - F_ice)$
Draw: Straight lines of P versus T for different constant volumes, intersecting the temperature axis at -273.15 C.
Interpret: Extrapolation of all isochores to zero pressure yields a common temperature intercept of absolute zero (-273.15 C), establishing the absolute zero concept.
Exam use: Frequently tested in conceptual MCQs regarding absolute zero and gas behavior.
Constant Volume Gas Thermometer Calibration Curve
Draw: Plot of measured gas pressure P against standard temperature T, showing a linear dependence.
Interpret: Confirms Gay-Lussac's Law and shows that the extrapolated pressure vanishes at absolute zero Kelvin.
Exam use: Basis of primary thermometry questions in PGT examinations.
Triple Point Cell of Water
Draw: Schematic diagram showing solid ice, liquid water, and water vapor coexisting in equilibrium in a sealed glass cell.
Interpret: Defines the exact thermodynamic fixed point (273.16 K, 611.65 Pa) for the Kelvin scale independent of atmospheric fluctuations.
Exam use: Theoretical questions regarding modern definition of absolute temperature scale.
Solved numericals
TGT core
At what temperature do the numerical values of the Celsius and Fahrenheit scales coincide?
Let the reading on both scales be x. Set C = x and F = x in the conversion formula.
Use the relation: C/$5 = (F - 32)/9 -> x/5 = (x - 32)/9$
Cross multiply and solve for x: 9$x = 5(x - 32) -> 9x = 5x - 160 -> 4x = -160 -> x = -40$
Answer: -40 degrees
PGT extension
Calculate the root mean square velocity of oxygen molecules ($M = 32 g/mol) at a temperature of 27 degrees C$. Take $R = 8.314 J/(mol K)$.
Convert temperature to Kelvin: $T = 27 + 273.15 = 300.15 K (or 300 K$ for approximation).
Convert molar mass to SI units: $M = 32 x 10^-3 kg/mol$.
Apply the formula $v_rms = \sqrt(3RT / M) = \sqrt((3 x 8.314 x 300) / (32 x 10^-3))$.
A Carnot engine operates between reservoirs at 500 K and 300 K. If the source temperature is increased by 50 K while the sink temperature remains constant, find the percentage change in its efficiency.
Percentage change: ((eta_2 - eta_1) / eta_1) x $100 = ((0.4545 - 0.4) / 0.4) x 100 = (0.0545 / 0.4) x 100 = 13.63%$.
Answer: 13.63% increase
Assumptions and limitations
Temperature defined via the ideal gas equation is strictly valid only in the limit of zero pressure (ideal gas limit).
Real gases liquefy or solidify before reaching absolute zero, making absolute zero an asymptotic limit unattainable in finite steps (Third Law of Thermodynamics).
Common misconceptions
Common error: Temperature is a measure of the total heat energy contained in a body.
Correct: Temperature measures the average kinetic energy of molecular motion, not the total thermal energy.
A small cup of boiling water has a high temperature but less total thermal energy than a swimming pool filled with warm water.
Common error: Heat flows from a body with higher thermal energy to one with lower thermal energy.
Correct: Heat flows spontaneously from a body at a higher temperature to one at a lower temperature.
Direction of heat transfer is strictly governed by temperature gradients, independent of the relative mass or total internal energy.
Quick revision
Temperature dictates the direction of spontaneous thermal energy flow and is quantified using empirical and thermodynamic scales.
Average Translational Kinetic Energy: $E_{avg} = \frac{3}{2} k_B T$
Celsius to Kelvin Conversion: $T = t + 273.15$
Must remember
The Zeroth Law of Thermodynamics provides the logical foundation for the existence of temperature as a universal property shared by systems in thermal equilibrium.
The absolute temperature $T$ of an ideal gas is directly proportional to the average translational kinetic energy per molecule: $E_{avg} = \frac{3}{2} k_B T$.
The Kelvin scale sets its zero at absolute zero ($-273.15^{\circ}\text{C}$) and uses the triple point of water ($273.16\text{ K}$) as its single defining fixed point.
The thermodynamic temperature scale is independent of the specific thermometric substance or property used to measure it.
Heat flows spontaneously from a body at a higher temperature to one at a lower temperature, governed strictly by temperature gradients rather than total thermal energy.
Glossary
Diathermal WallA wall that allows thermal energy (heat) to pass freely between two thermodynamic systems.
Thermal EquilibriumA state in which two systems in thermal contact do not undergo any changes in macroscopic properties over time.
Absolute ZeroThe theoretical temperature limit ($0\text{ K}$ or $-273.15^{\circ}\text{C}$) at which the entropy of a pure crystalline substance vanishes.
Triple Point of WaterThe single temperature and pressure ($273.16\text{ K}$ and $0.01^{\circ}\text{C}$) at which ice, liquid water, and water vapor coexist in equilibrium.
Equipartition TheoremA theorem stating that in thermal equilibrium, energy is shared equally among all active degrees of freedom, each contributing $\frac{1}{2}k_BT$ per molecule.
Comparison of Temperature Scales
Parameter
Celsius Scale
Kelvin Scale
Ice Point
$0^{\circ}\text{C}$
$273.15\text{ K}$
Steam Point
$100^{\circ}\text{C}$
$373.15\text{ K}$
Defining Fixed Point
Ice and steam points at standard pressure
Triple point of water ($273.16\text{ K}$)
Absolute Zero
$-273.15^{\circ}\text{C}$
$0\text{ K}$
Practice Quiz
Choose an option to reveal the explanation.
1. Which law of thermodynamics forms the basis of temperature measurement?
Explanation: The Zeroth Law of Thermodynamics establishes that thermal equilibrium is an equivalence relation, providing the logical foundation for temperature measurement.
2. Two systems A and B are in thermal equilibrium with a third system C. According to the Zeroth Law, what is the relation between A and B?
Explanation: The Zeroth Law states that if two systems are each in thermal equilibrium with a third system, they are also in thermal equilibrium with each other.
3. What does temperature physically measure in a system according to kinetic theory?
Explanation: Temperature is a measure of the intensity of thermal agitation, specifically proportional to the average translational kinetic energy per molecule.
4. What is the average translational kinetic energy of a molecule in an ideal gas at absolute temperature $T$?
Explanation: By kinetic theory and equipartition across 3 translational degrees of freedom, the average translational kinetic energy is $\frac{3}{2} k_B T$.
5. On the Celsius scale, what are the standard fixed points used for calibration?
Explanation: The Celsius scale is constructed using the ice point ($0^{\circ}\text{C}$) and steam point ($100^{\circ}\text{C}$) of pure water at standard atmospheric pressure.
6. What is the single defining fixed point of the modern Kelvin thermodynamic temperature scale?
Explanation: The modern Kelvin scale uses the triple point of water ($273.16\text{ K}$) as its single defining fixed point.
7. Convert $27^{\circ}\text{C}$ to Kelvin.
Explanation: Using $T = t + 273.15$, $T = 27 + 273.15 = 300.15\text{ K}$.
8. What is the relation between Fahrenheit ($t_F$) and Celsius ($t_C$) temperature scales?
Explanation: The correct conversion formula is $t_F = \frac{9}{5}t_C + 32$.
9. At what temperature do the Celsius and Fahrenheit scales read the exact same numerical value?
10. What does absolute zero ($0\text{ K}$) signify regarding entropy according to the Third Law of Thermodynamics?
Explanation: According to the Third Law of Thermodynamics, absolute zero represents the theoretical limit at which the entropy of a pure crystalline substance vanishes.
11. Why is a small cup of boiling water unable to melt as much ice as a large pool of warm water, even though its temperature is higher?
Explanation: Temperature measures the intensity of thermal agitation (average kinetic energy), not the total thermal energy contained in the body.
12. What is the formula for the RMS velocity of gas molecules in terms of absolute temperature $T$?
Explanation: The root-mean-square velocity is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$, where $M$ is the molar mass.
13. How does resistance vary with temperature for a metallic conductor in the linear approximation?
Explanation: The resistance thermometry equation in the linear approximation is $R_t = R_0(1 + \alpha t)$.
14. What is the thermodynamic efficiency $\eta$ of a reversible Carnot engine operating between source $T_1$ and sink $T_2$?
Explanation: The efficiency of a Carnot engine is defined on the thermodynamic temperature scale as $\eta = 1 - \frac{T_2}{T_1}$.
15. Which of the following statements about heat flow is correct?
Explanation: Heat flows spontaneously from a body at a higher temperature to one at a lower temperature, governed strictly by temperature differences.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If two bodies are in thermal equilibrium with a third body, they are in thermal equilibrium with each other. This is known as:
Explanation: This statement is the direct definition and formulation of the Zeroth Law of Thermodynamics.
2. The temperature of an ideal gas is increased from $27^{\circ}\text{C}$ to $327^{\circ}\text{C}$. The root-mean-square velocity of its molecules becomes:
Explanation: $T_1 = 27 + 273 = 300\text{ K}$, $T_2 = 327 + 273 = 600\text{ K}$. Since $v_{rms} \propto \sqrt{T}$, the ratio is $\sqrt{600/300} = \sqrt{2}$, wait! Let's recheck: $327^{\circ}\text{C} = 600\text{ K}$, $27^{\circ}\text{C} = 300\text{ K}$. $\sqrt{600/300} = \sqrt{2}$? Ah, let's recalculate: $600/300 = 2$, so $\sqrt{2}$ times. Wait, let's make $T_2 = 927^{\circ}\text{C}$ ($1200\text{ K}$) for twice. Let's fix option or calculation: If $T_1 = 300\text{ K}$ and $T_2 = 1200\text{ K}$ ($927^{\circ}\text{C}$), then $v_{rms}$ is twice. Let's change question to $927^{\circ}\text{C}$ or keep $327^{\circ}\text{C}$ and correct answer to $\sqrt{2}$. Let's adjust question: If temperature increases from $0^{\circ}\text{C}$ ($273\text{ K}$) to $819^{\circ}\text{C}$ ($1092\text{ K}$), $1092/273 = 4$, $\sqrt{4} = 2$. Let's use: temperature increased from $27^{\circ}\text{C}$ to $927^{\circ}\text{C}$, $T_1 = 300\text{ K}$, $T_2 = 1200\text{ K}$, ratio is $\sqrt{1200/300} = 2$.
3. What is the kelvin temperature corresponding to $30^{\circ}\text{C}$?
4. The thermodynamic temperature scale is advantageous because:
Explanation: The thermodynamic Kelvin scale is independent of any specific thermometric substance or property.
5. According to the equipartition theorem, how much energy does each translational degree of freedom contribute per molecule?
Explanation: Each translational, rotational, or vibrational quadratic degree of freedom contributes $\frac{1}{2}k_BT$ per molecule.
6. If the absolute temperature of a gas is doubled, what happens to its root-mean-square velocity?
Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, doubling $T$ increases $v_{rms}$ by a factor of $\sqrt{2}$.
7. Two systems are separated by an adiabatic wall. Can they be in thermal equilibrium?
Explanation: Thermal equilibrium is a state condition; systems can be in thermal equilibrium even if isolated by adiabatic walls (as tested via a third system per Zeroth Law).
8. What is the value of the universal gas constant $R$ approximately equal to?
Explanation: The universal gas constant $R$ is approximately $8.314\text{ J/(mol}\cdot\text{K)}$. ($k_B$ is $1.38\times 10^{-23}\text{ J/K}$).
9. Which law states that the entropy of a pure crystalline substance approaches zero as temperature approaches absolute zero?
Explanation: The Third Law of Thermodynamics states that the entropy of a pure crystalline substance vanishes at absolute zero ($0\text{ K}$).ุ
10. In an ideal gas, the product of pressure and volume ($PV$) is related to temperature $T$ by:
Explanation: The ideal gas law relating pressure, volume, moles, and temperature is $PV = nRT$.