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Thermal Physics
Absolute Temperature and the Kelvin Scale
Absolute temperature is a thermodynamic measure of temperature scaled from absolute zero, the theoretical state where particles possess minimum thermal motion and zero entropy. It is fundamentally defined via the second law of thermodynamics using Carnot engines, making it independent of the working substance.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of absolute zero, conversion between Celsius, Fahrenheit, and Kelvin temperature scales, triple point of water, and basic gas law relationships based on absolute temperature.
PGT extension
Thermodynamic temperature scale definition via Carnot efficiency, Kelvin's work function, limitations at absolute zero (Third Law of Thermodynamics), and kinetic theory interpretation of absolute temperature.
Prerequisites
Zeroth and First Laws of Thermodynamics
Ideal Gas Equation and Kinetic Theory of Gases
Carnot Theorem and Reversible Engines
Definition and Thermodynamic Basis
Absolute temperature is measured on the Kelvin scale, where 0 K represents absolute zero.
It is independent of the thermometric properties of any particular substance, unlike mercury-in-glass or constant-volume gas thermometers.
The thermodynamic temperature scale is established using the efficiencies of reversible Carnot engines operating between two heat reservoirs.
Ideal Gas Scale vs Thermodynamic Scale
In the limit of zero pressure, an ideal gas thermometer defines an absolute temperature scale that is identical to the thermodynamic Kelvin scale.
The pressure of a given mass of gas at constant volume extrapolates to zero at absolute zero temperature (-273.15 deg C).
Real gases deviate from this ideal behavior at high pressures and low temperatures due to intermolecular forces.
Triple Point of Water Reference
Since 1954 and reaffirmed in the 2019 SI redefinition, the Kelvin is defined by fixing the Boltzmann constant k to exactly 1.380649 x 10-23 J K-1.
Historically and practically, it is anchored to the triple point of water, where ice, liquid water, and water vapor coexist in equilibrium.
The temperature of the triple point of water is defined as exactly 273.16 K.
Kinetic Interpretation
According to the kinetic theory of gases, the absolute temperature of an ideal gas is directly proportional to the average translational kinetic energy per molecule.
Equipartition of energy states that each active degree of freedom contributes (1/2)kBT to the internal energy per molecule.
At absolute zero, molecular motion ceases entirely, though quantum zero-point energy remains in harmonic oscillators.
Derivations
Thermodynamic Temperature Scale from Carnot Efficiency
Conditions: Reversible Carnot engine operating between temperatures T_1 and T_2
The efficiency eta of any reversible engine operating between two heat reservoirs at empirical temperatures theta_1 and theta_2 depends only on these temperatures: et$a = 1 - Q_2/Q_1 = f(\theta_1$, theta_2).
Consider three engines operating with heat exchanges Q_1, Q_2, and Q_3 at temperatures theta_1, theta_2, and theta_3 such that $Q_2/Q_1 = f(\theta_1$, theta_2), $Q_3/Q_2 = f(\theta_2$, theta_3), and $Q_3/Q_1 = f(\theta_1$, theta_3).
Since $Q_3/Q_1 = (Q_3/Q_2) x (Q_2/Q_1)$, it follows that f(theta_1, thet$a_3) = f(\theta_1$, theta_2) x f(theta_2, theta_3).
This functional equation implies that the ratio of heats can be written as the ratio of a universal function of temperature alone: $Q_2/Q_1 = T_2/T_1$.
Defining T as this absolute temperature scale gives et$a = 1 - T_2/T_1$.
Result: frac{T_2}{$T_1 = \frac{Q_2}{Q_1}$
Relation between Absolute Temperature and Mean Kinetic Energy
Conditions: Ideal gas containing N molecules
The pressure of an ideal gas from kinetic theory is given by $P = (1/3)(Nm/V)v_{rms}^2$.
Using the ideal gas law P$V = N k_B T$, equate the two expressions: (1/3)Nm v_{rms}^2 = N k_B T$.
Rearrange to solve for the total translational kinetic energy $E_k = (3/2)N k_B T$.
Divide by N to find the average translational kinetic energy per molecule: <$E_k> = (3/2)k_B T$.
Result: $T = \frac{2}{3k_B} \langle E_k \rangle$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Celsius to Kelvin Conversion
$$T = t + 273.15$$
T = Absolute temperature (K), t = Celsius temperature (°C) K, °C
Standard barometric scale
Fahrenheit to Absolute Temperature
$$T = \frac{5}{9}(F - 32) + 273.15$$
T = Kelvin temperature, F = Fahrenheit temperature K, °F
Standard temperature conversion
Carnot Efficiency using Absolute Temperatures
$$\eta = 1 - \frac{T_C}{T_H}$$
eta = Efficiency, $T_C = Sink temperature$, $T_H = Source temperature$ Dimensionless (fractions or percentage)
Reversible Carnot cycle
Average Translational Kinetic Energy per Molecule
$$\langle E \rangle = \frac{3}{2}k_B T$$
$k_B = Boltzmann constant$, T = Absolute temperature J
Monatomic ideal gas
Root Mean Square Velocity
$$v_{rms} = \sqrt{\frac{3RT}{M}}$$
R = Universal gas constant, T = Absolute temperature, M = Molar mass m s^-1
Ideal gas behavior
Constant Volume Gas Thermometer Relation
$$T = 273.16 \left(\frac{P}{P_{tp}}\right)$$
P = Pressure at temperature T, $P_{tp} = Pressure at water triple point$ K, Pa
In the limit of vanishing gas pressure (P -> 0)
Energy per Degree of Freedom
$$E_{DOF} = \frac{1}{2}k_B T$$
$E_{DOF} = Energy associated with a single quadratic degree of freedom$ J
Law of equipartition of energy in thermal equilibrium
Internal Energy of an Ideal Gas
$$U = \frac{f}{2} n R T$$
f = Degrees of freedom, n = Number of moles, R = Gas constant, T = Temperature J
Ideal gas
Graphs and experiments
Pressure vs Celsius Temperature for Real and Ideal Gases
Draw: A set of straight lines for different gases on a P vs t (°C) plot, all converging to a single intercept on the temperature axis at -273.15 °C when extrapolated.
Interpret: Demonstrates that pressure vanishes at absolute zero for all dilute gases, establishing the universal absolute zero reference point.
Exam use: Frequently tested in TGT/PGT to check understanding of temperature axis intercept extrapolation in gas thermometry.
Constant-Volume Gas Thermometer Experiment
Draw: Manometer height difference versus temperature bath calibration curve showing linear response across ice and steam points.
Interpret: Shows experimental realization of absolute temperature scale via gas pressure proportionality.
Exam use: Conceptual questions on corrections needed for real gas behavior at high densities.
Carnot Efficiency vs Source Temperature
Draw: A hyperbolic-type curve plotting efficiency eta against T_H for a fixed sink temperature T_C, asymptotically approaching 1 as T_H approaches infinity.
Interpret: Illustrates the second law limitation that 100% efficiency requires an infinitely hot source or absolute zero sink.
Exam use: Used in numerical problems assessing changes in heat engine efficiencies.
Solved numericals
TGT core
At what temperature is the Fahrenheit scale reading equal to twice the Celsius scale reading?
Let the Celsius temperature be t. Then the Fahrenheit temperature is F = 2t.
Use the conversion formula: (F - 32)/$9 = t/5$.
Substitute $F = 2t into the equation: (2t - 32)/9 = t/5$.
Cross-multiply and solve: 5(2t - 32) = 9$t => 10t - 160 = 9t => t = 160 °C$.
Verify Fahrenheit reading: F = 2(160) = 320°F. Check: (320 - 32)/$9 = 288/9 = 32$, and 160/5 = 32. Both match.
Answer: 160 °C (or 433.15 K)
PGT extension
Calculate the root mean square velocity of oxygen molecules ($M = 32 g/mol) at an absolute temperature of 300 K$. Given $R = 8.314 J mol^-1 K^-1$.
Identify given parameters: T = 300 K, $M = 32 g/mol = 32 x 10^-3 kg/mol$, $R = 8.314 J mol^-1 K^-1$.
Write the formula for root mean square velocity: $v_{rms} = \sqrt(3RT / M)$.
Substitute values: $v_{rms} = \sqrt((3 x 8.314 x 300) / (32 x 10^-3))$.
Calculate numerator: 3 x 8.314 x 300 = 7482.6.
Divide by molar mass: 7482.6 / 0.032 = 233831.25.
Take square root: sqrt(233831.25) approx 483.56 m/s.
Answer: 483.56 m s^-1
PGT extension
A Carnot engine absorbs 1200 kJ of heat from a reservoir at 500 K and rejects heat to a sink at 300 K. Calculate the work done by the engine and the entropy change of the universe during one cycle.
Absolute zero is defined as $0\text{ K}$, which corresponds to $-273.15^{\circ}\text{C}$.
The thermodynamic Kelvin scale is independent of the thermometric properties of any particular substance.
The triple point of water is defined as exactly $273.16\text{ K}$ where ice, liquid water, and water vapor coexist.
According to kinetic theory, the average translational kinetic energy per molecule is given by $\langle E \rangle = \frac{3}{2}k_B T$.
In the 2019 SI redefinition, the Kelvin is fixed by assigning an exact numerical value to the Boltzmann constant $k_B$.
Glossary
Absolute TemperatureTemperature measured on the Kelvin scale starting from absolute zero ($0\text{ K}$).
Triple Point of WaterThe unique temperature ($273.16\text{ K}$) and pressure at which ice, liquid water, and water vapor are in equilibrium.
Carnot EfficiencyThe maximum theoretical efficiency of a heat engine operating between two absolute temperatures $T_C$ and $T_H$, given by $\eta = 1 - \frac{T_C}{T_H}$.
Zero-point EnergyThe lowest possible energy that a quantum mechanical physical system may possess, remaining present even at absolute zero ($0\text{ K}$).
Ideal Gas Scale vs Thermodynamic Scale
Feature
Ideal Gas Scale
Thermodynamic Kelvin Scale
Substance Dependence
Depends on dilute gas behavior
Completely independent of working substance
Theoretical Basis
Extrapolation of gas pressure to zero pressure ($P \to 0$)
Efficiencies of reversible Carnot engines between heat reservoirs
High Pressure / Low Temp
Deviates due to intermolecular forces
Universally valid across all phases and systems
Practice Quiz
Choose an option to reveal the explanation.
1. What is the temperature of the triple point of water on the Kelvin scale?
Explanation: By definition, the triple point of water is set at exactly $273.16\text{ K}$.
2. Which of the following statements about absolute zero ($0\text{ K}$) is scientifically accurate?
Explanation: At absolute zero, classical translational kinetic energy vanishes, but harmonic oscillators retain quantum zero-point energy due to Heisenberg's Uncertainty Principle.
3. The thermodynamic temperature scale is fundamentally established using:
Explanation: The thermodynamic temperature scale is defined independently of working substances via reversible Carnot engine efficiencies.
4. What is the conversion formula between Celsius temperature ($t$) and Kelvin absolute temperature ($T$)?
Explanation: The standard conversion relation is $T = t + 273.15$.
5. According to the kinetic theory of gases, the average translational kinetic energy per molecule of a monatomic ideal gas is:
Explanation: The average translational kinetic energy per molecule is $\frac{3}{2}k_B T$ (while $\frac{3}{2}RT$ is per mole).
6. Real gases deviate from ideal gas absolute temperature behavior at high pressures and low temperatures due to:
Explanation: Real gases deviate from ideality at high pressures and low temperatures because intermolecular attractive/repulsive forces and finite molecular volumes become significant.
7. What is the root mean square velocity ($v_{rms}$) formula for an ideal gas of molar mass $M$ at absolute temperature $T$?
Explanation: The standard expression derived from kinetic theory is $v_{rms} = \sqrt{\frac{3RT}{M}}$.
8. In a constant volume gas thermometer, the absolute temperature $T$ in the limit of vanishing pressure is given by:
Explanation: The relation is $T = 273.16 \left(\frac{P}{P_{tp}}\right)$ in the limit as $P \to 0$.
9. How is the Kelvin defined in the 2019 SI system redefinition?
Explanation: Since the 2019 SI redefinition, the Kelvin is defined by fixing the numerical value of the Boltzmann constant $k_B$ to $1.380649 \times 10^{-23}\text{ J K}^{-1}$.
10. According to the law of equipartition of energy, how much energy is associated with each quadratic degree of freedom per molecule in thermal equilibrium?
Explanation: Each active quadratic degree of freedom contributes an average energy of $\frac{1}{2}k_B T$ per molecule.
11. Convert a Fahrenheit temperature of $98.6^{\circ}\text{F}$ to absolute temperature in Kelvin.
13. What is the internal energy of an ideal gas with $f$ degrees of freedom containing $n$ moles at absolute temperature $T$?
Explanation: The total internal energy is given by $U = \frac{f}{2}nRT$.
14. Which temperature scale property distinguishes the thermodynamic scale from mercury-in-glass thermometer scales?
Explanation: The thermodynamic temperature scale is universal and independent of any particular thermometric substance.
15. If the absolute temperature of an ideal gas is doubled, what happens to the root mean square velocity $v_{rms}$ of its molecules?
Explanation: Since $v_{rms} \propto \sqrt{T}$, doubling $T$ increases $v_{rms}$ by a factor of $\sqrt{2}$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The pressure of a gas at constant volume is $200\text{ Pa}$ at the triple point of water. When immersed in a thermostat, the pressure becomes $250\text{ Pa}$. What is the temperature of the thermostat?
4. If the root mean square velocity of hydrogen molecules at a given temperature is $v$, what is the rms velocity of oxygen molecules at the same absolute temperature?
Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, $v_{rms} \propto \frac{1}{\sqrt{M}}$. The molar mass of $O_2$ is $32\text{ g/mol}$ and $H_2$ is $2\text{ g/mol}$. Thus, $\frac{v_{O_2}}{v_{H_2}} = \sqrt{\frac{2}{32}} = \sqrt{\frac{1}{16}} = \frac{1}{4}$.
5. What is the average translational kinetic energy of $2\text{ moles}$ of an ideal gas at temperature $300\text{ K}$? (Take $R = 8.314\text{ J mol}^{-1}\text{ K}^{-1}$)
Explanation: Total translational kinetic energy for $n$ moles is $E = \frac{3}{2}nRT = \frac{3}{2} \times 2 \times 8.314 \times 300 = 3 \times 8.314 \times 300 = 7482.6\text{ J}$? Wait: $3 \times 8.314 = 24.942$, $24.942 \times 300 = 7482.6$. Let's check options: Option 1 is $3741.3$ (which is half) or Option 2 is $7482.6$. Correct index is 1 ($7482.6\text{ J}$).
6. Which misconception about absolute zero is corrected by quantum mechanics?
Explanation: Classical physics assumed total cessation of motion and zero energy at absolute zero, but quantum mechanics proves zero-point energy persists.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. A centigrade thermometer and a Kelvin thermometer are used to measure the temperature of a system. If the Kelvin thermometer reads $300\text{ K}$, what does the centigrade thermometer read?
3. What is the internal energy of $1\text{ mole}$ of a rigid diatomic gas at absolute temperature $T$?
Explanation: A rigid diatomic gas has $5$ degrees of freedom ($3$ translational and $2$ rotational), so $U = \frac{5}{2}nRT = \frac{5}{2}RT$.
4. Two ideal gas thermometers $A$ and $B$ use different gases. As the reference pressure at the triple point of water goes to zero, what can be said about their temperature readings at any given thermal state?
Explanation: In the limit of vanishing pressure ($P \to 0$), all ideal gas thermometers yield identical absolute temperature scales matching the thermodynamic Kelvin scale.
5. If a Carnot engine operates between $400\text{ K}$ and $200\text{ K}$, what is its efficiency?
6. What is the ratio of root mean square speeds of oxygen ($O_2$) and helium ($He$) gases at the same absolute temperature? (Atomic mass of $He = 4$, Molar mass of $O_2 = 32$)
7. Which of the following physical quantities is directly proportional to the absolute temperature of an ideal gas?
Explanation: The average translational kinetic energy per mole is $\frac{3}{2}RT$, which is directly proportional to $T$.
8. The absolute zero temperature on the Fahrenheit scale is approximately:
Explanation: Using $T = \frac{5}{9}(F - 32) + 273.15 = 0 \implies \frac{5}{9}(F - 32) = -273.15 \implies F - 32 = -491.67 \implies F = -459.67^{\circ}\text{F}$.
9. Why can absolute zero ($0\text{ K}$) never be attained experimentally in a finite number of steps?
Explanation: The Third Law of Thermodynamics states that it is impossible by any procedure, no matter how idealized, to reduce any system to absolute zero in a finite number of operations.
10. In the relation $T = 273.16 \left(\frac{P}{P_{tp}}\right)$, the requirement $P \to 0$ is necessary because:
Explanation: Extrapolating to zero pressure eliminates the effects of intermolecular interactions, ensuring the gas behaves ideally and matches the thermodynamic temperature scale.