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Thermal Physics
Concept of Heat in Thermal Physics
Heat is the form of energy transferred between systems or objects due to a temperature difference. It is a path-dependent transient quantity measured in joules or calories.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of heat, units and conversion (calorie to joule), mechanical equivalent of heat, specific heat capacity, molar heat capacity, and calorimetry principle.
PGT extension
First law of thermodynamics, path dependence of heat, latent heat, phase transitions, microscopic interpretation via kinetic theory, and degrees of freedom.
Prerequisites
Concept of Temperature and Thermal Equilibrium
Kinetic Theory of Gases
Law of Equipartition of Energy
Nature and Definition of Heat
Heat is energy in transit across the boundary of a system due to a temperature gradient.
It is not a state function; rather, it is a path function depending on the thermodynamic path taken.
SI unit is the Joule (J), though the calorie (cal) is frequently used, where 1 cal = 4.186 J.
Mechanical Equivalent of Heat
Joule's experiment established that mechanical work (W) and heat (Q) are mutually convertible.
The relation is expressed as W = JQ, where J is the mechanical equivalent of heat.
J is a conversion factor equal to approximately 4.186 J/cal, not a physical quantity with independent dimensions.
Specific Heat Capacity and Molar Heat Capacity
Specific heat capacity (s) is the amount of heat required to raise the temperature of unit mass of a substance by one Kelvin or degree Celsius.
Molar heat capacity (C) refers to one mole of a substance undergoing the same temperature change.
For gases, specific heat is distinguished between constant pressure (C_p) and constant volume (C_v), linked by Mayer's relation C_p - $C_v = R$.
Latent Heat and Phase Change
Latent heat (L) is the heat absorbed or released per unit mass during a constant-temperature phase transition.
During phase change, the heat supplied does not increase kinetic energy or temperature; it alters intermolecular potential energy.
Expressed as Q = mL, where m is mass and L is latent heat of fusion or vaporization.
Derivations
Mayer's Relation between Principal Specific Heats
Conditions: Ideal gas undergoing quasi-static isobaric and isochoric processes
Consider 1 mole of an ideal gas heated at constant volume: d$Q_1 = C_v dT = dU$.
Consider heating the same gas at constant pressure: d$Q_2 = C_p dT = dU + dW$.
For an isobaric process, external work done is dW = P dV = R dT using the ideal gas equation PV = RT.
Substitute dU and dW into the constant pressure heat equation: C_p d$T = C_v dT + R dT$.
Result: C_p - $C_v = R$
Work Done and Heat Transferred in an Isothermal Expansion
Conditions: Reversible isothermal expansion of an ideal gas from volume V_1 to V_2
Write the first law of thermodynamics: d$Q = dU + dW$.
For an isothermal process in an ideal gas, temperature is constant, implying internal energy change dU = 0.
Integrate the work done equation: $W = \int_{V_1}^{V_2} P dV$.
Substitute P from the ideal gas law P = \frac{RT}{V} to get $W = RT \int_{V_1}^{V_2} \frac{dV}{V}$.
Result: $Q = W = RT \ln\left(\frac{V_2}{V_1}\right)$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Heat Capacity Equation
$$Q = ms\Delta T$$
Q = Heat added, m = Mass, s = Specific heat capacity, \Delta T = Temperature change J, kg, J kg^{-1} K^{-1}, K
No phase change occurring within temperature range
Molar Heat Capacity
$$Q = nC\Delta T$$
n = Number of moles, C = Molar heat capacity mol, J mol^{-1} K^{-1}
Constant volume or constant pressure conditions
Latent Heat Equation
$$Q = mL$$
m = Mass undergoing phase change, L = Latent heat kg, J kg^{-1}
Constant temperature phase transition
Mayer's Relation
$$C_p - C_v = R$$
$C_p = Molar heat capacity at constant pressure$, $C_v = Molar heat capacity at constant volume$, R = Universal gas constant J mol^{-1} K^{-1}
Ideal gas behavior
Ratio of Specific Heats
$$\gamma = \frac{C_p}{C_v} = 1 + \frac{2}{f}$$
\gamma = Adiabatic index, f = Degrees of freedom Dimensionless
Rigid or non-rigid gas molecules
Mechanical Equivalent of Heat
$$W = JQ$$
W = Mechanical work, J = Joule's constant, Q = Heat J, J cal^{-1}, cal
Macroscopic frictional or electrical heating systems
Principle of Calorimetry
$$Q_{lost} = Q_{gained}$$
Heat lost by hotter body equals heat gained by colder body J or cal
Heating Curve of Water (Temperature vs Heat Added)
Draw: A plot showing temperature on the y-axis and heat added on the x-axis, featuring sloped lines for temperature rise and flat horizontal plateaus during phase changes (melting and vaporization).
Interpret: Sloped regions represent single phases where Q = ms\Delta T. Flat plateaus indicate latent heat absorption (Q = mL) at constant temperature where phases coexist.
Exam use: Tested conceptually to evaluate understanding of specific heat versus latent heat.
Indicator Diagram (P-V Diagram for Isothermal and Adiabatic Processes)
Draw: P-V curves starting from the same initial state, where the adiabatic curve is steeper than the isothermal curve.
Interpret: Slope of isothermal curve is -P/V, while adiabatic slope is -\gamma P/V. Steeper adiabatic slope reflects temperature increase during compression without heat loss.
Exam use: Directly asked in PGT and TGT examinations regarding work and heat exchange characteristics.
Joule's Paddle Wheel Experiment
Draw: Schematic setup of falling weights turning a paddle wheel inside an insulated water container with a thermometer.
Interpret: Gravitational potential energy of falling weights is converted into mechanical work, raising the kinetic energy of water molecules and increasing temperature.
Exam use: Conceptual verification of the mechanical equivalent of heat (W = JQ).
Solved numericals
TGT core
Calculate the amount of heat required to raise the temperature of 2 kg of water from 20 °C to 80 °C. (Specific heat of wate$r = 4200 J kg^{-1} K^{-1})$
For a particular gas, the specific heat at constant pressure is $C_p = 0.24 cal g^{-1} K^{-1}$ and at constant volume is $C_v = 0.17 cal g^{-1} K^{-1}$. Find the molecular weight of the gas and its degree of freedom (assume $R = 2 cal mol^{-1} K^{-1})$.
Molar heat capacities: $C_P = M \times c_p$ and $C_V = M \times c_v$, where M is molecular weight.
Use Mayer's relation in molar terms: C_P - $C_V = R \implies M(c_p - c_v) = R$.
Substitute values: M(0.24 - 0.17) = 2 \implies M \times $0.07 = 2 \implies M = \frac{2}{0.07} = 28.57 g mol^{-1}$.
Calculate ratio $\gamma = \frac{c_p}{c_v} = \frac{0.24}{0.17} = 1.41$.
Relate \gamma to degrees of freedom f: $\gamma = 1 + \frac{2}{f} \implies 1.41 = 1 + \frac{2}{f} \implies f \approx 5$.
Answer: Molecular weight \approx 28.6 g mol^{-1}, Degrees of freedom f = 5
PGT extension
100 g of ice at 0 °C is mixed with 100 g of water at 100 °C. Find the final temperature of the mixture. (Latent heat of ice $L_f = 80 cal g^{-1}$, specific heat of wate$r = 1 cal g^{-1} K^{-1})$.
Calculate heat required by 100 g ice to melt completely at 0 °C: $Q_1 = m_i L_f = 100 \times 80 = 8000 cal$.
Calculate heat released by 100 g water cooling from 100 °C to 0 °C: $Q_2 = m_w s \Delta T = 100 \times 1 \times 100 = 10000 cal$.
Since Q_2 > Q_1, all ice melts completely and the excess heat raises the temperature of the resulting 200 g water above 0 °C.
Excess heat available after melting: $Q_{excess} = 10000 - 8000 = 2000 cal$.
Let final temperature be T. $Q_{excess} = (m_i + m_w) s T \implies 2000 = (100 + 100) \times 1 \times T \implies 2000 = 200 T \implies T = 10 °C$.
Answer: 10 °C
Assumptions and limitations
Specific heat capacities are assumed constant over small temperature intervals, though they vary with temperature at cryogenic regimes.
Calorimetry calculations assume perfect thermal isolation with zero heat exchange with the surrounding environment.
Mayer's relation assumes ideal gas behavior; deviations occur at high pressures and low temperatures (real gases).
Common misconceptions
Common error: Heat is a substance stored inside a body.
Correct: Heat is energy in transit across system boundaries due to temperature differences; a body stores internal energy, not heat.
Heat only exists during transfer; once inside, it contributes to internal kinetic and potential energy.
Common error: Temperature and heat are identical concepts.
Correct: Temperature is a state property measuring average kinetic energy, whereas heat is total energy transferred.
A small cup of boiling water has a higher temperature than a warm swimming pool, but the pool contains far more thermal energy (heat content).
Common error: Adding heat to a substance always increases its temperature.
Correct: During a phase transition, heat added does not change the temperature.
Latent heat is entirely utilized to alter intermolecular potential energy and change phase.
Quick revision
Heat is transient energy crossing a boundary driven by temperature difference, measured in Joules or calories.
Mechanical equivalent of heat $J = 4.186 J/cal bridges mechanical work$ and thermal energy.
Mayer's relation C_p - $C_v = R holds strictly$ for ideal gases.
Latent heat operates at constant temperature during phase transitions (Q = mL).
Must remember
Heat is energy in transit across the boundary of a system due to a temperature gradient and is a path function, not a state function.
The mechanical equivalent of heat $J$ is a conversion factor equal to approximately $4.186\text{ J/cal}$, not a physical quantity with independent dimensions ($W = JQ$).
Mayer's relation for ideal gases is given by $C_p - C_v = R$, linking molar heat capacities at constant pressure and constant volume.
Latent heat ($Q = mL$) represents heat absorbed or released during a constant-temperature phase transition, altering intermolecular potential energy rather than kinetic energy.
The principle of calorimetry is based on the conservation of energy in a thermally isolated system: $Q_{\text{lost}} = Q_{\text{gained}}$.
Glossary
Heat ($Q$) Energy in transit across a system boundary due to a temperature difference.
Specific Heat Capacity ($s$)Amount of heat required to raise the temperature of unit mass of a substance by one Kelvin or degree Celsius.
Molar Heat Capacity ($C$)Amount of heat required to raise the temperature of one mole of a substance by one Kelvin.
Mechanical Equivalent of Heat ($J$)Conversion factor relating mechanical work to thermal energy, $J \approx 4.186\text{ J/cal}$.
Latent Heat ($L$)Heat absorbed or released per unit mass during a constant-temperature phase transition.
Adiabatic Index ($\gamma$)Ratio of molar heat capacities $\frac{C_p}{C_v}$, which depends on the degrees of freedom of gas molecules.
Heat vs. Temperature
Parameter
Heat ($Q$)
Temperature ($T$)
Nature
Energy in transit (path function)
State function measuring average kinetic energy
Units
Joule (J) or Calorie (cal)
Kelvin (K) or Celsius ($^\circ\text{C}$)
Additivity
Extensive property (depends on mass/size)
Intensive property (independent of mass)
Specific Heat vs. Latent Heat
Parameter
Specific Heat ($s$)
Latent Heat ($L$)
Thermal Effect
Changes the temperature of a substance
Changes the phase at constant temperature
Formula
$Q = ms\Delta T$
$Q = mL$
Molecular Change
Alters average kinetic energy of molecules
Alters intermolecular potential energy
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following best describes heat in thermodynamics?
Explanation: Heat is strictly defined as energy in transit across the boundary of a system due to a temperature difference, making it a path function rather than a state function.
2. What is the SI unit of heat, and how does it relate to the calorie?
Explanation: The SI unit of heat is the Joule (J). The relation with the calorie is established as $1\text{ cal} = 4.186\text{ J}$.
3. In the relation $W = JQ$, what is the physical nature of $J$ (the mechanical equivalent of heat)?
Explanation: $J$ is a conversion factor equal to approximately $4.186\text{ J/cal}$ used to convert mechanical work units to thermal energy units; it does not possess independent fundamental dimensions.
4. Mayer's relation for an ideal gas is given by $C_p - C_v = R$. What do $C_p$ and $C_v$ represent?
Explanation: $C_p$ and $C_v$ represent molar heat capacities at constant pressure and constant volume respectively for one mole of an ideal gas.
5. What happens to the temperature of a substance during a phase change, such as melting or boiling?
Explanation: During a phase change, all the supplied heat (latent heat) is utilized to change the intermolecular potential energy rather than increasing molecular kinetic energy, keeping the temperature constant.
6. The equation $Q = ms\Delta T$ is valid under which of the following conditions?
Explanation: The formula $Q = ms\Delta T$ is used to calculate heat transfer when temperature changes without any accompanying phase transition.
7. Why is heat considered a path function rather than a state function?
Explanation: Heat depends on the intermediate steps and processes (the path) taken to go from one thermodynamic state to another, unlike state functions like internal energy.
8. What is the ratio of specific heats $\gamma = \frac{C_p}{C_v}$ for a monoatomic ideal gas with $3$ degrees of freedom?
Explanation: Using $\gamma = 1 + \frac{2}{f}$, for a monoatomic gas $f = 3$, so $\gamma = 1 + \frac{2}{3} = \frac{5}{3} \approx 1.67$.
9. Which misconception about heat and temperature is scientifically correct?
Explanation: Temperature is an intensive state property measuring average molecular kinetic energy, whereas heat is energy transferred across boundaries due to temperature differences.
10. The principle of calorimetry ($Q_{\text{lost}} = Q_{\text{gained}}$) is a direct consequence of which fundamental law?
Explanation: Calorimetry relies on the conservation of energy within a thermally isolated system, which is formulated by the First Law of Thermodynamics.
11. If $2\text{ kg}$ of a substance requires $10\text{ kJ}$ of heat to raise its temperature by $5^\circ\text{C}$ (with no phase change), what is its specific heat capacity $s$?
12. How does the molar heat capacity at constant pressure ($C_p$) compare to that at constant volume ($C_v$) for any ideal gas?
Explanation: According to Mayer's relation $C_p - C_v = R$, $C_p$ is greater than $C_v$ by the universal gas constant $R$ because work is done when heating at constant pressure.
13. What is latent heat of vaporization?
Explanation: Latent heat of vaporization is the thermal energy absorbed per unit mass when a substance changes phase from liquid to gas at its boiling point without temperature change.
14. What is the differential definition of heat capacity $C$ for an arbitrary thermodynamic process?
Explanation: The differential heat capacity is defined as $C = \frac{dQ}{dT}$, representing the infinitesimal heat input required for an infinitesimal temperature change.
15. Which of the following gases has the highest adiabatic index $\gamma$?
Explanation: For a monoatomic gas, $f=3$ and $\gamma = 1.67$. Diatomic gases have $\gamma = 1.40$, and polyatomic gases have even lower $\gamma$ values due to increased degrees of freedom.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The mechanical equivalent of heat $J$ establishes a relation between mechanical work $W$ and heat $Q$ as $W = JQ$. The value of $J$ is approximately:
Explanation: Joule's experiments showed that work and heat are mutually convertible with the conversion factor $J \approx 4.186\text{ J/cal}$.
2. For an ideal gas, the difference between molar heat capacity at constant pressure ($C_p$) and constant volume ($C_v$) is equal to:
Explanation: Mayer's relation states that $C_p - C_v = R$, where $R$ is the universal gas constant.
3. During the melting of ice at $0^\circ\text{C}$, heat supplied to the ice-water mixture is used to:
Explanation: Latent heat supplied during phase change is entirely utilized to change intermolecular potential energy, keeping the kinetic energy and temperature constant.
4. What is the ratio of specific heats $\gamma = \frac{C_p}{C_v}$ for a rigid diatomic gas molecule (e.g., $N_2$ or $O_2$ at room temperature)?
Explanation: A rigid diatomic gas has $5$ degrees of freedom ($3$ translational and $2$ rotational). Thus, $\gamma = 1 + \frac{2}{5} = 1.40$.
5. Which of the following thermodynamic quantities is a path function?
Explanation: Heat and work are path functions depending on the thermodynamic path, whereas internal energy, temperature, and pressure are state functions.
6. A metal rod of mass $m$ and specific heat $s$ is heated by supplying $Q$ amount of heat. If its temperature rises by $\Delta T$ without any phase change, the heat equation is:
Explanation: The standard heat capacity equation for sensible heating is $Q = ms\Delta T$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. A block of ice of mass $100\text{ g}$ at $0^\circ\text{C}$ melts completely into water at $0^\circ\text{C}$. If the latent heat of fusion of ice is $3.34 \times 10^5\text{ J kg}^{-1}$, what is the total heat absorbed?
Explanation: Using $Q = mL$, $m = 0.1\text{ kg}$ and $L = 3.34 \times 10^5\text{ J kg}^{-1}$, so $Q = 0.1 \times 3.34 \times 10^5 = 3.34 \times 10^4\text{ J}$.
2. Why is it incorrect to state that 'a body contains a large amount of heat'?
Explanation: Heat is not stored in a body; bodies store internal energy. Heat is exclusively energy in transit across system boundaries.
3. For a gas, if $C_p = \frac{7}{2}R$, what is the degree of freedom $f$ of the gas molecules?
Explanation: From $C_p = \left(1 + \frac{f}{2}\right)R$, we have $\frac{7}{2}R = \left(1 + \frac{f}{2}\right)R$, giving $1 + \frac{f}{2} = 3.5 \implies \frac{f}{2} = 2.5 \implies f = 5$. Wait, let's check: if $f=5$, $C_p = \frac{7}{2}R$. Thus $f=5$ (diatmic rigid). Let's re-verify: $C_v = \frac{f}{2}R = \frac{5}{2}R$, $C_p = C_v + R = \frac{7}{2}R$. Correct degree of freedom for $C_p = \frac{7}{2}R$ is $5$.
4. What is the physical meaning of the principle of calorimetry $Q_{\text{lost}} = Q_{\text{gained}}$?
Explanation: Calorimetry is based on the principle of conservation of energy applied to a thermally isolated system where heat lost by hot bodies equals heat gained by cold bodies.
5. If $500\text{ J}$ of mechanical work is done on a system and $200\text{ J}$ of heat is released by the system, what is the change in internal energy?
Explanation: By the First Law of Thermodynamics, $\Delta U = Q + W$. Here work is done *on* the system ($W = +500\text{ J}$) and heat is released ($Q = -200\text{ J}$). Thus $\Delta U = -200 + 500 = +300\text{ J}$.
6. What is the specific heat capacity of a substance if an infinite amount of heat is supplied with zero temperature change (e.g., during phase change)?
Explanation: From $s = \frac{Q}{m\Delta T}$, if $\Delta T = 0$ while heat $Q$ is supplied (phase change), the specific heat capacity approaches infinity ($s \to \infty$).
7. Which of the following pairs correctly matches the thermodynamic variable to its property type (State vs Path function)?
Explanation: Work and heat are path functions, whereas internal energy, temperature, and pressure are state functions.
8. A liquid of mass $m$ and specific heat $s_1$ at temperature $T_1$ is mixed with another liquid of mass $2m$ and specific heat $s_2$ ($s_2 = 2s_1$) at temperature $T_2$. What is the final equilibrium temperature of the mixture?
9. In Joule's experiment, falling weights are used to churn water in a calorimeter via paddle wheels. This demonstrates that:
Explanation: Joule's experiment demonstrated the mechanical equivalence of heat by showing that loss in mechanical potential energy of falling weights equals the thermal energy gained by water.
10. What is the molar heat capacity of a monoatomic ideal gas at constant volume ($C_v$)?
Explanation: For a monoatomic ideal gas, $f=3$, and $C_v = \frac{f}{2}R = \frac{3}{2}R$.