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Thermal Physics
Isothermal Process
An isothermal process is a thermodynamic process in which the temperature of a system remains constant, maintained by thermal contact with a heat reservoir. During this process, internal energy remains constant for an ideal gas, and all heat added is completely converted into work.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition, Boyle's law verification, P-V characteristics, work done calculation for an ideal gas, and basic isothermal expansion or compression problems.
PGT extension
Advanced derivation of isothermal work done, isothermal elasticity, slope of isothermal curve on P-V diagrams compared to adiabatic curves, and deviation of real gases from isothermal behavior.
Prerequisites
Concept of Temperature
Kinetic theory of gases
First Law of Thermodynamics
Definition and Thermal Equilibrium
The system temperature remains strictly constant throughout the process (dT = 0).
Achieved practically by keeping the system in contact with a large heat reservoir (constant temperature bath) and allowing very slow (quasi-static) changes.
For an ideal gas, internal energy depends solely on temperature, hence change in internal energy (dU = 0).
Equation of State (Boyle's Law)
Derived from the ideal gas equation PV = nRT when T is constant.
Yields PV = constant, meaning pressure is inversely proportional to volume.
Represents a rectangular hyperbola on a P-V (Indicator) diagram.
Work Done in an Isothermal Process
Since temperature is constant, work done is entirely at the expense of heat supplied to the system (Q = W).
Calculated by integrating the pressure-volume function along the isothermal path from initial volume V1 to final volume V2.
Can be expressed in terms of pressures as P1V1 ln(P1 / P2).
Isothermal Elasticity
Defined as the ratio of stress to volumetric strain under constant temperature conditions.
Mathematically equal to the operating pressure P of the gas.
Differs from adiabatic elasticity, which is given by gamma P.
Derivations
Work Done During Isothermal Expansion/Compression
Conditions: n moles of an ideal gas undergoing a reversible isothermal process from (P_1, V_1) to (P_2, V_2) at temperature T.
Start from the definition of mechanical work: $W = \int_{V_1}^{V_2} P dV$.
Substitute pressure from the ideal gas equation: P = \frac{nRT}{V}.
Factor out constant terms: $W = nRT \int_{V_1}^{V_2} \frac{1}{V} dV$.
Evaluate the integral: \int \frac{1}{V} d$V = \ln\left(\frac{V_2}{V_1}\right)$.
Apply Boyle's law substitution to express in pressures if required: \frac{V_2}{$V_1 = \frac{P_1}{P_2}$.
P_1, $P_2 = Initial$ and final pressures; V_1, $V_2 = Initial$ and final volumes Pa or atm, m^3
Fixed mass of ideal gas at constant temperature
First Law for Isothermal Process
$$Q = W$$
Q = Heat supplied to the system; W = Work done by the system Joules (J)
Ideal gas (Delta U = 0)
Isothermal Work Done (Volume form)
$$W = nRT \ln\left(\frac{V_2}{V_1}\right)$$
n = Number of moles, R = Universal gas constant, T = Absolute temperature Joules (J)
Reversible isothermal process
Isothermal Work Done (Pressure form)
$$W = nRT \ln\left(\frac{P_1}{P_2}\right)$$
P_1, $P_2 = Initial$ and final pressures Joules (J)
Reversible isothermal process
Change in Internal Energy
$$\Delta U = 0$$
\Delta U = Change in internal energy Joules (J)
Constant temperature for an ideal gas
Isothermal Bulk Modulus (Elasticity)
$$E_T = P$$
$E_T = Isothermal elasticity$, P = Pressure Pascal (Pa)
Constant temperature
Molar Heat Capacity in Isothermal Process
$$C = \infty$$
C = Molar heat capacity J mol^-1 K^-1
Temperature change is zero for non-zero heat exchange
Indicator Diagram Curve Equation
$$P = \frac{nRT}{V}$$
P = Pressure, V = Volume Pa, m^3
Isotherm at temperature T
Graphs and experiments
P-V Indicator Diagram for Isothermal Process
Draw: A downward-sloping rectangular hyperbola curve on a P-V graph, showing pressure decreasing as volume increases.
Interpret: Represents Boyle's law; different curves (isotherms) at different temperatures lie higher for higher temperatures (T_2 > T_1).
Exam use: Frequently tested in identifying isothermal vs. adiabatic curves (isotherm is less steep than adiabat).
P-T and V-T Diagrams for Isothermal Process
Draw: A straight line parallel to the pressure axis on a T-P graph, and a straight line parallel to the volume axis on a T-V graph.
Interpret: Explicitly highlights that temperature remains constant regardless of variations in pressure or volume.
Exam use: Used in objective questions to test graph recognition for thermodynamic processes.
Clément and Desormes' Apparatus Principle
Draw: Schematic setup of a large glass flask with a wide mouth, stopcock, manometer, and pressure pump.
Interpret: Used to determine the ratio of specific heats (gamma) by combining slow (isothermal) and rapid (adiabatic) operations.
Exam use: Conceptual questions on experimental verification of thermodynamic constants in PGT physics.
Solved numericals
TGT core
2 moles of an ideal gas expand isothermally at 300 K from an initial volume of 20 litres to 60 litres. Calculate the work done by the gas. ($R = 8.314 J mol^-1 K^-1$, \ln 3 = 1.0986)
Identify given parameters: n = 2 mol, T = 300 K, $V_1 = 20 L$, $V_2 = 60 L$, $R = 8.314 J/(mol K)$.
Use the isothermal work formula: $W = nRT \ln(V_2 / V_1)$.
Calculate final product: W = 2 \times 8.314 \times 300 \times 1.0986 = 5497.56 \text{ J}.
Answer: 5497.56 J
PGT extension
An ideal gas at 27 °C is compressed isothermally to one-fourth of its initial volume. If the initial pressure is 100 kPa, find the final pressure and the total heat released by the gas per mole.
Find final pressure using Boyle's law: P_1 $V_1 = P_2 V_2$, where $V_2 = V_1 / 4$. Thus, $P_2 = 4 P_1$.
Answer: Final pressure = 400 kPa; Heat released = 3456.9 J
PGT extension
Compare the slope of an isothermal curve with that of an adiabatic curve for an ideal gas at a given state (P, V). Given the adiabatic exponent is \gamma.
Recall the isothermal curve slope: \left(\frac{dP}{dV}\right)_{\text{iso}} = -\frac{P}{V}.
Recall the adiabatic curve relation: P$V^\gamma = \text{constant}$.
Differentiate implicitly: P \gamma V^{\gamma-1} dV + V^\gamma dP = 0.
Simplify to find adiabatic slope: \left(\frac{dP}{dV}\right)_{\text{ad}} = -\gamma \frac{P}{V}.
Take the ratio of adiabatic slope to isothermal slope: \frac{\text{Slope}_{\text{ad}}}{\text{Slope}_{\text{iso}}} = \gamma.
Answer: The adiabatic curve is \gamma times steeper than the isothermal curve.
Assumptions and limitations
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Common misconceptions
Common error: An isothermal process means the container must be insulated.
Correct: An isothermal process requires a highly conducting container in contact with a constant temperature heat reservoir.
Insulation prevents heat exchange, leading instead to an adiabatic process.
Common error: Since temperature is constant, no heat flows into or out of the system.
Correct: Heat flows freely to maintain constant temperature when work is done.
By the First Law, Q = W; if a gas expands isothermally, it must absorb heat from the reservoir to keep its temperature from falling.
Common error: The isothermal slope is steeper than the adiabatic slope on a P-V diagram.
Correct: The adiabatic curve is steeper by a factor of gamma.
Gamma (>1) multiplies the pressure term in the adiabatic derivative, making its absolute slope larger.
Quick revision
Isothermal process implies dT = 0 and delta U = 0 for an ideal gas.
Governed by Boyle's Law: PV = constant.
Work done during isothermal expansion/compression is $W = nRT \ln(V_2 / V_1)$.
Molar heat capacity in an isothermal process is infinite (C = infinity).
Isothermal elasticity equals the current pressure ($E_T = P)$.
On a P-V indicator diagram, isotherms are rectangular hyperbolas.
Must remember
In an isothermal process, the system temperature remains strictly constant throughout the process ($dT = 0$).
For an ideal gas, internal energy depends solely on temperature, hence the change in internal energy is zero ($\Delta U = 0$).
The equation of state governing an isothermal process for an ideal gas is Boyle's Law ($PV = \text{constant}$).
On a P-V indicator diagram, an isothermal process is represented by a rectangular hyperbola.
The first law of thermodynamics simplifies to $Q = W$ during an isothermal process.
The work done in a reversible isothermal expansion or compression is given by $W = nRT \ln(V_2 / V_1)$ or $nRT \ln(P_1 / P_2)$.
The isothermal elasticity (or bulk modulus) of an ideal gas is exactly equal to its operating pressure ($E_T = P$).
The molar heat capacity of a system undergoing an isothermal process is infinite ($C = \infty$) because $dT = 0$ while $Q \neq 0$.
Achieving an isothermal process practically requires a highly conducting container in contact with a large heat reservoir and very slow (quasi-static) changes.
An adiabatic curve is steeper than an isothermal curve on a P-V diagram by a factor of $\gamma$ (the ratio of heat capacities).
Glossary
Isothermal ProcessA thermodynamic process in which the temperature of a system remains constant ($dT = 0$).
Boyle's LawThe pressure of a given mass of an ideal gas is inversely proportional to its volume at constant temperature ($PV = \text{constant}$).
Isothermal Elasticity ($E_T$)The ratio of stress to volumetric strain under constant temperature conditions, numerically equal to pressure $P$.
Indicator DiagramA P-V (Pressure-Volume) diagram used to represent thermodynamic states and work done during processes.
Molar Heat Capacity ($C$)The amount of heat required to raise the temperature of one mole of a substance by one Kelvin; infinite for an isothermal process.
Isothermal vs Adiabatic Processes
Parameter
Isothermal Process
Adiabatic Process
Temperature change
Constant ($dT = 0$)
Variable ($dT \neq 0$)
Heat exchange
Allowed ($Q = W$)
None ($Q = 0$)
Equation of state
$PV = \text{constant}$
$PV^\gamma = \text{constant}$
Elasticity
$E_T = P$
$E_S = \gamma P$
P-V curve slope
$-P/V$
$-\gamma P / V$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following conditions must be satisfied for a process to be considered strictly isothermal for an ideal gas?
Explanation: An isothermal process is defined by a constant temperature ($dT = 0$). For an ideal gas, internal energy is a function of temperature alone, hence $dT = 0$ implies $\Delta U = 0$.
2. What is the shape of the curve representing an isothermal process on a P-V (Indicator) diagram for an ideal gas?
Explanation: Since $PV = \text{constant}$ (Boyle's Law) for an isothermal process, pressure is inversely proportional to volume ($P \propto 1/V$), which graphically represents a rectangular hyperbola.
3. According to the First Law of Thermodynamics, what is the relationship between heat supplied ($Q$) and work done ($W$) in an isothermal process involving an ideal gas?
Explanation: The first law is $Q = \Delta U + W$. For an isothermal process of an ideal gas, $\Delta U = 0$, which yields $Q = W$.
4. What is the expression for the work done during a reversible isothermal expansion of $n$ moles of an ideal gas from volume $V_1$ to $V_2$?
Explanation: Integrating $P dV$ along the isothermal path $P = nRT/V$ from $V_1$ to $V_2$ gives $W = nRT \ln(V_2 / V_1)$.
5. What is the isothermal elasticity (bulk modulus) of an ideal gas at pressure $P$?
Explanation: Isothermal elasticity is defined as $E_T = -V \frac{dP}{dV}$. Differentiating $PV = \text{constant}$ gives $P dV + V dP = 0$, leading to $-V \frac{dP}{dV} = P$.
6. What is the molar heat capacity of an ideal gas during an isothermal process?
Explanation: Molar heat capacity is given by $C = \frac{Q}{n \Delta T}$. Since $\Delta T = 0$ for a non-zero heat exchange $Q$, $C$ approaches infinity.
7. Which of the following statements is a common misconception about an isothermal process?
Explanation: Insulating the container prevents heat exchange, leading instead to an adiabatic process. An isothermal process requires a highly conducting container in contact with a constant temperature bath.
8. If an ideal gas expands isothermally, how does its pressure change?
Explanation: By Boyle's Law ($PV = \text{constant}$), pressure is inversely proportional to volume ($P \propto 1/V$). As volume increases, pressure decreases accordingly.
9. If 2 moles of an ideal gas undergo an isothermal expansion at $300\text{ K}$ such that its volume doubles, what is the work done? (Given $R = 8.314\text{ J mol}^{-1}\text{ K}^{-1}$)
10. How does the slope of an isothermal curve compare to that of an adiabatic curve on a P-V diagram at the same point?
Explanation: The isothermal slope is $-P/V$, whereas the adiabatic slope is $-\gamma P/V$. Thus, the adiabatic curve is steeper by a factor of $\gamma$.
11. For a real gas undergoing an isothermal process, which thermodynamic quantity generally changes along with volume?
Explanation: Unlike ideal gases whose internal energy depends solely on temperature, real gas internal energy also depends on intermolecular forces (volume/intermolecular separation), so $\Delta U \neq 0$ even if $dT = 0$ during isothermal expansion/compression.
12. Which law serves as the governing equation of state for an isothermal process of an ideal gas?
Explanation: Boyle's Law states that at constant temperature, the product of pressure and volume for a fixed mass of gas remains constant ($PV = \text{constant}$).
13. What happens to the internal energy of an ideal gas when it undergoes an isothermal compression?
Explanation: Since temperature remains constant ($dT = 0$) during an isothermal process, the internal energy of an ideal gas ($\Delta U = n C_v \Delta T$) remains unchanged.
14. Calculate the work done when a gas expands isothermally from an initial pressure of $4\text{ atm}$ to $1\text{ atm}$, given $nRT = 500\text{ J}$.
15. Which experimental condition is essential to ensure a process remains isothermal?
Explanation: High thermal conductivity allows rapid heat exchange with a thermal reservoir, and very slow (quasi-static) changes ensure thermal equilibrium is maintained continuously.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. An ideal gas is compressed isothermally to half of its initial volume. The work done on the gas is:
Explanation: Work done by the gas is $W = RT \ln(V_2 / V_1) = RT \ln(V/2 / V) = -RT \ln 2$. Therefore, the work done ON the gas is positive $RT \ln 2$, meaning work done BY the gas is $-RT \ln 2$.
2. The ratio of isothermal elasticity to adiabatic elasticity for a diatomic gas (with $\gamma = 1.4$) is:
Explanation: Isothermal elasticity $E_T = P$, and adiabatic elasticity $E_S = \gamma P$. The ratio $E_T / E_S = P / (\gamma P) = 1 / \gamma = 1 / 1.4 \approx 0.71$.
3. If the temperature of an ideal gas is kept constant while its pressure is tripled, what happens to its volume?
Explanation: By Boyle's Law ($PV = \text{constant}$), if pressure is multiplied by 3 ($P_2 = 3P_1$), the final volume must become one-third of the initial volume ($V_2 = V_1 / 3$).
4. During an isothermal expansion of an ideal gas, heat energy supplied to the gas is completely converted into:
Explanation: From the first law $Q = \Delta U + W$. Since $\Delta U = 0$ for an isothermal process of an ideal gas, $Q = W$, meaning all supplied heat is converted into external work done by the gas.
5. What is the slope of the isothermal curve on a P-V diagram at a point $(P, V)$?
Explanation: Differentiating $PV = C$ gives $P dV + V dP = 0 \implies \frac{dP}{dV} = -\frac{P}{V}$.
6. Which of the following processes represents an isothermal change for an ideal gas?
Explanation: A copper cylinder is a good conductor, and an ice-water bath acts as a constant temperature reservoir. Very slow compression ensures thermal equilibrium is maintained constantly (isothermal process).
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. An ideal gas undergoes an isothermal process from $(P_1, V_1)$ to $(P_2, V_2)$. Which of the following is correct?
Explanation: An isothermal process follows Boyle's Law, which states $P_1 V_1 = P_2 V_2$.
2. If 1 mole of a monatomic ideal gas expands isothermally at $300\text{ K}$ from $10\text{ L}$ to $20\text{ L}$, what is the change in internal energy?
Explanation: For an ideal gas, internal energy depends solely on temperature. Since temperature is constant ($dT = 0$), the change in internal energy $\Delta U = 0$.
3. The molar heat capacity of an ideal gas during an isothermal expansion is:
Explanation: Molar heat capacity $C = Q / (n \Delta T)$. Since $\Delta T = 0$ and $Q \neq 0$, $C$ is infinite.
4. What is the ratio of slopes (adiabatic to isothermal) at the same point on a P-V diagram?
Explanation: Adiabatic slope is $-\gamma P / V$ and isothermal slope is $-P / V$. The ratio of adiabatic slope to isothermal slope is $\gamma$.
5. In an isothermal compression of a gas, the heat released to the surroundings is:
Explanation: By $Q = \Delta U + W$, since $\Delta U = 0$, the heat rejected $Q$ equals the work done on the system $W$.
6. Which of the following parameters remains constant during an isothermal process of a fixed mass of an ideal gas?
Explanation: By definition, an isothermal process is one where temperature remains constant ($dT = 0$).
7. Calculate the work done when $2\text{ moles}$ of an ideal gas expand isothermally at $400\text{ K}$ to four times its initial volume. (Use $\ln 2 \approx 0.693$, $R = 8.314\text{ J mol}^{-1}\text{ K}^{-1}$)
8. The isothermal bulk modulus of a gas at normal atmospheric pressure ($1.013 \times 10^5\text{ Pa}$) is:
Explanation: Isothermal bulk modulus (elasticity) $E_T = P$. Thus, it equals the operating pressure of $1.013 \times 10^5\text{ Pa}$.
9. Why does a gas get cooled when it expands adiabatically, but does not cool when it expands isothermally?
Explanation: In an isothermal process, heat is absorbed from the thermal bath to offset the energy lost as work, keeping temperature constant. In adiabatic expansion, $Q = 0$, so internal energy drops, causing cooling.
10. Which thermodynamic variable is zero for an isothermal process of an ideal gas?
Explanation: Since internal energy depends solely on temperature for an ideal gas, $dT = 0$ dictates that $\Delta U = 0$.