An isothermal process occurs at constant temperature. For an ideal gas, internal energy remains constant, so heat supplied equals work done by the gas.
Learning Outcomes
Temperature remains constant: ΔT = 0.
For an ideal gas, ΔU = 0.
The first law reduces to Q = W.
A slow expansion permits heat transfer that maintains temperature.
For fixed gas amount, PV = constant.
The P–V curve is called an isotherm; area under it is work.
Intermediate Level
Condition and First-Law Reduction
For an ideal gas, internal energy depends only on temperature. Because temperature is unchanged in an isothermal process, ΔU = 0.
Key equation
Δ T=0
Key equation
Δ U=0
Key equation
Q=Δ U+W=0+W
Key equation
Q=W
Pressure–Volume Relation
A fixed amount of ideal gas follows Boyle’s law during an isothermal change.
Key equation
PV=constant
Key equation
P1V1=P2V2
Graph and Rate
An isotherm slopes downward on a P–V graph, and the area under the curve represents work. The process should occur slowly enough for heat to pass between the system and surroundings and hold temperature constant.
Isothermal expansion
Isothermal compression
V rises and P falls
V falls and P rises
gas does positive work
work by gas is negative
heat enters gas
heat leaves gas
Energy Balance
An ideal gas expands isothermally and does 240 J of work. Find ΔU and Q.
1
Working
Isothermal ideal gas: ΔU = 0
2
Step 2
Q = ΔU + W
3
Step 3
Q = 0 + 240 J
Final answer:ΔU = 0 and Q = +240 J
Examples
Melting ice at 0 °C during the phase change
Melting and evaporation carried out at constant temperature
Slow gas expansion while in thermal contact with a reservoir
Common Mistakes
Constant temperature does not mean Q = 0.
For an ideal gas, ΔU = 0 because T is constant.
An isothermal gas change should be slow enough for heat exchange.
Do not draw an isotherm as a horizontal line.
Short-Question Preparation
Q. Define an isothermal process.
A. A thermodynamic process in which temperature remains constant.
Q. How does the first law reduce?
A.For an ideal gas ΔU = 0, so Q = W.
Q. Why is the process slow?
A. Heat needs time to flow and keep temperature constant.
Q. What is an isotherm?
A. The P–V curve representing an isothermal process.
Sticky Notes
Temperature remains constant: ΔT = 0.
For an ideal gas, ΔU = 0.
The first law reduces to Q = W.
A slow expansion permits heat transfer that maintains temperature.
For fixed gas amount, PV = constant.
The P–V curve is called an isotherm; area under it is work.
60-Second Revision
Isothermal means ΔT = 0. For an ideal gas, ΔU = 0, so Q = W. The gas follows PV = constant; its P–V curve is an isotherm and the area under it is work.
Summary
An isothermal process occurs at constant temperature. For an ideal gas, internal energy remains constant, so heat supplied equals work done by the gas.
Ask-the-Tutor Modes
Explain the energy transfers simply.
Apply the textbook sign convention.
Derive the reduced equation.
Solve a numerical step by step.
Explain the P–V graph.
Test me with an MCQ.
Did you understand this topic?
Your answer is saved with your learning record. It does not mark the topic as mastered.
Exam preparation
Exam Focus
Important points
Temperature remains constant: ΔT = 0.
For an ideal gas, ΔU = 0.
The first law reduces to Q = W.
A slow expansion permits heat transfer that maintains temperature.
For fixed gas amount, PV = constant.
The P–V curve is called an isotherm; area under it is work.
Common mistakes
Constant temperature does not mean Q = 0.
For an ideal gas, ΔU = 0 because T is constant.
An isothermal gas change should be slow enough for heat exchange.
Do not draw an isotherm as a horizontal line.
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.
Isothermal Process
Sticky Notes · Quick Revision
1.
Key Point 1
Temperature remains constant: ΔT = 0.
2.
Key Point 2
For an ideal gas, ΔU = 0.
3.
Key Point 3
The first law reduces to Q = W.
4.
Key Point 4
A slow expansion permits heat transfer that maintains temperature.
