Isochoric Process (Constant Volume): Q = dU, No Work Done
Physics · Thermodynamics · NEET
An isochoric process is one where the volume of the gas stays constant. Since volume does not change, the gas does no work (W = 0), so by the first law all the heat goes into internal energy: Q = dU = n Cv dT. Memory hook: "Iso-CHORIC = volume stays, so no room to push, no work done - heat only heats the gas."
On a P-V diagram an isochoric process is a vertical line: volume is fixed, so the area under the line (the work) is zero, and all heat added raises pressure and temperature through Q = dU = n Cv dT.
Your doubts, answered
Why is work done zero in an isochoric process?
Work done by a gas is W = P dV, meaning force times distance the piston moves. In an isochoric process the volume is constant, so dV = 0 and the piston does not move at all. If dV = 0 then W = P times 0 = 0. On a P-V graph the process is a vertical line, and a vertical line has no area under it, which again means zero work.
If work is zero, where does the heat go?
By the first law of thermodynamics, Q = dU + W. Since W = 0 in an isochoric process, Q = dU. So every bit of heat you add goes straight into raising the internal energy (and therefore the temperature) of the gas. Adding heat at constant volume is the most efficient way to raise a gas's temperature because none of the energy is lost as work.
Is an isochoric process the same as an adiabatic process?
No. In an isochoric process volume is constant and heat CAN flow (Q = n Cv dT, usually not zero). In an adiabatic process no heat flows (Q = 0) but volume and pressure both change. Do not confuse them: isochoric means constant volume, adiabatic means no heat exchange. In isochoric, W = 0; in adiabatic, Q = 0.
Which specific heat, Cp or Cv, is used for isochoric heat?
Use Cv, the molar specific heat at constant volume, because the volume is held constant. The heat is Q = n Cv dT. You use Cp only when pressure is constant (isobaric). A common mistake is using Cp in an isochoric problem, which gives the wrong answer.
What does the P-V graph of an isochoric process look like?
It is a straight vertical line, because volume (on the x-axis) is fixed while pressure (on the y-axis) changes as you heat or cool the gas. Heating raises the pressure straight up; cooling lowers it straight down. The area under a vertical line is zero, matching W = 0.
⚠️ The NEET trap ✗ Since the pressure changes during heating, the gas must do some work. ✓ Work depends on volume change, not pressure change. In an isochoric process volume is fixed (dV = 0), so W = P dV = 0 no matter how much the pressure rises. 🧠 NTA loves to change the pressure and hope you think that means work. Work needs the piston to MOVE. No dV, no work.
Real NEET questions
NEET 2019
In which of the following processes, heat is neither absorbed nor released by a system?
A · Isothermal
B · Adiabatic ✓
C · Isobaric
D · Isochoric
Solution: Step 1: Read what each process means. Isochoric here is a distractor - in an isochoric process heat is exchanged, Q = n Cv dT, which is not zero. Step 2: Isothermal exchanges heat to keep temperature constant. Isobaric exchanges heat, Q = n Cp dT. Step 3: Only the adiabatic process has Q = 0 by definition (no heat enters or leaves). Answer: Adiabatic (B). Note the trap: isochoric has W = 0, not Q = 0.
NEET 2017
A P-V diagram shows three isotherms at 700 K, 500 K and 300 K. Four processes I, II, III, IV start from a common initial state. Match each with its type: (a) Adiabatic, (b) Isobaric, (c) Isochoric, (d) Isothermal.
A · I to a, II to c, III to d, IV to b
B · I to c, II to a, III to d, IV to b ✓
C · I to d, II to a, III to c, IV to b
D · I to b, II to c, III to a, IV to d
Solution: Step 1: Identify by shape on the P-V graph. A vertical line means constant volume, so Process I (vertical) is isochoric (c). Step 2: A steep falling curve that is steeper than the isotherm is adiabatic, so Process II is adiabatic (a). Step 3: A curve that stays on the same isotherm is isothermal, so Process III is isothermal (d). Step 4: A horizontal line means constant pressure, so Process IV is isobaric (b). Matching: I to c, II to a, III to d, IV to b. Answer: B.
Solved Thermodynamics NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Volume V is constant, work done W = 0, heat Q = dU = n Cv dT, and pressure follows Gay-Lussac's law P/T = constant. All the added heat becomes internal energy.
Is isochoric the same as isovolumetric?
Yes. Isochoric, isovolumetric and constant-volume process all mean the same thing: the volume of the gas does not change.
Does temperature change in an isochoric process?
Yes. When you add heat at constant volume, internal energy rises, so temperature rises. Pressure rises too, following P/T = constant. It is not the same as isothermal, where temperature stays fixed.
Why does Q = dU in an isochoric process?
From the first law Q = dU + W. Because volume is constant, W = 0, which leaves Q = dU. So heat added equals the change in internal energy exactly.
What real example shows an isochoric process?
Heating a gas inside a sealed rigid container (like a closed pressure cooker with a fixed lid, or a bomb calorimeter) is isochoric - the walls do not move, so volume is fixed and pressure rises.
