Physics · Ch 8 — Heat and Thermodynamics

Isochoric Process

8.8.4

Isochoric Process

An isochoric process holds volume fixed while pressure, temperature and internal energy vary. From $PV=\mu RT$ with $V$ fixed, $P\propto T$ , giving a straight $P$ - $T$ line through the origin; on a P-V diagram, an isochoric process is a vertical line (Figure 8.37). Since $\Delta V=0$ , no work is done at all ( $W=0$ ), so by the first law $\Delta U=Q$ : all supplied (or released) heat goes entirely into changing internal energy, raising (or lowering) both temperature and pressure together. Everyday examples include a tightly closed cooking vessel, where the trapped steam's rising pressure eventually pushes the lid upward (Figure 8.38), and the two constant-volume heat-addition/heat-rejection strokes of a petrol engine's four-stroke cycle, bracketed between its two adiabatic strokes. Table 8.4 collects the equation of state, work formula and sign behaviour for isot …

Figure 8.37PV diagram for an isochoric process

What this figure shows. Two small P-V diagrams, each showing a single vertical straight line parallel to the pressure axis (since volume is constant throughout). Diagram (a) shows the line running from a lower initial pressure Pi up to a higher final pressure Pf, representing an isochoric process where pressure increases (heat is added). Diagram (b) shows the line running from a higher initial pressure Pi down to a lower final pressure Pf, representing an isochoric process where pressure decreases (heat is removed). Both lines sit at the same fixed position V on the volume axis, confirming that an isochoric process …

Figure 8.38Isochoric process in a sealed rigid container

What this figure shows. A rigid, sealed container of gas shown in an 'Initial' state at a lower pressure P1 and a 'Final' state at a higher pressure P2, with the container's fixed volume outline identical in both states (drawn with the exact same rigid boundary, showing the volume genuinely cannot change). The figure is paired with the everyday example of food cooking in a tightly closed vessel: as heat is continuously supplied, the pressure inside builds up (with volume unable to change) until eventually the rising steam pressure is strong enoug …

Table 8.4Summary of the four thermodynamic processes

Process	Direction	Heat Q	Temperature & internal energy	Pressure	Volume	Equation of state	Work done (ideal gas)
Isothermal	Expansion	Q > 0	Constant	decreases	increases	PV = constant	$W=\mu RT\ln(V_f/V_i) > 0$
Isothermal	Compression	Q < 0	Constant	increases	decreases	PV = constant	$W=\mu RT\ln(V_f/V_i) < 0$
Isobaric	Expansion	Q > 0	increases	Constant	increases	V/T = constant	$W=P[V_f-V_i]=P\Delta V > 0$
Isobaric	Compression	Q < 0	decreases	Constant	decreases	V/T = constant	$W=P[V_f-V_i]=P\Delta V < 0$
Isochoric	--	Q > 0 / Q < 0	Increases / Decreases	increases / decreases	Constant	P/T = constant	Zero

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