Q.Boyle's law is applicable for an

Concept: Boyle's law and thermodynamic processes

Boyle's law states that for a fixed mass of gas, pressure and volume are inversely proportional: $PV = \text{constant}$, or $P \propto \frac{1}{V}$.

This relationship holds only when temperature remains constant. From the ideal gas equation $PV = nRT$, if $n$ (amount of gas) and $T$ (temperature) are both fixed, then $PV$ must indeed be constant.

An isothermal process is defined as one in which temperature stays constant throughout. During such a process, any change in pressure is accompanied by an inverse change in volume, exactly as Boyle's law describes.

In contrast:

• An adiabatic process has $PV^\gamma = \text{constant}$ (temperature changes).
• Isobaric means constant pressure (volume can change, but $P$ does not).
• Isochoric means constant volume (pressure can change, but $V$ does not).

Understanding Boyle's Law

Boyle's law states that for a fixed amount of gas, the product of pressure and volume remains constant:

This can also be written as $P_1 V_1 = P_2 V_2$ when comparing two states.

The critical question is: under what conditions does this relationship hold? The answer lies in what must be kept fixed for the law to work.

Derivation from the Ideal Gas Equation

Start with the ideal gas equation:

For a fixed amount of gas ($n$ constant), if we want $PV$ to remain constant, we need:

This is only possible when $T$ remains constant. When temperature is held fixed, the right side of the equation doesn't change, so the left side ($PV$) cannot change either.

Examining Each Process Type

Let me walk through why each option does or doesn't work:

1. Isothermal process ($T = \text{constant}$): Since $PV = nRT$ and both $n$ and $T$ are fixed, we immediately get $PV = \text{constant}$. This is exactly Boyle's law.

2. Adiabatic process (no heat exchange): Here the relationship is $PV^\gamma = \text{constant}$, where $\gamma > 1$ (typically 1.4 for diatomic gases, 1.67 for monatomic). Temperature changes during compression or expansion, so $PV$ is not constant.

3. Isobaric process ($P = \text{constant}$): Pressure doesn't change at all, so there's no inverse relationship with volume to speak of. Instead, $V \propto T$.

4. Isochoric process ($V = \text{constant}$): Volume doesn't change, so again Boyle's law (which describes how $P$ and $V$ vary together) is irrelevant. Instead, $P \propto T$.

Physical Intuition

When you compress a gas isothermally, you're doing work on it, which would normally raise its temperature. But because heat flows out to keep $T$ constant, the only effect of reducing volume is to increase pressure proportionally. The molecules hit the walls more often (smaller volume) but with the same average speed (same temperature), giving the inverse $P \propto 1/V$ relationship.

Memory hook, faster than recalling the law by name: work backwards from the ideal gas equation $PV = nRT$ and freeze one variable at a time. Freeze $T$ and you get Boyle's $PV = \text{const}$; freeze $P$ and you get Charles's $V \propto T$; freeze $V$ and you get Gay-Lussac's $P \propto T$. The process name always describes the variable held constant, not the one that changes — so 'isothermal' (constant $T$) is immediately Boyle's law without needing to recall which historical name attaches to which law.