In elementary thermodynamics and gas laws, which of the following equations correctly represents Boyle's law for an ideal gas undergoing a change at constant temperature?

Introduction / Context:
This question checks your understanding of the basic gas laws that together lead to the ideal gas equation. In school level physics and chemistry, you study individual relationships such as Boyle's law, Charles law and Gay Lussac law. Each describes how two state variables (pressure, volume and temperature) are related when the third is held constant. Recognising the correct mathematical form of Boyle's law is important because it is used repeatedly in numerical problems and also forms part of the derivation of the ideal gas law PV = nRT.

Given Data / Assumptions:

• We are dealing with an ideal gas undergoing a change from an initial state (P1, V1, T1) to a final state (P2, V2, T2).
• Boyle's law applies when temperature is kept constant (isothermal process).
• The options show different possible relationships between pressure, volume and temperature.
• We assume the amount of gas (number of moles) remains constant during the process.

Concept / Approach:
Boyle's law states that for a fixed mass of an ideal gas at constant temperature, the pressure of the gas is inversely proportional to its volume. In symbols, P is proportional to 1 / V when temperature T is constant. This proportionality can be written in equation form as P * V = constant. When the gas changes from one state to another isothermally, this constant remains the same, so we obtain P1V1 = P2V2. The other equations involving temperature belong to different gas laws and do not represent Boyle's law.

Step-by-Step Solution:
Step 1: Recall the verbal statement of Boyle's law: at constant temperature, pressure is inversely proportional to volume for a fixed mass of gas. Step 2: Express this as P * V = constant when temperature is unchanged. Step 3: For two different isothermal states, write P1V1 = constant and P2V2 = constant. Step 4: Equate the two constants to obtain P1V1 = P2V2, which is the standard equation for Boyle's law. Step 5: Compare this with the answer choices and identify the option that matches exactly.

Verification / Alternative check:
A quick mental check is to think about what happens when you compress a gas slowly at constant temperature. If the volume is halved (V2 = V1 / 2), the equation P1V1 = P2V2 predicts that P2 = 2P1, meaning pressure doubles. This matches the idea of inverse proportionality from Boyle's law. Any equation involving T1 and T2 cannot represent Boyle's law because temperature must remain constant in an isothermal process. Thus, only P1V1 = P2V2 satisfies both the law and this simple test.

Why Other Options Are Wrong:
P1T1 = P2T2 suggests a direct relation between pressure and temperature, which is closer to Gay Lussac type behaviour, not Boyle's law. P1 / V1 = P2 / V2 implies pressure is directly proportional to volume, which contradicts the inverse relationship of Boyle's law. T1 / V1 = T2 / V2 mixes temperature and volume and relates to a different type of gas law, not Boyle's law at constant temperature.

Common Pitfalls:
Students often confuse the separate gas laws because they all involve P, V and T. A simple way to avoid this is to remember one key phrase: Boyle's law connects P and V with T constant, Charles law connects V and T with P constant and Gay Lussac law connects P and T with V constant. Matching these verbal statements to the correct formulas ensures that you do not pick an option just because it looks algebraically neat. Always check which variable is supposed to remain constant and whether the relationship is direct or inverse.

Final Answer:
The equation that correctly represents Boyle's law for an ideal gas at constant temperature is P1V1 = P2V2.