The ideal gas law is PV = mRT, where P is the pressure, V is the volume, T is the temperature, m is the mass and R is the gas constant. At constant temperature, if the new volume of an ideal gas is three times the original volume, how does the pressure change? (P1 is the original pressure, P2 is the new pressure.)

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Question restatement: The ideal gas law is PV = mRT. With P as pressure, V as volume, T as temperature, m as mass and R as the gas constant. At constant temperature, if the new volume is three times the original volume, how does the pressure change? (P1 is original pressure, P2 is new pressure.)

Option-by-option analysis:

Option 1: 'P2 = 9 P1' This would imply pressure increases by a factor of 9 when volume triples. Since PV = mRT and at fixed m, R, T, the product PV must stay constant, increasing V should decrease P, not increase it. A factor of 9 increase contradicts the inverse relationship between pressure and volume at constant temperature. Therefore this is incorrect.

Option 2: 'P2 = 3 P1' This suggests pressure triples when volume triples. Again, PV = constant (for fixed m, R, T), so P ∝ 1/V. Tripling V should reduce P, not raise it. Hence this option is incorrect.

Option 3: 'P2 = P1/3' This matches the inverse proportionality P ∝ 1/V at constant temperature and constant amount of gas. If V becomes three times larger, P must become one-third of its original value to keep PV constant, so P2 = P1/3 is consistent with the ideal gas relation. This option is correct in principle.

Option 4: 'P2 = P1/9' This would imply a much larger drop in pressure (one-ninth of original) when volume triples. The ideal gas relation P ∝ 1/V would give a factor of 1/3, not 1/9, under a threefold increase in volume, so this option is incorrect.

Additional clarifications:
- The key idea is that for a fixed amount of gas and constant temperature, the product PV remains constant (PV = mRT with m, R, T constant). Therefore, when V increases by a factor of 3, P must decrease by the same factor to keep the product constant. This directly leads to P2 = P1/3.
- It's important not to confuse the situation with changes in temperature or amount of gas, which would alter the relation and potentially change the proportionality constant; here those are held fixed by the problem statement.

Summary: Among the given options, the one that aligns with the PV = mRT constraint at constant T and m is P2 = P1/3, since pressure inversely scales with volume. The other options fail to reflect the inverse relationship dictated by the ideal gas law under the stated conditions.

Math & Science Prep Program Placement Assessment

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