Which of the following represents a combination of Boyle's Law and Charle's Law?

P_{1} V_{1} T_{1} = P_{2} V_{2} T_{2}

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\frac{P_{1} T_{2}}{V_{2}} = \frac{P_{2} T_{1}}{V_{1}}

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\frac{P_{1} T_{2}}{V_{1}} = \frac{P_{2} T_{1}}{V_{2}}

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\frac{T_{2} V_{1}}{P_{1}} = \frac{T_{1} V_{2}}{P_{2}}

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Solution

The correct option is C $\frac{P_{1} T_{2}}{V_{2}} = \frac{P_{2} T_{1}}{V_{1}}$
Charles' law is the change in volume with respect to temperature at constant pressure, while Boyle's law is the change in volume with respect to pressure at a constant temperature.

Taking a gas of volume $V_{1}$ , pressure $P_{1}$ and temperature $T_{1}$ , and let its change have a state ( $V_{2}, P_{2}, T_{2}$ ), then according to Boyle's law,

$P_{1} V_{1} = P_{2} V_{2}$ ..........(1)

Then keeping this constant pressure, move to state ( $V_{2}, P_{2}, T_{2}$ ) using Charles' law,

$\frac{V_{2}}{T_{1}} = \frac{V_{2}}{T_{2}}$ .........(2)

Solving for $V_{2}$ , in both (1) and (2) and then equating them, we get-

$\frac{P_{1} V_{1}}{T_{1}} = \frac{P_{2} V_{2}}{T_{2}}$

After changing the pressure, volume and temperature of the gas, still their product is equal, suggesting that the relation is constant: $\frac{P V}{T} = k$ .

Hence, the correct answer is option $B$ .

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Q. Which gas law is summarized with the following equation?

P_{1} V_{1} = P_{2} V_{2}

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