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Ideal Gas Equation

Two closed bu...

Question

Two closed bulbs of equal volumes (V) containing an ideal gas initially at pressure $p_{i}$ and temperature $T_{1}$ are
connected through a narrow tube of negligible volume as shown in the figure below. The temperature of one of the bulbs is then raised to $T_{2}$ . The final pressure $p_{f}$ is:

2 p_{i} (\frac{T_{1}}{T_{1} + T_{2}})

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2 p_{i} (\frac{T_{2}}{T_{1} + T_{2}})

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2 p_{i} (\frac{T_{1} T_{2}}{T_{1} + T_{2}})

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p_{i} (\frac{T_{1} T_{2}}{T_{1} + T_{2}})

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Solution

The correct option is B $2 p_{i} (\frac{T_{2}}{T_{1} + T_{2}})$
Initial total moles:

$n = \frac{p_{i} V}{R T_{1}} + \frac{p_{i} V}{R T_{1}} = \frac{2 p_{i} V}{R T_{1}} \dots \dots (1)$

Final total moles:

$n = \frac{p_{f} V}{R T_{1}} + \frac{p_{f} V}{R T_{2}} \dots \dots (2)$

Equating the two:

$\frac{2 p_{i}}{T_{1}} = \frac{p_{f}}{T_{1}} + \frac{p_{f}}{T_{2}}$

$p_{f} = \frac{2 p_{i}}{T_{1}} \times \frac{T_{1} T_{2}}{(T_{1} + T_{2})}$

$p_{f} = 2 p_{i} \times \frac{T_{2}}{(T_{1} + T_{2})}$

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