A small bubble rises from the bottom of a lake, where the temperature and pressure are 8°C and 6.4 atm, to the water’s surface, where the temperature is 25°C and the pressure is 1.0 atm. Calculate the final volume (in mL) of the bubble if its initial volume was 2.1 mL. Strategy In solving this

Question

A small bubble rises from the bottom of a lake, where the temperature and pressure are 8°C and 6.4 atm, to the water’s surface, where the temperature is 25°C and the pressure is 1.0 atm. Calculate the final volume (in mL) of the bubble if its initial volume was 2.1 mL.

Strategy In solving this kind of problem where a lot of information is provided, it is often helpful to organize the information. The given information is tabulated here:

[latex]\begin{matrix} Initial Conditions & & Final Conditions\\hline \ P_{1}=6.4 amt & & P_{2}=1.0 atm \ V_{1}=2.1 mL & & V_{2}=? \T_{1}=(8+273) K=281 K & &T_{2}=(25+273) K=298 K\end{matrix} [/latex]

What temperature unit should be used in the calculation?

Accepted Answer

According to Equation (5.9)

[latex]\frac{P_{1}V_{1}}{n_{1}T_{1}} =\frac{P_{2}V_{2}}{n_{2}T_{2}} [/latex]

We assume that the amount of air in the bubble remains constant, that is, [latex]n_{1} = n_{2}[/latex] so that

[latex]\frac{P_{1}V_{1}}{T_{1}} =\frac{P_{2}V_{2}}{T_{2}} [/latex]

which is Equation (5.10). Rearranging Equation (5.10) gives

[latex]V_{2}=V_{1} imes \frac{P_{1}}{T_{1}} =\frac{P_{2}}{T_{2}} [/latex]

[latex]= 2.1 mL imes \frac{6.4 atm}{1.0 atm} imes \frac{298 K}{281 K} [/latex]

= 14 mL

Check We can use any appropriate units for volume (or pressure) as long as we use the same units on both sides of the equation. We see that the final volume involves multiplying the initial volume by a ratio of pressures [latex](P_{1} ⁄P_{2} )[/latex] and a ratio of temperatures [latex](T_{1} ⁄T_{2} )[/latex]. Recall that volume is inversely proportional to pressure, and volume is directly proportional to temperature. Because the pressure decreases and temperature increases as the bubble rises, we expect the bubble’s volume to increase. In fact, here the change in pressure plays a greater role in the volume change.

Question 5.7: A small bubble rises from the bottom of a lake, where the te...

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