Some scientists were staring into their beer glasses and noticed that small bubb

Question

Some scientists were staring into their beer glasses and noticed that small bubbles begin at the bottom of a glass of beer and get bigger as they travel to the top of the glass at 15.6cm.

bubble radius (cm)

0.017 ± 0.004

0.020 ± 0.004

0.026 ± 0.004

0.027 ± 0.004

0.030 ± 0.004

0.031 ± 0.004

0.034 ± 0.004

height of bubble in glass (cm)

0.0 ± 0.2

1.2 ± 0.2

3.4 ± 0.2

7.0 ± 0.2

9.6 ± 0.2

12.4 ± 0.2

15.6 ± 0.2

There are two mechanisms for the bubble getting bigger as it rises. The first is that the pressure at the bottom of the glass is bigger. The second is that as the bubble accumulates carbon dioxide gas dissolved in the beer as it rises. If the change in the bubble’s volume is only due to a change in pressure (the first mechanism), what would the pressure have to be at the bottom of the glass?

Explanation / Answer

the bubble consists of a gas medium ( CO2)

we assume it is ideal gas then

PV= nRT,

through the glass T is const

P1V1 = P2V2

at the top of the glass the pressure is atmosphereic pressure P1 = 101 kPa

volume V1   = 4/3 *pi * (0.034/100)3

volume at the bottom V2 = 4/3 *pi * (0.017/100)3

                            Pressure at the bottom P2 = P1V1/V2 = 101*(0.034/100)3 /(0.017)3

                                                                                           = 808 kpa , assuming the bubble expands only due to the pressure change.

uncertianity in P =P1 V1/V2 = p1 r13/r23

            uncertanity in P , we take it as 0, atmospheric pressure is known correctly

   r1 = 0.034+/- 0.004   perc. unc. = 100*0.004/0.034 = 11.76 %

   r2 = 0.017 +/- 0.004     perc.unc. = 22.76%

Total perc. uncertianity in P = 11.76*3 + 22.76*3 = 105.8%

Pressure at the bottom = 808 kPa +/- 855 kPa

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