Banks Ferrell
2/6/13
IB Physics 2A
Exponential Decay of Beer Foam
Experiment
In order model the decay constant of nuclear material, beer foam can be used
to create a similar exponential digression as the foam subsides upward into liquid
over time. To conduct this experiment I filled a graduated cylinder with enough beer
to produce foam up to the top of the cylinder. I then timed the liquid’s progression
to the top and marked the location of the liquid at an integral of five seconds for a
1.5 minutes. I repeated this process two more times to compensate for the decrease
in foaming potential of opened beer. I will then use the equation produced from a
lineation of this data to calculate the decay constant of the beer foam.
Data Collection
This uncertainty was derived from the millimeter scale division on the ruler used to
obtain the height of the beer foam.
Distance from the final height in cm +/- 0.1cm
Time in sec Trial 1
Trial 2
Trial 3
0
7.1
7.3
7.0
5
7.0
7.1
6.9
10
6.9
6.9
6.8
15
6.7
6.8
6.7
20
6.6
6.6
6.5
25
6.3
6.4
6.3
30
6.0
6.1
6.1
35
5.9
5.9
5.8
40
5.7
5.7
5.4
45
5.5
5.5
5.2
50
5.0
5.2
5.0
55
4.7
4.9
4.6
60
4.2
4.6
4.4
65
3.8
4.1
3.9
70
3.1
3.4
3.2
75
2.5
2.5
2.3
80
1.8
1.7
1.5
85
1.2
0.9
1.2
90
0.2
0.1
0.0
Average
Height
7.13
7.00
6.87
6.73
6.57
6.33
6.07
5.87
5.60
5.40
5.07
4.73
4.40
3.93
3.23
2.43
1.67
1.10
0.10
ln of Avg.
Height
1.96
1.95
1.93
1.91
1.88
1.85
1.80
1.77
1.72
1.69
1.62
1.55
1.48
1.37
1.17
0.89
0.51
0.10
-2.30
Banks Ferrell
2/6/13
IB Physics 2A
Banks Ferrell
2/6/13
IB Physics 2A
Banks Ferrell
2/6/13
IB Physics 2A
Processing
Based on the Law of Exponential Decay the traditional equation can be modified to
fit this experiment by substituting 𝑁0 and N for ℎ0 and h, where h is the height of the
beer foam at a given time and ℎ0 is the initial height of the foam at t=0.
N=𝑁0 𝑒 −𝜆𝑡 → h=ℎ0 𝑒 −𝜆𝑡
This exponential relationship can be converted to a linear equation by taking the
natural logarithm of the foam height. ln(H(t)) vs time is a straight line with slope –𝜆.
ln(H(t)) = ln(H(0)) – 𝜆t
d(ln(H))/dt = –𝜆
Thus the slope from a linear best fit line will yield a proportion that can be equated
to the decay constant:
∆𝑦
∆𝑥
→
∆ln⁡(ℎ)
∆𝑡
= –𝜆
According to the linear model of the data, the best-fit slope is -0.02673⁡𝑐𝑚⁄𝑠𝑒𝑐 with
an uncertainty of 0.005852 which is equivalent to the decay constant. Alternatively
the equation can be solved numerically by plugging in the correct numbers for slope
manually:
ln⁡(7.13𝑐𝑚−0.1𝑐𝑚)
= -0.021669𝑐𝑚⁄𝑠𝑒𝑐
90𝑠𝑒𝑐
This variation in decay constant is likely due to the uncertainty of the original data.
This is compensated by the graph’s uncertainty which is very close to the difference
between the manual slope and best-fit slope.
Using the equation 𝜆𝑇1/2 =ln2 the half-life of this beer foam can be calculated as
ln⁡(2)
follows: 𝑇1/2=
𝜆
ln⁡(2)
→ 𝑇1/2=0.02673⁡+/−⁡⁡0.005852⁡= 25.9314 +/- 5.7sec
Conclusion
Based on the data obtained from measuring the foam height of the beer, it appears
that beer foam bubbles decay at a rate similar to radioactive material. The equation
N=𝑁0 𝑒 −𝜆𝑡 is a product of the Exponential Decay Law and can be rearranged to equal
the slope of its best-fit gradient. This has been done to determine the decay constant
of the foam which is 0.02673. Because of the transparency demonstrated between
beer foam decay and nuclear decay the function of beer foam decay can be used to
realistically model the decay of nuclear material in a common environment. With
this model, the half-life of the foam was found to be 26 sec using the calculated
decay constant. This calculated decay constant can be further extrapolated to
determine the original amount of material and remaining amount after a particular
time in seconds.
Banks Ferrell
2/6/13
IB Physics 2A
Evaluation
The primary uncertainty associated with the height measurements, once
propagated, ultimately skews the data. This is primarily due to the systematic error
associated with the limited scale division of the ruler used to measure the height of
foam, as it could only read down to the whole millimeter. To compensate, three
trials were conducted and averaged in order to obtain more accurate data.
Nevertheless, the reduced uncertainty yielded by the natural logarithm of the
average heights made the application of error bars and max/min gradients
irrelevant as they were all to small to see on the graph. But the large RMSE of
0.6986cm demonstrates that the linearized data fits loosely with the best-fit
primarily because of the outlining value of -2.3 which is a result of taking the natural
logarithm of 0.1. In addition, the procedure could be improved to reduce the random
errors associated with manually marking measurements over time by utilizing video
tracking software that can more accurately collect measurements over time via a
stationary video against a origin of measurement. This is a more tedious process,
however it would provide a smaller uncertainty with precise data.