INTERNATIONAL BACCALAUREATE

BIOLOGY LAB REPORT

EFFECT OF SURFACE AREA TO VOLUME RATIO ON THE RATE OF DIFFUSION

Introduction

Diffusion is an important physiological process for cells that allows important molecules such as
gases, water, nutrients and waste products to move from their region of higher concentration to a
region of lower concentration. The relationship between surface area and volume is nonlinear
and in biology this has profound implications. This experiment aims to quantify the relationship
between SA/V ratio and the rate of diffusion, using an agar cube as a model for a cell and hence
exhibiting how cellular size and structure impacts diffusion efficiency.

Background Information

In biochemistry, when two regions with different concertation potentials come in contact with
one another (one at a higher concentration than another), a concentration gradient is generated.
The rate of diffusion of molecules between two substances is directly proportional to the
concentration in between them. Diffusion can be defined as the process of movement of
molecules from their region of higher concentration to their region of lower concentration.
Diffusion is a process that occurs in all cells, prokaryotic and eukaryotic.

Since diffusion is a passive process and does not use energy to occur, cells prefer it to any other
physiological to obtain useful products such as water, proteins, amino acids, glucose and oxygen
as well as get rid of waste substances such as urea and carbon dioxide. Many factors influence
the rate of diffusion of molecules-temperature, humidity, thickness of the membrane, size of the
molecules, concentration gradient and most importantly, surface area to volume ratio.

Cells are bound by geometric restrictions such as their surface area by volume ratio, which is
important for many cellular functions. A mammalian cell is near spherical in nature and so, the
same way experimentation has proved that surface area by volume ratio has an inverse relation
with the size of a sphere, a cell is expected to mimic this property. Experiments on surface area
to volume ratio can date as far back as the 17th century, when early work on osmosis and
diffusion done by a German Botanist, Wilhelm Pfeffer studied how solvents and solutes moved
in and out of a plant cell and hence laid the groundwork for further experiments on cell size and
indirectly highlighted the importance of surface area. In the 19th century, William Thompson and
others formalized the concept in thermodynamics and geometry that described how surface area
increases by square of the radius of a sphere why volume increases by a cube. This principle was
later essential for understanding the similar behavior in cells. By the late 19th and early 20th
century, biologists began experimenting on how cellular size impacts cell functions and
efficiency. Smaller cells proved to be more efficient in exchange of substances across their
membrane as compared to larger cells.

Many laboratory experiments have been developed which help us demonstrate the relationship
between surface area and volume ratio and diffusion, such as the potato osmosis experiment,
gelatin diffusion with food coloring, yeast cell respiration experiment, dialysis tubing experiment
and most notably, the agar cube diffusion experiment. The agar cube diffusion experiment is a
simple and effective way to demonstrate the relationship between surface area-to-volume ratio
and diffusion. It involves placing agar cubes of different sizes, infused with a pH indicator, into
hydrochloric acid and recording the time they take to decolorize. The cubes represent cells, with
varying SA/V ratio affecting the rate of diffusion. This experiment is inexpensive, safe, and
allows for precise control of variables, making it replicable and quantifiable, and a great tool for
studying diffusion.

Variables

Variable Description
Independent Variable The size of the Agar Cubes Manipulate the independent variable
(SA/V ratio) by changing the dimensions of the
agar cubes and so, changing the
SA/V ratio of each of these cubes. It
is expected to have an impact on the
dependent variable.
Dependent Variable Time taken in seconds for Measure the dependent variable, the
the Agar cube to decolorize rate of diffusion measured in terms
(rate of diffusion) of time taken in seconds for an agar
cube to decolorized. It is expected to
change with change in the dependent
variable (different SA/V ratio with
different sizes).
Controlled Variables
Controlled Variable Description Impact
Temperature of the The temperature at which the Temperature affects diffusion rate,
surroundings entire experiment should be with higher temperatures speeding it
conducted should be constant, up and lower temperatures slowing it
in this case it was room down. Inconsistent temperatures
temperature- 24 ℃. across cube sizes make it harder to
link SA/V ratio with diffusion rate.
Concentration of The concentration of the HCl A higher or lower concentration of
HCl used used in every test tube with HCl would change the rate of
different sizes of the cubes diffusion. A higher concentration of
should be kept constant, the HCl increases the diffusion rate,
concentration was 0.1M (mol causing the cube in the stronger
−3
dm ) solution to decolorize faster. This can
make it harder to observe the direct
relationship between SA/V ratio and
diffusion rate.

Volume of HCl used The volume of HCl in each If the volume of the HCl varies, the
test tube should be the same time taken for each cube to
and every cube should be decolorize would vary. If some cubes
immersed in the HCL. are immersed and the others are not,
Volume was 10cm3 . it reduces diffusion efficiency and
hence impact the experiment.
The kind of agar The kind of agar used to Different types of agar cubes or gels
 used prepare all different sizes of have different porosity or
the cube should be kept permeability. Denser types would
constant. slow down diffusion while less dense
agar allows the diffusion to occur
faster.
Initial pH level and The volume of the If the initial pH or color of the agar
color intensity phenolphthalein used while cubes differs, it becomes difficult to
(volume of preparing the different sizes accurately measure decolorization
phenolphthalein of the cube needs to be kept time. Larger cubes with more
used) constant. They should have phenolphthalein will take longer to
the same intensity of color. decolorize than smaller ones.

Table 1: Variables in the experiment

Research Question

What is the effect of surface area to volume ratio on the rate of diffusion measured in terms of
time taken in seconds for an agar cube to decolorize when placed in HCl?

Hypothesis:

H 0- As the size of the agar cube and hence the surface area to volume ratio of the cube changes,
there is no effect on the rate of diffusion measured in terms of time taken in seconds for the agar
cube to decolorize. All cubes irrespective of their size take the same amount of time to
decolorize.

H 1- As the size of the agar cube increases, the surface area to volume ratio decreases and the rate
of diffusion decreases i.e. the time taken for the agar cubes to decolorize increases. Smaller
cubes take lesser time to decolorize as compared to bigger cubes.

Apparatus:
 Apparatus Quantity
Agar gel of 1cm x1
thickness in a petri
dish
Agar gel of 0.5cm x1
thickness in a petri
dish
Scalpel x1
Ruler (30cm) x1
White Tile x1
Plastic Graph Sheet x1
Stopwatch x1
Measuring cylinder x1
(10cm3 ¿
Plastic dropper x1
Goggles x1
Lab coat x1
Gloves x1
Table 2: Apparatus

Methodology:

1. Carefully remove the agar gel from the petri dish with the help of a scalpel.
2. Place your plastic graph sheet on a white tile.
3. Place the agar gel on the white tile with the plastic graph sheet .
4. Using a ruler and scalpel, cut the agar gel into cubes of different dimensions:
- 2cm x 2cm x 1cm
- 2cm x 1cm x 0.5cm
- 1cm x 1cm x 1cm
- 1x1x0.5
- 1cm x 0.5cm x 0.5cm
- 0.5cm x 0.5cm x 0.5cm
5. Measure 10cm3 of 0.1 M HCl solution using a measuring cylinder. This solution will
serve as the diffusing agent.
6. Add 10cm3 to 5 different test tubes.
7. Immerse the largest cube in HCl first.
8. Start the stopwatch.
9. At 10 second intervals, add each consequent cube to a test tube.
10. Note down the times at which each cube was immersed.
11. Observe the colour change. As the HCl diffuses into the cubes, they begin to turn
colourless from pink.
12. Note down the time that each cube turns completely colourless.
13. Repeat the experiment 5 to 6 times for accurate results.

Safety and Ethical Considerations

 - Wear gloves and goggles when handling Hydrochloric Acid (HCl); it is corrosive
even at low concentrations. Work in a well-ventilated area to avoid inhaling fumes.
Disposal of chemicals after completing the experiment should be done properly
following the laboratory’s guidelines. Do not pour HCl down the drain unless your
lab supervisor permits you to.
- Use caution with sharp tools like scalpels. Cut on a stable surface and wear gloves to
avoid injury
- Handle glassware carefully to prevent breakage. Clean up and dispose of broken glass
immediately.
- There are no ethical concerns as the experiment involves non-living materials.. .
- Dispose of waste responsibly and minimize environmental impact. Use only
necessary amounts and preserve materials if possible.

Analysis

Qualitative data

- The agar cubes that were originally pink in color due to the presence of
phenolphthalein, when immersed in HCL of concentration 0.1M, begin to decolorize.
- The outer edges of the cube turn colorless first and this decolorization moves inward
towards the center of the cube till the entire thing turns colorless.
- The smaller cubes turn colorless much faster as compared to the larger cubes, which
suggests higher diffusion efficiency. It takes the HCl longer time to penetrate into the
center of the larger cubes.
- The agar cubes remain firm and maintain their shape throughout the experiment,
There is no noticeable change in the structure, size or texture of the agar cubes other
than the color fading,
- The HCl remains colorless throughout the entire experiment. There is no bubbling,
precipitation or any other chemical reactions that can be observed.

Quantitative Data

SA:VOL Time taken for each agar block to turn completely colourless/ sec
Ratio Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7
6:1 782 893 790 840 623 680 720
7:1 505 505 235 225 240 438 327
8:1 457 457 265 271 256 358 464
10:1 307 307 192 193 155 250 335
12:1 246 246 163 157 162 194 145
Table 3: Raw Data Collection

SA:VOL Time taken for each agar block to turn completely white/ sec Average Standard
Ratio ( ±0.01 sec) (±0.01 sec) Deviation
 Trial (±0.01
Trial 1 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7 sec)
761.142857
6:01 782 893 790 840 623 680 720 1 93.303141
353.571428
7:01 505 505 235 225 240 438 327 6 127.26856
361.142857
8:01 457 457 265 271 256 358 464 1 97.801548
248.428571
10:01 307 307 192 193 155 250 335 4 69.93296
187.571428
12:01 246 246 163 157 162 194 145 6 42.570949
Table 4: Processed data with outliers

SA:VR AVERAGE Time/

800 sec(±0.01)
750
700
650
Average time taken to decolorize

600
550
500
450
400
350
±0.01s

300
250
200
150
100
50
0
1 2 3 4 5
Surface area to volume ratio
Graph 1: SA/V ratio by Average time taken for each cube to decolorize (with outliers)

Time taken for each agar block to turn completely colourless/sec

(±0.01 sec)
SA:VOL
Standard
Ratio Average
Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Trial 7 deviation
(±0.01)
(±0.01)
739.166
6:01 782 893 790 840 623 680 720
667 79.93602
7:01 505 505 235 225 240 438 327 375 124.8183
8:01 457 457 265 271 256 358 464 344 94.91891
10:01 307 307 192 193 155 250 335 234 64.187223
177.833
12:01 246 246 163 157 162 194 145
333 37.12367
Table 4: Processed data without outliers
 SA:VR AVERAGE Time
sec(±0.01)
750
700
Average Time taken to decolorize ±0.01 )

650
600
550
500
450
400
350
300
250
200
150
100
50
0
1 2 3 4 5
Surface Area to Volume Ratio
Graph 2: SA/V ratio by Average time taken for decolorization (without outliers)

Table 5: Calculation of f-value from ANOVA test (processed data without outliers)
F value= 39.54; P value= 0

Statistical Analysis

When outliers are included in the data, it does not show a reducing change with the change in
SA/V ratio to volume ratio and instead shows an irregular pattern. When the data is calculated
without the outliers, you can observe a decreasing trend in the average time taken for the
cube to decolorize as the SA/V ratio increases or the size of the cube decreases.

Between ratios, there are notable differences in the standard deviation values. For example, even
in the absence of outliers, the 7:1 ratio exhibits significant variability in diffusion durations, as
seen by its high standard deviation of 124.82 seconds. Conversely, the 12:1 ratio exhibits a
significantly lower standard deviation (37.12 seconds), suggesting more uniform diffusion times.

Multiple trials are used to establish the average time for each SA ratio. The average takes into
account the uncertainty from each individual measurement, even if each trial has an error of
±0.01 seconds. Nonetheless, the average results are not greatly impacted by this uncertainty
because it is relatively modest. The variability in the time measurements is already taken into
account by the standard deviation. Greater variety in the times observed is indicated by larger
standard deviations (e.g., 124.82 seconds for the 7:1 ratio), which probably exceed the miniscule
measurement error of ±0.01 seconds.

The error bars are similar in length and do not overlap with one another indicating a natural
variation in the experiment. A high F value suggests that variance between groups is much
higher than variance within the groups. A p value of 0 suggests that the hypothesis is not null.

Conclusion

The results we obtain from this experiment successfully demonstrate the significant impact that
surface area to volume ratio has on the rate of diffusion. The smaller agar cubes had a smaller
SA/V ratio and hence showed faster diffusion. Conversely, the larger cubes with a SA/V ratio
showed lesser diffusion efficiency. This confirms the hypothesis that smaller cubes with a higher
SA would allow faster diffusion. The greater surface area compared to the inner volume of the
cubes means that the HCl takes longer to penetrate to the center of the larger cubes; this can be
qualitatively observed through observation as well. Therefore, with decrease in size of the cells
(inversely related to SA/V ratio) the diffusion rate increases. The rate of diffusion can be given
by the formula:
1
Rate of diffusion= decolorize when placed∈ HCl ¿
timetaken for the cube ¿

The relationship between surface area to volume ratio and time taken for diffusion can
quantitatively be expressed as:
1
Time taken for diffusion∝ volume ratio ¿
surface area ¿

These findings have important implications in a biological context. Cells rely primarily on
diffusion for the exchange of gases, nutrients and waste products and other important substance.
Smaller cells, such as red blood cells, alveoli, and microvilli, have high surface area-to-volume
ratios, which improve diffusion for gas exchange, nutrient absorption, and waste removal. As
cells grow larger, their SA/V ratio decreases, slowing down diffusion, which can hinder their
ability to meet metabolic demands. To overcome this, cells may divide or develop adaptations to
maintain a high SA/V ratio. This ratio is also important for heat dissipation in larger organisms,
helping them regulate temperature more effectively.

To conclude with, this experiment reinforces the critical relationship between the surface area,
volume and rate of diffusion of a substance. The results align with already established biological
principles but the visual representation helps us understand why exactly is it that cells maintain a
SA/V ratio. The findings of the experiment emphasize the importance of diffusion in both
microscopic and macroscopic biological processes and delves deeper into the concept of
maintaining diffusion efficiency.

Evaluation

Although this experiment successfully demonstrates the linear relationship between surface area
and volume ratio and rate of diffusion and gives us a great understanding of both biological and
physical processes, it has several strengths as well as limitations. To begin with, the experiment
has a clear hypothesis and we can clearly state that as the SA/V ratio increases (sizes of the agar
cubes decreases) time taken for diffusion (time taken in seconds for the agar cube to decolorize).
The results align with theoretical expectations and hence the analysis of data is straightforward
and fast.

Another strength is that this experiment is ideal to understand the relationship between these
factors visually. The use of phenolphthalein allowed for easy observation of the process of
diffusion and hence draw a comparison between the diffusion rates of different sizes of the cubes
and quantify the relation between the two variables in question. It was also extremely helpful to
collect qualitative data.

Additionally, this experiment allows us to control many variables which if not moderated can
have an unwanted effect on the independent and hence the dependent variable. This hence helped
us understand that any change in the dependent variable (time taken for diffusion measured in
terms of time in seconds) is due to manipulation of the independent variable (surface rea to
volume ratio by change in the dimensions of the cube) and no confounding variable.

However, a limitation would be that it is difficult to achieve complete consistency in size and
shape when the cubes were cut with a scalpel. Differences in cube dimensions, even if modest,
might cause disparities in the diffusion findings. Furthermore, the idea that the cubes were
precisely geometric forms would not accurately capture the complexity of diffusion in biological
systems, where uniform structures are unusual. There is hence a large scope for random errors in
this experiment.

Even if the HCl content and volume were regulated, the outcomes might still be impacted by
uncontrollable elements such air currents or variations in the agar's composition that cause
irregularities in the diffusion. Although they weren't regulated in this experiment, these variables
might have an impact on the data's variability.

Overall, this experiment is a well designed and efficient experiment. The limitations would not
impact the overall trend of the rates of diffusion unless they are extremely distinct. It has many
merits and hence answers the research question appropriately. As surface area to volume ratio
increases (size decreases) the time taken in seconds for an agar cube to turn colorless decreases.
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