investigate the effect of changing surface area to volume ratio on diffusion using agar blocks of different sizes

Movement into and out of Cells (Cambridge AS & A Level Biology – Topic 4.2)

1. Objective

To investigate how the surface‑area‑to‑volume (SA:V) ratio of agar blocks influences the rate of diffusion of a solute into the block, and to relate the findings to the wider concepts of simple diffusion, facilitated diffusion, osmosis, active transport and vesicular transport.

2. Syllabus Coverage

3. Background Theory

3.1 Simple Diffusion

• Definition: Net movement of particles from an area of higher concentration to an area of lower concentration, occurring directly through a permeable membrane or a gel matrix.
• Energy/gradient link (AO1): No external energy is required; the concentration gradient provides the driving force.

3.2 Factors affecting the rate of diffusion (AO1)

• Concentration gradient (ΔC)
• Temperature (higher temperature ↑ kinetic energy)
• Size and polarity of the diffusing molecule
• Surface‑area‑to‑volume ratio of the diffusing object
• Medium through which diffusion occurs (e.g., agar, water, air)

3.3 Quantitative SA:V relationship

For a regular geometric shape the SA:V ratio can be expressed mathematically. The larger the ratio, the shorter the average diffusion path, so the faster the net flux (AO2).

Cube (side = l)

\[

A = 6l^{2},\qquad V = l^{3},\qquad \frac{A}{V}= \frac{6}{l}

\]

Sphere (radius = r)

\[

A = 4\pi r^{2},\qquad V = \frac{4}{3}\pi r^{3},\qquad \frac{A}{V}= \frac{3}{r}

\]

Red blood cell (approximated as a biconcave disc)

• Typical dimensions: diameter ≈ 8 µm, thickness ≈ 2 µm.
• Surface area ≈ 140 µm², volume ≈ 90 µm³ (calculated using the disc‑volume formula).
• SA:V ≈ \(\dfrac{140}{90}\) ≈ 1.56 µm⁻¹. Rounded to 0.75 µm⁻¹ in many textbooks because the cell is slightly flattened; the worked example below shows the calculation.

Worked example (RBC)

\[

\begin{aligned}

\text{Surface area (approx.)}&= 2\pi r\,(r+h) \\

&\approx 2\pi (4\,\mu\text{m})(4\,\mu\text{m}+1\,\mu\text{m}) \\

&\approx 140\,\mu\text{m}^{2} \\

\text{Volume (approx.)}&= \pi r^{2}h \\

&\approx \pi (4\,\mu\text{m})^{2}(2\,\mu\text{m}) \\

&\approx 100\,\mu\text{m}^{3} \\

\frac{A}{V}&\approx \frac{140}{100}=1.4\,\mu\text{m}^{-1}

\end{aligned}

\]

3.4 Facilitated Diffusion

• Uses specific carrier or channel proteins to allow polar or large molecules (e.g., glucose, ions) to cross the membrane down their concentration gradient.
• Energy/gradient link: No ATP is required; the concentration gradient provides the driving force, but the rate is limited by the number and affinity of carriers.

3.5 Osmosis

Diffusion of water through a selectively permeable membrane from a region of higher water potential (Ψ) to lower Ψ.

\[

\Psi = \Psi{s} + \Psi{p}

\]

• Solute potential (Ψ_s) \( \Psi_{s}= -i C R T\) (Units: bar).
  

  i = ionisation factor (≈ 1 for non‑electrolytes), C = molarity (mol L⁻¹), R = 0.0831 L bar mol⁻¹ K⁻¹, T = temperature in kelvin.

• Pressure potential (Ψ_p) turgor pressure in living cells; in the agar‑cube set‑up it can be assumed to be 0 bar.

3.6 Active Transport

• Moves substances against their concentration gradient and therefore requires energy, usually supplied by ATP.
• Typical example required by the syllabus: the Na⁺/K⁺‑pump (3 Na⁺ out, 2 K⁺ in per ATP hydrolysed).
• Energy/gradient link: ATP provides the energy to create an electrochemical gradient that the cell can then use for secondary active transport.
• Relevance to the practical: The agar‑cube diffusion experiment does not involve a plasma membrane or ATP, so active transport cannot be observed directly. It is mentioned to illustrate why diffusion is the dominant mechanism for small, uncharged molecules in this set‑up.

3.7 Endocytosis & Exocytosis

• Endocytosis – uptake of material by invagination of the plasma membrane (phagocytosis, pinocytosis). Requires ATP to remodel the membrane.
• Exocytosis – release of material by fusion of intracellular vesicles with the plasma membrane (e.g., hormone secretion). Also ATP‑dependent.
• These vesicular processes are much slower than simple diffusion for small molecules and cannot be demonstrated with agar blocks, which lack a lipid bilayer.

4. Hypothesis

If the SA:V ratio of the agar blocks is increased (by using smaller cubes or a spherical shape), then the diffusion of the solute into the blocks will be faster, producing a greater colour/intensity change over a fixed time period.

5. Materials

• Agar powder (1 % w/v solution)
• Distilled water
• Food‑colouring (blue) or potassium permanganate solution (diffusible solute)
• Cubic moulds: 1 cm, 2 cm, 3 cm, 4 cm sides
• Silicone beads or other moulds for ~1 cm diameter spheres (optional extension)
• Ruler or digital caliper (0.1 mm accuracy)
• Stopwatch or timer
• Clear plastic cuvettes or beakers (≥ 50 mL)
• Digital camera or smartphone (for colour‑intensity recording)
• Analytical balance (0.01 g)
• Thermometer (record temperature, 20 ± 2 °C)
• Safety equipment: lab coat, goggles, heat‑resistant gloves

6. Method

6.1 Core Practical – Diffusion & SA:V

1. Prepare a 1 % agar solution (10 g agar in 1 L water). Heat until fully dissolved, then cool to ≈ 45 °C.
2. Pour the molten agar into the four sets of cubic moulds (1 cm, 2 cm, 3 cm, 4 cm). Allow to set at room temperature (≈ 20 °C) for at least 30 min.
3. Measure the actual side length l of each cube to the nearest 0.1 mm and record.
4. Place each cube in a separate cuvette containing 50 mL of the dye solution (same concentration for all cuvettes).
5. Start the timer as soon as the cubes are immersed.
6. At 5‑minute intervals (5, 10, 15, 20, 25, 30 min):

   • Gently remove a cube, blot excess surface liquid with filter paper.
   • Measure the depth of colour penetration from the surface toward the centre:

     • Option A: use a ruler placed against the side of the cube.
     • Option B: photograph the cube against a white background, then analyse the image with free software (e.g., ImageJ) to obtain the distance.

   • Return the cube to its cuvette.

7. Stop the experiment when the colour reaches the centre of the smallest cube or after 30 min, whichever occurs first.

6.2 Extension – Facilitated Diffusion (optional)

1. Prepare two additional agar blocks (2 cm × 2 cm × 2 cm):

   • Block A – plain agar.
   • Block B – agar containing 0.5 % w/v glucose (provides substrate for a glucose‑specific carrier).

2. Immerse both blocks in 0.1 M glucose solution that also contains a non‑reactive blue dye.
3. Record colour penetration as in 6.1. Faster penetration in Block B would suggest the involvement of a carrier protein (facilitated diffusion).

6.3 Extension – Osmosis (optional)

1. Prepare three sets of identical agar cubes (2 cm side).
2. Place each set in 50 mL of sucrose solution of 0 %, 5 % and 10 % (w/v) respectively.
3. After 15 min, gently blot each cube and weigh to the nearest 0.01 g.
4. Calculate the mass change (Δm) as an indicator of water uptake (Δm > 0) or loss (Δm < 0).
5. Convert each sucrose % w/v to molarity (M) using the density of water (≈ 1 g mL⁻¹):

   • 5 % w/v = 5 g / 100 mL = 0.05 g mL⁻¹ → 0.05 g mL⁻¹ ÷ 342 g mol⁻¹ ≈ 0.146 M
   • 10 % w/v = 0.10 g mL⁻¹ → 0.10 ÷ 342 ≈ 0.293 M

6. Compute the external solute potential (Ψ_s) for each solution using

   \[

   \Psi_{s}= -i C R T

   \]

   (i = 1, R = 0.0831 L bar mol⁻¹ K⁻¹, T = 298 K). Record Ψ_s in bar.

7. Assume the internal pressure potential (Ψ_p) of the agar cube is 0 bar, so Ψ_cube = Ψ_s(internal). Compare the calculated Ψ values with the direction of water movement inferred from Δm.

7. Variables

• Independent variable: SA:V ratio (changed by cube size or shape).
• Dependent variable: Rate of diffusion – measured as depth of colour penetration (cm) per unit time.
• Controlled variables: Agar concentration, temperature, dye concentration, volume of external solution, observation intervals, and accuracy of side‑length measurement.

8. Data Collection Tables

8.1 Cube Diffusion Data (core practical)

8.2 Facilitated Diffusion (optional)

8.3 Osmosis (optional)

9. Calculations

1. Surface area, volume and SA:V ratio (cubes):

   \[

   A = 6l^{2},\qquad V = l^{3},\qquad \frac{A}{V}= \frac{6}{l}

   \]

2. Sphere (if used):

   \[

   A = 4\pi r^{2},\qquad V = \frac{4}{3}\pi r^{3},\qquad \frac{A}{V}= \frac{3}{r}

   \]

3. Rate of diffusion (approximation):

   \[

   \text{Rate} = \frac{\text{Depth of colour (cm)}}{\text{Time (min)}}

   \]

   Calculate for each cube at 15 min and 30 min, then plot against the SA:V ratio.

4. Osmosis – Solute potential:

   \[

   \Psi_{s}= -i C R T

   \]

   • Units: bar (1 bar ≈ 10⁵ Pa).
   • Convert % w/v to molarity as shown in 6.3.

5. Water potential of agar cube:

   \[

   \Psi{\text{cube}} = \Psi{s}(\text{internal}) + \Psi_{p}(\text{internal})

   \]

   • For the experiment, Ψ_p ≈ 0 bar, so Ψ_cube ≈ Ψ_s(internal).
   • Compare this value with the external Ψ (calculated from the sucrose solution) to predict the direction of water movement.

10. Graphs and Data Presentation

• Scatter plot: SA:V ratio (x‑axis) vs. depth of colour after 15 min (y‑axis). Add a line of best fit; a positive linear relationship supports the hypothesis.
• Bar chart (optional): Diffusion depth for plain agar vs. glucose‑enriched agar (facilitated diffusion).
• Line graph (osmosis): Mass change (Δm) on the y‑axis against sucrose concentration (% w/v) on the x‑axis, with calculated Ψ_s values indicated.

11. Safety Considerations

• Handle hot agar (≈ 80 °C) with heat‑resistant gloves.
• Wear safety goggles and a lab coat when working with dyes or potassium permanganate.
• Use tongs or forceps to avoid direct skin contact with hot agar.
• Dispose of dye solutions according to school chemical‑waste procedures.
• Clean spills immediately to prevent staining of work surfaces.

12. Analysis and Discussion (AO2)

Answer the following questions in your lab report:

1. Describe the relationship between SA:V ratio and diffusion rate. Does your data support the hypothesis? Use the scatter plot to justify your answer.
2. Explain any deviations from the expected linear trend (e.g., measurement error, temperature fluctuations, edge effects, or differences in agar density).
3. For the osmosis extension, calculate Ψ_s for each sucrose solution, state the assumed Ψ_p, and discuss whether the direction of water movement inferred from Δm matches the predicted direction based on water‑potential gradients.
4. Explain why active transport, endocytosis and exocytosis are not observed in this agar‑cube experiment, linking each process to the requirement for a lipid bilayer and ATP.
5. Evaluate the facilitated‑diffusion extension: does the presence of glucose in the agar block increase the rate of dye penetration? Relate your observation to the role of carrier proteins.

13. Quick Audit of Syllabus Alignment