Limiting Factors, Inverse Square Law & Greenhouse Economics (HT)

Exam questions often feature rate-of-reaction graphs with a plateau.

• The sloping part of the graph shows that the factor on the x-axis (e.g., light) is the limiting factor because increasing it increases the rate.
• The point where the line levels off (plateaus) indicates that the x-axis variable is no longer limiting; instead, another factor like temperature or $CO_2$ has become the limiting factor.
• In multi-line graphs comparing two temperatures (e.g., 15°C and 25°C), if the 25°C line plateaus at a higher rate than the 15°C line, it proves that temperature was the limiting factor at the lower temperature plateau.

The Inverse Square Law and Light Intensity

Every time you walk away from a street lamp at night, the light does not just fade gradually—it drops off extremely quickly. This happens because light intensity and the distance from a light source are inversely proportional. As distance increases, the light intensity decreases.

This relationship is non-linear because it follows the Inverse Square Law. This means the intensity of light is inversely proportional to the square of the distance from the source.

• The Doubling Rule: If you double the distance ($2 \times d$), the light intensity does NOT halve; it quarters ($1/2^2 = 1/4$ of the original intensity).
• The Tripling Rule: If you triple the distance ($3 \times d$), the intensity becomes nine times smaller ($1/3^2 = 1/9$).
• The Halving Rule: If you halve the distance ($1/2 \times d$), the light intensity increases by a factor of four ($2^2 = 4 \times$ original intensity).

Worked Example: Calculating Light Intensity

Calculate the light intensity at a distance of 20 cm from a lamp.

Step 1: State the general formula.

Step 2: Substitute the distance ($d = 20$) into the formula.

Step 3: Calculate the final answer, ensuring you use arbitrary units (au).

Greenhouse Economics and Yield Optimization

Why do farmers spend thousands of pounds heating greenhouses when natural sunlight is completely free? Commercial horticulturists manipulate limiting factors in controlled environments to achieve the maximum rate of photosynthesis. This faster growth results in a higher yield and greater profit.

Farmers manipulate several specific conditions inside a greenhouse:

• Heating: Paraffin or electric heaters are used in winter to increase the kinetic energy of enzymes.
• Cooling: Ventilation (vents or fans) or shades are used in summer to prevent enzymes from denaturing.
• Carbon dioxide enrichment: This is often achieved using paraffin heaters, which provide both heat and $CO_2$ (as a byproduct of combustion) simultaneously.
• Lighting: Artificial lamps extend daylight hours or maintain high light intensity on cloudy days.

Controlling these conditions is only economically viable if the cost of providing the equipment and fuel is lower than the additional profit generated. Farmers must conduct a cost-benefit analysis to find the financial balance between expensive inputs and the income from larger, faster-growing crops.

Increasing a factor beyond the point where the rate of photosynthesis plateaus is a complete waste of money. For example, if increasing $CO_2$ from 500 ppm to 600 ppm costs an extra £20 per week, but the yield graph has already plateaued, it adds cost without increasing the yield. This means the overall profit decreases.