An environmental scientist models air quality with the equation C = C₀e^(-kt). If C₀ = 100 µg/m³, k = 0.1/year, and t = 3 years, what is C? - Coaching Toolbox

Understanding the Context

Where:

• C is the air pollutant concentration at time t
  
• C₀ is the initial concentration (100 µg/m³ in this case)
  
• k is the decay constant (0.1 per year here)
  
• t is the time elapsed (3 years)

Understanding the Equation

This equation describes how concentrations of pollutants decline over time through natural processes such as chemical reactions, deposition, or dispersion—modeling a first-order decay. It’s widely used in environmental studies to forecast pollution levels and assess risks to human health and ecosystems.

Plugging in the Numbers

Key Insights

Given:

• C₀ = 100 µg/m³
  
• k = 0.1/year
  
• t = 3 years

We substitute into the formula:

C = 100 × e^(-0.1 × 3)
C = 100 × e^(-0.3)

Using a calculator, e^(-0.3) ≈ 0.7408

Thus:
C ≈ 100 × 0.7408 = 74.08 µg/m³

Final Thoughts

Interpreting the Result

After 3 years under these conditions, the modeled concentration of the pollutant drops to approximately 74.08 µg/m³. This reduction highlights how rapid natural dispersion and chemical breakdown can lower harmful levels, but also shows lingering pollution above safe thresholds—underscoring the need for continuous monitoring and effective environmental policies.

Why This Matters

Accurate modeling using equations like C = C₀e^(-kt) empowers scientists to:

• Predict future air quality based on current emissions
  
• Evaluate the effectiveness of pollution control measures
  
• Support public health interventions and urban planning

For environmental scientists, refining parameters such as k—based on local meteorology, terrain, and emission sources—remains essential for building reliable predictive models.