Calculate average and instantaneous rates from concentration-time data
Calculate average rate, instantaneous rate, or concentration changes over time.
Average Rate = -Δ[A]/Δt = -([A]₂ - [A]₁)/(t₂ - t₁)
Rate is positive (by convention, negative of concentration decrease)
Average Rate:
The rate of change in concentration over a time interval. Calculated as the slope of a secant line on the concentration vs. time graph: Rate = -Δ[A]/Δt
Instantaneous Rate:
The rate at a specific moment in time. Calculated as the slope of the tangent line to the concentration vs. time curve: Rate = -d[A]/dt (requires calculus or graphical analysis)
Sign Convention:
Rates are always positive. For reactants (decreasing concentration), we use a negative sign: Rate = -Δ[Reactant]/Δt. For products (increasing), Rate = +Δ[Product]/Δt.
Reaction rate measures how quickly reactants are consumed or products are formed in a chemical reaction. It quantifies the speed of chemical change and is expressed as the change in concentration per unit time.
Key characteristics of reaction rates:
Key Concept:
The rate of a reaction can be measured by monitoring the disappearance of a reactant or the appearance of a product over time. Different species in the same reaction may have different rates based on their stoichiometric coefficients.
The rate over a finite time interval. It's the slope of the secant line connecting two points on the concentration vs. time curve.
Formula:
Rate = -Δ[A]/Δt
= -([A]₂ - [A]₁)/(t₂ - t₁)
When to use:
The rate at a specific moment in time. It's the slope of the tangent line to the concentration vs. time curve at that point.
Formula:
Rate = -d[A]/dt
(requires calculus)
When to use:
Important Note:
As the time interval Δt becomes smaller and smaller (approaches zero), the average rate approaches the instantaneous rate. Mathematically: instantaneous rate = lim(Δt→0) Δ[A]/Δt
The decomposition of N₂O₅ in CCl₄ solution was studied. The following data were obtained:
| Time (s) | [N₂O₅] (M) |
|---|---|
| 0 | 0.200 |
| 100 | 0.180 |
| 200 | 0.162 |
| 300 | 0.146 |
Calculate the average rate of decomposition during each time interval.
Interval 1: 0-100 s
Rate = -Δ[N₂O₅]/Δt = -(0.180 - 0.200)/(100 - 0)
Rate = -(-0.020)/100 = 2.0 × 10⁻⁴ M/s
Interval 2: 100-200 s
Rate = -(0.162 - 0.180)/(200 - 100)
Rate = -(-0.018)/100 = 1.8 × 10⁻⁴ M/s
Interval 3: 200-300 s
Rate = -(0.146 - 0.162)/(300 - 200)
Rate = -(-0.016)/100 = 1.6 × 10⁻⁴ M/s
Observation:
The rate decreases as the reaction proceeds: 2.0 × 10⁻⁴ → 1.8 × 10⁻⁴ → 1.6 × 10⁻⁴ M/s. This is typical for reactions where the rate depends on reactant concentration.
For a general reaction: aA + bB → cC + dD
The rates of change for different species are related by stoichiometry:
Rate = -(1/a)Δ[A]/Δt = -(1/b)Δ[B]/Δt = (1/c)Δ[C]/Δt = (1/d)Δ[D]/Δt
Negative for reactants (decreasing), positive for products (increasing)
If N₂O₅ is disappearing at 2.0 × 10⁻⁴ M/s, then:
Pharmacologists measure how quickly drugs are metabolized (rate of disappearance from blood). This determines dosing schedules and drug effectiveness over time.
Rates of pollutant degradation help predict how long contaminants persist in water or soil. Crucial for environmental cleanup and remediation planning.
Chemical engineers optimize reaction rates to maximize product formation while minimizing costs. Reaction rates determine reactor size and production capacity.
Understanding rates of food spoilage (oxidation, bacterial growth) helps determine shelf life and optimal storage conditions. Temperature control slows reaction rates.
Rates of ozone formation and depletion in the atmosphere determine air quality. Understanding these rates is essential for pollution control and climate modeling.
Measuring reaction rates catalyzed by enzymes reveals enzyme efficiency and mechanism. Critical for drug design, diagnostics, and understanding metabolism.
Forgetting the negative sign when calculating rate from reactant disappearance.
Correct: Rate = -Δ[A]/Δt (negative sign makes it positive)
Mixing seconds and minutes, or using time in hours with concentration change per second.
Correct: Convert all times to the same unit before calculating
Not dividing by coefficients when relating rates of different species.
Correct: For 2A → B, if A disappears at 0.4 M/s, B appears at 0.2 M/s
Using average rate formula when instantaneous is needed, or vice versa.
Correct: Read the problem carefully to determine which is requested
Average Rate = -Δ[A]/Δt
= -([A]₂ - [A]₁)/(t₂ - t₁)
Instantaneous Rate = -d[A]/dt
(Negative for reactants,
positive for products)
Concentration: M (mol/L)
Time: s, min, h
Rate:
Reactants: Rate = -Δ[A]/Δt
(Negative sign needed)
Products: Rate = +Δ[P]/Δt
(Already positive)
All rates are positive!
aA + bB → cC
Rate = -(1/a)Δ[A]/Δt
= -(1/b)Δ[B]/Δt
= (1/c)Δ[C]/Δt