Pharmaceutics
Drug Stability & Kinetics Calculator
Turns a Time vs Absorbance stability practical into % drug remaining, fits zero-, first- and second-order plots, picks the order by R², and works out k, half-life and shelf life (t90). Opens with the Procaine HCl practical at 50°C.
Experimental inputs
Printed on the result card and lab record.
An experimental condition — shown throughout, not used in the maths.
The absorbance taken as 100 % drug.
k, t½ and t90 are reported in this unit.
Experimental data
Time | Absorbance — edit any value, add or delete rows. Everything below recalculates as you type.
| # | Time(min) | Absorbance(AU) | Remove |
|---|---|---|---|
| T1 | |||
| T2 | |||
| T3 | |||
| T4 | |||
| T5 | |||
| T6 |
Load an example data set (you can edit every value)
Stability result
Procaine HCl stability — 50°C
Determined kinetic order
Zero order
R² = 0.9927
Shelf life (t90) at 50°C
8.839min
- Temperature
- 50°C
- Initial absorbance
- 0.597
- Final % drug remaining
- 50.08 %
- Determined kinetic order
- Zero order
- R²
- 0.9927
- Rate constant
- 1.1314 % min⁻¹
- Half-life
- 44.19 min
- Shelf life (t90)
- 8.839 min
Experimental estimate at 50°C — not the labeled product shelf life.
% drug remaining
% Drug Remaining = (Aₜ / A₀) × 100 — Beer–Lambert, A ∝ C.
| Time (min) | Absorbance | % Drug Remaining |
|---|---|---|
| 5 | 0.470 | 78.73 % |
| 10 | 0.425 | 71.19 % |
| 15 | 0.398 | 66.67 % |
| 20 | 0.371 | 62.14 % |
| 25 | 0.325 | 54.44 % |
| 30 | 0.299 | 50.08 % |
Transformed data
Y₀ = % Remaining (zero order) · Y₁ = ln(% Remaining) (first order) · Y₂ = 1/% Remaining (second order).
| Time (min) | % Remaining | ln(% Remaining) | 1/% Remaining |
|---|---|---|---|
| 5 | 78.73 | 4.36599 | 0.0127021 |
| 10 | 71.19 | 4.26534 | 0.0140471 |
| 15 | 66.67 | 4.19971 | 0.015 |
| 20 | 62.14 | 4.12946 | 0.0160916 |
| 25 | 54.44 | 3.99708 | 0.0183692 |
| 30 | 50.08 | 3.9137 | 0.0199666 |
Rounded for display; the regressions use full precision.
Kinetic Model Analysis
Each plot is fitted by least squares; the highest R² decides the order.
| Order | Plot | Slope | Intercept | R² | k |
|---|---|---|---|---|---|
| Zero | % Remaining vs Time | −1.13137 | 83.6739 | 0.992717 | 1.1314 % min⁻¹ |
| First | ln(% Remaining) vs Time | −0.0179228 | 4.45886 | 0.990160 | 0.017923 min⁻¹ |
| Second | 1/% Remaining vs Time | 0.000287887 | 0.0109914 | 0.978933 | 0.00028789 %⁻¹ min⁻¹ |
Determined Order of Kinetics: Zero order
Best R²: 0.992717 · ahead of first order by 0.0026
Graphical Kinetic Analysis
Experimental data at 50°C, plotted from the values above. Hover or tap a point for its exact value.
Effect of Time on % Procaine HCl Remaining at 50°C
Trend: the zero-order regression, drawn back as % remaining.
ExperimentalZero-order trendZero-Order Kinetic PlotBest fit · highest R²
y = −1.13137x + 83.6739 · R² = 0.9927y = %R, x = time (min)
ExperimentalRegression lineFirst-Order Kinetic Plot
y = −0.0179228x + 4.45886 · R² = 0.9902y = ln(%R), x = time (min)
ExperimentalRegression lineSecond-Order Kinetic Plot
y = 0.000287887x + 0.0109914 · R² = 0.9789y = 1/%R, x = time (min)
ExperimentalRegression lineGraph Analysis
- Zero-order R²
- 0.9927
- First-order R²
- 0.9902
- Second-order R²
- 0.9789
Best-fit model
Zero order
Highest R²
0.9927
Regression equation
y = −1.13137x + 83.6739y = %R, x = time (min)
The kinetic model with the highest R² provides the best linear fit to the experimental data and is therefore selected as the experimentally determined reaction order.
Rate constant, half-life and shelf life
From the zero-order slope, with C₀ = 100 % because the data are % remaining.
Rate Constant Calculation
Slope = −1.13137
Therefore:
k = −slope
k = −(−1.13137)
k = 1.13137
Units = % min⁻¹
Half-life (t½)
t½ = C₀ / 2k
t½ = 100 / 2k
t½ = 100 / (2 × 1.13137)
t½ = 44.1942 min
Shelf life (t90)
t90 = 0.10 C₀ / k
t90 = 10 / k
t90 = 10 / 1.13137
t90 = 8.83883 min
Shelf Life (t90) at 50°C
8.839 min
Final interpretation
At 50°C, the % drug remaining of Procaine HCl decreases with increasing time, indicating degradation of the drug. The reaction order is determined by comparing the linearity (R²) of zero-, first-, and second-order kinetic plots.
The experimentally determined order is zero order with R² = 0.9927.
At 50°C, the calculated rate constant is k = 1.1314 % min⁻¹.
The estimated half-life is 44.19 min.
The estimated shelf life (t90) is 8.839 min.
Experimental estimate, not a labeled shelf life
Complete Practical Calculation
Every step, exactly as it goes into the lab record.
01Step 1 — Initial absorbance
02Step 2 — Calculate % drug remaining = (Aₜ / A₀) × 100
At 5 min: % Remaining = (0.470 / 0.597) × 100 = 78.73 %
At 10 min: % Remaining = (0.425 / 0.597) × 100 = 71.19 %
At 15 min: % Remaining = (0.398 / 0.597) × 100 = 66.67 %
At 20 min: % Remaining = (0.371 / 0.597) × 100 = 62.14 %
At 25 min: % Remaining = (0.325 / 0.597) × 100 = 54.44 %
At 30 min: % Remaining = (0.299 / 0.597) × 100 = 50.08 %
Beer–Lambert law under constant analytical conditions: A ∝ C, so the absorbance ratio is the concentration ratio.
03Step 3 — Transformed data
| Time (min) | % Remaining | ln(% Remaining) | 1/% Remaining |
|---|---|---|---|
| 5 | 78.73 | 4.36599 | 0.0127021 |
| 10 | 71.19 | 4.26534 | 0.0140471 |
| 15 | 66.67 | 4.19971 | 0.015 |
| 20 | 62.14 | 4.12946 | 0.0160916 |
| 25 | 54.44 | 3.99708 | 0.0183692 |
| 30 | 50.08 | 3.9137 | 0.0199666 |
Values are rounded here for display only; the regression uses them at full precision.
04Step 4 — Linear regression (least squares, x = time)
Zero order — % Remaining vs Time (n = 6)
Σx = 105 Σy = 383.25 Σxy = 6,211.89 Σx² = 2,275
m = [nΣxy − (Σx)(Σy)] / [nΣx² − (Σx)²] = −2,969.85 / 2,625 = −1.13137
b = (Σy − mΣx) / n = 83.6739
y = −1.13137x + 83.6739 (y = %R, x = time (min))
SSE = Σ(y − ŷ)² = 4.10817 SST = Σ(y − ȳ)² = 564.108
R² = 1 − SSE/SST = 1 − 4.10817 / 564.108 = 0.992717
First order — ln(% Remaining) vs Time (n = 6)
Σx = 105 Σy = 24.8713 Σxy = 427.406 Σx² = 2,275
m = [nΣxy − (Σx)(Σy)] / [nΣx² − (Σx)²] = −47.0473 / 2,625 = −0.0179228
b = (Σy − mΣx) / n = 4.45886
y = −0.0179228x + 4.45886 (y = ln(%R), x = time (min))
SSE = Σ(y − ŷ)² = 0.00139659 SST = Σ(y − ȳ)² = 0.141933
R² = 1 − SSE/SST = 1 − 0.00139659 / 0.141933 = 0.990160
Second order — 1/% Remaining vs Time (n = 6)
Σx = 105 Σy = 0.0961766 Σxy = 1.80904 Σx² = 2,275
m = [nΣxy − (Σx)(Σy)] / [nΣx² − (Σx)²] = 0.755704 / 2,625 = 0.000287887
b = (Σy − mΣx) / n = 0.0109914
y = 0.000287887x + 0.0109914 (y = 1/%R, x = time (min))
SSE = Σ(y − ŷ)² = 7.80321 × 10⁻⁷ SST = Σ(y − ȳ)² = 3.70400 × 10⁻⁵
R² = 1 − SSE/SST = 1 − 7.80321 × 10⁻⁷ / 3.70400 × 10⁻⁵ = 0.978933
| Time (min) | Residual, zero (y − ŷ) | Residual, first (y − ŷ) | Residual, second (y − ŷ) |
|---|---|---|---|
| 5 | 0.7099 | −0.00326 | 0.0002713 |
| 10 | −1.171 | −0.01429 | 0.0001768 |
| 15 | −0.03669 | 0.009688 | −0.0003097 |
| 20 | 1.098 | 0.02905 | −0.0006575 |
| 25 | −0.9508 | −0.01371 | 0.0001806 |
| 30 | 0.351 | −0.007479 | 0.0003385 |
05Step 5 — Compare R²
| Order | Plot | Slope | Intercept | R² | k |
|---|---|---|---|---|---|
| Zero | % Remaining vs Time | −1.13137 | 83.6739 | 0.992717 | 1.1314 % min⁻¹ |
| First | ln(% Remaining) vs Time | −0.0179228 | 4.45886 | 0.990160 | 0.017923 min⁻¹ |
| Second | 1/% Remaining vs Time | 0.000287887 | 0.0109914 | 0.978933 | 0.00028789 %⁻¹ min⁻¹ |
Ranked by R²: Zero 0.992717 > First 0.990160 > Second 0.978933.
Zero order has the highest R², so its plot is the most linear.
06Step 6 — Determine kinetic order
07Step 7 — Calculate rate constant
Slope = −1.13137
Therefore:
k = −slope
k = −(−1.13137)
k = 1.13137
Units = % min⁻¹
08Step 8 — Calculate half-life (C₀ = 100 %)
t½ = C₀ / 2k
t½ = 100 / 2k
t½ = 100 / (2 × 1.13137)
t½ = 44.1942 min
09Step 9 — Calculate t90 (time to 90 % remaining)
t90 = 0.10 C₀ / k
t90 = 10 / k
t90 = 10 / 1.13137
t90 = 8.83883 min
10Step 10 — Scientific interpretation
At 50°C, the % drug remaining of Procaine HCl decreases with increasing time, indicating degradation of the drug. The reaction order is determined by comparing the linearity (R²) of zero-, first-, and second-order kinetic plots.
The experimentally determined order is zero order with R² = 0.9927.
At 50°C, the calculated rate constant is k = 1.1314 % min⁻¹.
The estimated half-life is 44.19 min.
The estimated shelf life (t90) is 8.839 min.
This t90 represents an experimental degradation estimate under the specified 50°C laboratory conditions and should NOT be interpreted as the actual labeled shelf life or recommended storage stability of Procaine HCl.
Procaine HCl undergoes degradation when exposed to elevated temperature. A temperature of 50°C accelerates chemical degradation compared with normal storage conditions.
Under constant analytical conditions, the decrease in absorbance is assumed to represent a decrease in drug concentration according to Beer–Lambert behaviour:
Absorbance ↓ → Drug concentration ↓ → % Drug remaining ↓ → Drug degradation ↑
Kinetic analysis is then used to determine the mathematical order of degradation.
1. % drug remaining
% Remaining = (Aₜ / A₀) × 100
2. Three linearised plots, each fitted by least squares
Zero: C = C₀ − kt · First: ln C = ln C₀ − kt · Second: 1/C = 1/C₀ + kt
For each plot the slope m and intercept b of y = mx + b come from the least-squares sums, then the predicted ŷ, the residuals y − ŷ, SSE = Σ(y − ŷ)², SST = Σ(y − ȳ)² and R² = 1 − SSE/SST. k = −slope for zero and first order and k = slope for second order. The order with the highest R² is reported — nothing is assumed.
3. Half-life and shelf life (C₀ = 100 %)
Zero: t½ = 100 / 2k, t90 = 10 / k
First: t½ = 0.693 / k, t90 = 0.10536 / k
Second: t½ = 1 / (kC₀), t90 = 1/(k × 90) − 1/(k × 100)
These assume the drug starts at 100 % at t = 0. The fitted intercept is usually a little different — the first reading is taken after degradation has begun — so t90 is a rate-based estimate, not a value read off the table. The intercept is shown in the Kinetic Model Analysis table.
Temperature is recorded as the condition of the run. It does not change k: that needs runs at several temperatures and an Arrhenius plot.
Keep learning
References
- Connors KA, Amidon GL, Stella VJ. Chemical Stability of Pharmaceuticals: A Handbook for Pharmacists. 2nd ed. New York: Wiley; 1986.
- Sinko PJ, editor. Martin's Physical Pharmacy and Pharmaceutical Sciences. 8th ed. Philadelphia: Wolters Kluwer; 2023.
- Harris DC, Lucy CA. Quantitative Chemical Analysis. 10th ed. New York: W. H. Freeman (Macmillan Learning); 2019.
- International Council for Harmonisation. ICH Q1A(R2): Stability Testing of New Drug Substances and Products. 2003. Source
- International Council for Harmonisation. ICH Q1E: Evaluation for Stability Data. 2003. Source
Common questions
Because it has to be measured, not assumed. The calculator fits all three plots and reports the one with the highest R². With the Procaine HCl practical's data, zero order narrowly beats first order — a hard-coded first order would get that wrong.
The practical's rule is the highest R², and that is what the calculator reports. When the gap is under 0.01 it also says the models are hard to separate — over 50 % degradation and six points, zero and first order often look alike. Following the reaction further, or adding time points, separates them.
Because the data are expressed as % remaining, and the practical defines the starting amount as 100 %. The intercept of each fitted line is printed in the Kinetic Model Analysis table, so you can see how far it sits from 100.
No. It is an experimental degradation estimate at the test temperature. A labeled shelf life comes from long-term studies at the recommended storage condition, usually supported by Arrhenius extrapolation across several temperatures.
No. A single run gives k at that temperature only; the temperature is recorded and shown as the experimental condition. Predicting k at another temperature needs data at several temperatures.
Two points are the minimum for a straight line, but any two points fit every model exactly (R² = 1), so the order cannot be decided from them. The calculator warns below four readings; the practical uses six.
Prepared by the PharmaWallah student team. Verify results with your textbook or instructor. Report an error · Editorial policy