Beer-Lambert law with a lux meter
Discover the exponential law of light absorption by measuring transmission through an increasing number of translucent sheets.
Activity Summary
The student measures illuminance with decreasing numbers of tracing paper sheets and plots ln(T) versus N to verify the exponential Beer-Lambert law.
Introduction
Why does a stack of tracing paper become increasingly opaque? The answer lies in the Beer-Lambert law: transmitted light decreases exponentially with the thickness of absorbing material. Each sheet absorbs a constant fraction of the light it receives, so the total transmission follows T = exp(-αN). Using the smartphone's light sensor, you can verify this fundamental law of optics with nothing more than a lamp and a stack of paper.
Learning Objectives
- Calculate an optical transmission coefficient
- Experimentally verify the Beer-Lambert law
- Plot and interpret a ln(T) versus N graph
- Understand exponential decay in the context of light absorption
Scientific Concepts
Instruments and sensors
Scientific instruments
- Light meter (lux meter)
Sensors
- Light sensor
FizziQ Features
- Real-time graph and data table — Used to record each illuminance value I(N) and to plot T versus N and ln(T) versus N directly in the app to check the linear fit.
Required Materials
- - Smartphone or tablet with the FizziQ app - A constant light source (LED desk lamp or second phone's flash) - 8-10 sheets of tracing paper - A darkened room - Note: the protocol remains adaptable to any comparable light-measurement tool.
Experimental Protocol
Set up in a dark room or close the curtains. Ambient light is the main source of error.
Place a constant light source (LED desk lamp or a second phone's flash) about 30 cm above the smartphone's light sensor.
Open FizziQ and select the Brightness (lux meter) instrument. Place the smartphone flat, sensor facing up.
Record the brightness with no sheets: this is I₀, the reference incident intensity.
Place a first sheet of tracing paper on the sensor. Record I₁. Add a second sheet. Record I₂. Continue up to 8-10 sheets.
For each measurement, note the number of sheets N and the brightness I(N).
Calculate the transmission coefficient T(N) = I(N)/I₀ for each N.
Plot T versus N: you obtain an exponentially decreasing curve.
Plot ln(T) versus N: if the Beer-Lambert law holds, you obtain a straight line with slope -α, where α is the absorption coefficient per sheet.
The law is written: T(N) = exp(-αN), or equivalently I(N) = I₀ × exp(-αN). Calculate α and compare between different types of sheets.
Expected Results
The brightness visibly decreases with each sheet, but the relative decrease is constant. The graph of ln(T) versus N is a straight line with negative slope, confirming the exponential law. The linearity of ln(T) is typically excellent (R² > 0.98). Typical transmission per sheet: 82-92% for tracing paper, 40-60% for colored plastic, 95-98% for clear glass.
Scientific Questions
- What happens if the sheets are not all identical?
- How is this experiment related to spectrophotometry in chemistry?
- What is the difference between absorbance and transmittance?
- Could you use this method to measure the concentration of a colored solution?
- Why is ambient light the main source of error?
Scientific Background
The Beer-Lambert law describes the exponential attenuation of light passing through a material: **I = I₀ × exp(-αcx)**, where α is the absorption coefficient, c the concentration, and x the thickness.
Here, each sheet adds a constant thickness, so x = N × d, where N is the number of sheets and d the thickness of one sheet. The transmission simplifies to T = exp(-α'N) where α' = αd.
The exponential model means that each sheet absorbs a **constant fraction** of the light it receives, not a constant amount. If one sheet transmits 85%, two transmit 0.85² = 72%, three transmit 0.85³ = 61%, etc.
For standard tracing paper, the absorption coefficient is α ≈ 0.08 to 0.15 per sheet, corresponding to a transmission of about 85-92% per sheet.
The plot of **ln(T) versus N** should give a straight line of slope **-α**. The quality of the linear fit (R² > 0.99) confirms the exponential model.
This experiment naturally leads to spectrophotometry: by replacing the sheets with colored solutions of increasing concentration, one verifies Beer-Lambert as a function of concentration c.
The main source of error is stray ambient light. A dark background and a stable source are essential for reliable measurements.
Extensions
- Compare different materials (tracing paper, tissue paper, colored plastic, fabric)
- Investigate whether the wavelength of light affects the absorption coefficient by using colored filters
- Measure absorption of sunglasses and compare with their stated UV protection ratings
- Build a simple colorimeter to measure the concentration of an unknown solution
Frequently Asked Questions
iOS restricts access to the light sensor.
On iOS, FizziQ can use the camera as a brightness sensor. On Android, the native sensor is directly accessible.
The readings fluctuate even with a stable light source.
LED lamps powered by AC may flicker at 50/60 Hz. Use a DC-powered LED or a battery-operated flashlight for stable readings.
My ln(T) graph is not perfectly straight.
Check for ambient light leaking in. Also ensure the sheets are positioned consistently, fully covering the sensor.
How many sheets do I need for a convincing result?
At least 6-8 sheets. With fewer, random errors dominate and the exponential trend may not be clear.
Detailed Description
The student places their smartphone under a constant light source and measures illuminance with the FizziQ lux meter. They add sheets of tracing paper one by one and record the decreasing light intensity, then plot ln(T) versus N to verify the exponential relationship. FizziQ provides the built-in lux meter for direct illuminance readings and a real-time graphing tool to plot and linearize the T(N) and ln(T)(N) curves on the smartphone.
Ready to start?
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