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Sep 15
Beer-Lambert's Law, also known as the Beer-Lambert Law or the Beer-Lambert-Bouguer Law, is a relationship between the concentration of a solution and the amount of light absorbed by that solution. It is commonly used in spectroscopy to determine the concentration of a substance in a solution.
The Beer-Lambert Law is expressed as:
A = εcl
Where:
A = Absorbance of the solution
ε = Molar absorptivity (also known as molar extinction coefficient), a constant that depends on the substance being measured and the wavelength of light
c = Concentration of the solution
l = Path length of the light through the solution
To derive the Beer-Lambert Law, we start with the Beer-Lambert Law in terms of transmittance (T):
T = I/I₀
Where:
T = Transmittance of the solution
I = Intensity of light transmitted through the solution
I₀ = Intensity of incident light
The absorbance (A) of the solution is defined as:
A = -log(T) = log(1/T)
Using the definition of transmittance, we can rewrite the absorbance in terms of the intensities of light:
A = log(I₀/I)
The intensity of light transmitted through the solution (I) can be expressed as the product of the incident intensity (I₀) and the fraction of light absorbed by the solution (1 - T):
I = I₀ * (1 - T)
Substitute this expression for I into the equation for absorbance:
A = log(I₀ / (I₀ * (1 - T)))
A = log(1 / (1 - T))
Since T = I/I₀, we can rewrite the equation for absorbance in terms of transmittance:
A = log(1 / (1 - I/I₀))
A = log(1 / (1 - T))
Now, we can use the relationship between absorbance and concentration (A = εcl) to derive the Beer-Lambert Law:
A = εcl = log(1 / (1 - T))
Taking the antilog of both sides gives:
10^A = 1 / (1 - T)
Rearranging the equation gives:
1 - T = 1 / 10^A
T = 1 - 1 / 10^A
T = 1 - 10^(-A)
This is the Beer-Lambert Law in terms of transmittance.
The Beer-Lambert Law is expressed as:
A = εcl
Where:
A = Absorbance of the solution
ε = Molar absorptivity (also known as molar extinction coefficient), a constant that depends on the substance being measured and the wavelength of light
c = Concentration of the solution
l = Path length of the light through the solution
To derive the Beer-Lambert Law, we start with the Beer-Lambert Law in terms of transmittance (T):
T = I/I₀
Where:
T = Transmittance of the solution
I = Intensity of light transmitted through the solution
I₀ = Intensity of incident light
The absorbance (A) of the solution is defined as:
A = -log(T) = log(1/T)
Using the definition of transmittance, we can rewrite the absorbance in terms of the intensities of light:
A = log(I₀/I)
The intensity of light transmitted through the solution (I) can be expressed as the product of the incident intensity (I₀) and the fraction of light absorbed by the solution (1 - T):
I = I₀ * (1 - T)
Substitute this expression for I into the equation for absorbance:
A = log(I₀ / (I₀ * (1 - T)))
A = log(1 / (1 - T))
Since T = I/I₀, we can rewrite the equation for absorbance in terms of transmittance:
A = log(1 / (1 - I/I₀))
A = log(1 / (1 - T))
Now, we can use the relationship between absorbance and concentration (A = εcl) to derive the Beer-Lambert Law:
A = εcl = log(1 / (1 - T))
Taking the antilog of both sides gives:
10^A = 1 / (1 - T)
Rearranging the equation gives:
1 - T = 1 / 10^A
T = 1 - 1 / 10^A
T = 1 - 10^(-A)
This is the Beer-Lambert Law in terms of transmittance.
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Google Gemini AI
Sep 15
Beer-Lambert's Law
Beer-Lambert's Law, also known as the Beer-Lambert-Bouguer Law, is a fundamental law in spectroscopy that relates the absorbance of light to the concentration of an absorbing substance in a solution.
Derivation
The absorbance, A, of a solution is defined as the logarithmic ratio of the intensity of light incident on the solution, I0, to the intensity of light transmitted through the solution, I:
```
A = log10(I0/I)
```
According to Lambert's Law, the absorbance is directly proportional to the path length, b, through the absorbing medium:
```
A ∝ b
```
According to Beer's Law, the absorbance is directly proportional to the concentration of the absorbing substance, c:
```
A ∝ c
```
Combining these two relationships, we get:
```
A ∝ bc
```
The constant of proportionality, ε, is known as the molar absorptivity and is a characteristic of the absorbing substance. It is defined as the absorbance of a solution with a concentration of 1 mol/L and a path length of 1 cm.
Therefore, the full equation of Beer-Lambert's Law becomes:
```
A = εbc
```
Relationship
The relationship established by Beer-Lambert's Law can be used to determine the concentration of an absorbing substance in a solution by measuring its absorbance at a specific wavelength. The molar absorptivity, ε, is typically known or can be determined experimentally.
By measuring the absorbance and knowing the path length, the concentration of the absorbing substance can be calculated using the following formula:
```
c = A / (εb)
```
Beer-Lambert's Law, also known as the Beer-Lambert-Bouguer Law, is a fundamental law in spectroscopy that relates the absorbance of light to the concentration of an absorbing substance in a solution.
Derivation
The absorbance, A, of a solution is defined as the logarithmic ratio of the intensity of light incident on the solution, I0, to the intensity of light transmitted through the solution, I:
```
A = log10(I0/I)
```
According to Lambert's Law, the absorbance is directly proportional to the path length, b, through the absorbing medium:
```
A ∝ b
```
According to Beer's Law, the absorbance is directly proportional to the concentration of the absorbing substance, c:
```
A ∝ c
```
Combining these two relationships, we get:
```
A ∝ bc
```
The constant of proportionality, ε, is known as the molar absorptivity and is a characteristic of the absorbing substance. It is defined as the absorbance of a solution with a concentration of 1 mol/L and a path length of 1 cm.
Therefore, the full equation of Beer-Lambert's Law becomes:
```
A = εbc
```
Relationship
The relationship established by Beer-Lambert's Law can be used to determine the concentration of an absorbing substance in a solution by measuring its absorbance at a specific wavelength. The molar absorptivity, ε, is typically known or can be determined experimentally.
By measuring the absorbance and knowing the path length, the concentration of the absorbing substance can be calculated using the following formula:
```
c = A / (εb)
```
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