Answer:
Assume I1 = Io * T1 transmission proportional to thickness
I2 = Io * T2 I is intensity of light absorbed (3/8 absorbed)
I1 / I2 = T1 / T2
T2 = T1 * (I2 / I1) = 2 * (2/3 / 3/8) = 2 * 16/9 = 32/9 thickness for 2/3 absorbed
The length of the beaker when the transmittance is changed to one-third is 13.78 cm.
The given parameters;
To find:
The length of the beaker is calculated by applying Beer-Lambert law;
[tex]\frac{I}{I_o} = e^{-cl}[/tex]
[tex]\frac{I_1}{I_0} = e^{-cl_1} = \frac{3}{8} \ ----(1)\\\\\frac{I_2}{I_0} = e^{-cl_2}= \frac{1}{3} \ ---- (2)[/tex]
divide equation (1) by equation (2);
[tex]\frac{e^{-cl_1}}{e^{-cl_2}} = \frac{3}{8} \times \frac{3}{1} \\\\ (c \ is \ constant \ for \ both) \\\\e^{-l_1 + l_2} = \frac{9}{8} \\\\l_2 - l_1 = ln(\frac{9}{8} )\\\\l_2 - 0.02 = 0.1178\\\\l_2 = 0.1178 + 0.02\\\\l_2 = 0.1378 \ m\\\\l_2 = 13.78 \ cm[/tex]
Thus, the length of the beaker when the transmittance is changed to one-third is 13.78 cm.
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