/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 A solution's transmittance is \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 3

A solution's transmittance is \(35.0 \%\). What is the transmittance if you dilute \(25.0 \mathrm{~mL}\) of the solution to \(50.0 \mathrm{~mL}\) ?

Short Answer

Expert verified
The new transmittance is 59.4%.

Step by step solution

01

Understand the Problem

We have a solution with an initial transmittance of 35.0%. It is diluted by doubling its volume from 25.0 mL to 50.0 mL. We need to find the new transmittance of this diluted solution.
02

Recall the Beer-Lambert Law

The Beer-Lambert Law relates absorbance (A) to concentration (c) and has the formula: \( A = \epsilon \cdot c \cdot l \), where \( \epsilon \) is the molar absorptivity, \( c \) is the concentration, and \( l \) is the path length.
03

Determine the Relationship Between Transmittance and Absorbance

Transmittance (T) is related to absorbance (A) by the formula: \( T = 10^{-A} \), or equivalently \( A = -\log_{10}(T) \). For a 35.0% transmittance, \( T = 0.35 \).
04

Calculate the Initial Absorbance

Using the relation \( A = -\log_{10}(T) \), the initial absorbance \( A_1 \) is calculated as: \[ A_1 = -\log_{10}(0.35) = 0.456 \]
05

Understand the Effect of Dilution on Concentration and Absorbance

Diluting the solution decreases its concentration proportionally. Since the volume is doubled (25.0 mL to 50.0 mL), the concentration is halved. The new absorbance \( A_2 \) is half of \( A_1 \): \( A_2 = \frac{A_1}{2} = 0.228 \).
06

Calculate the New Transmittance

Now that we have \( A_2 = 0.228 \), calculate the new transmittance \( T_2 \) using \( T_2 = 10^{-A_2} \): \[ T_2 = 10^{-0.228} = 0.594 \] Expressing 0.594 as a percentage gives a new transmittance of 59.4%.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transmittance
Transmittance refers to the measure of the fraction of light that passes through a sample. It is expressed as a percentage and signifies how much light is not absorbed by the solution. For a transmittance of 35.0%, this means 35% of the light directed at the solution is passing through, while the remaining 65% is absorbed. This concept is crucial when studying light interactions with materials, as it provides insights into the concentration of substances within the solution. To calculate transmittance, the formula used is typically based on the absorbance value, where transmittance T is equals to:
  • \( T = 10^{-A} \)
Understanding this relationship is vital for determining the properties and behavior of different solutions when they are exposed to light.
Absorbance
Absorbance is a measure that indicates how much light a sample absorbs at a particular wavelength, and it is directly related to the concentration of the substance within the sample. The formula linking absorbance and transmittance is given by:
  • \( A = -\log_{10}(T) \)
where A is absorbance, and T is transmittance. An important aspect of absorbance is its logarithmic nature, meaning small changes in absorbance can result in significant alterations in transmittance. For an initial solution with 35.0% transmittance, the calculated absorbance would be approximately 0.456.
Absorbance helps us understand the Beer-Lambert Law's role in determining concentrations. When you understand that a higher absorbance means more light is absorbed by the sample, you start seeing the direct relationship between light behavior and sample characteristics.
Dilution Effect
Dilution occurs when you add a solvent to a solution, thus decreasing the concentration of the solute. Practically, this is seen when a solution is made less concentrated by increasing its volume. In the context of the Beer-Lambert Law, diluting a solution affects its absorbance and transmittance.
When a solution is diluted, as in the given exercise where the volume is doubled from 25.0 mL to 50.0 mL, the concentration decreases by half. Consequently, the absorbance is halved, and the new absorbance can be calculated using the relationship:
  • New absorbance: \( A_2 = \frac{A_1}{2} \)
This understanding allows us to predict changes in optical properties of the solution. The new transmittance, calculated from the lowered absorbance, increases to indicate more light passes through the solution post-dilution.
Concentration
Concentration describes the amount of solute per unit volume of solution and is a fundamental concept in chemistry. It directly affects the solution's optical properties, as described by the Beer-Lambert Law.
The concentration determines how much light a solution absorbs, and thus influences both the absorbance and transmittance values. In our example, by diluting the solution from 25.0 mL to 50.0 mL, the concentration of the solute is halved. This results in a halved absorbance, computing using the formula, as the number of absorbing molecules in the path of light has decreased by half.
Understanding this concept is crucial when using spectrophotometric methods to determine concentrations, as it provides a basis for correlating measured absorbance with actual solute concentrations in a detailed manner, guiding practical applications in laboratory settings.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The concentration of \(\mathrm{Na}\) in plant materials are determined by flame atomic emission. The material to be analyzed is prepared by grinding, homogenizing, and drying at \(103^{\circ} \mathrm{C}\). A sample of approximately \(4 \mathrm{~g}\) is transferred to a quartz crucible and heated on a hot plate to char the organic material. The sample is heated in a muffle furnace at \(550^{\circ} \mathrm{C}\) for several hours. After cooling to room temperature the residue is dissolved by adding \(2 \mathrm{~mL}\) of \(1: 1 \mathrm{HNO}_{3}\) and evaporated to dryness. The residue is redissolved in \(10 \mathrm{~mL}\) of \(1: 9 \mathrm{HNO}_{3},\) filtered and diluted to \(50 \mathrm{~mL}\) in a volumetric flask. The following data are obtained during a typical analysis for the concentration of \(\mathrm{Na}\) in a \(4.0264-\mathrm{g}\) sample of oat bran. $$ \begin{array}{lcc} {\text { sample }} & \mathrm{mg} \mathrm{Na} / \mathrm{L} & \text { emission (arbitrary units) } \\ \hline \text { blank } & 0.00 & 0.0 \\ \text { standard 1 } & 2.00 & 90.3 \\ \text { standard } 2 & 4.00 & 181 \\ \text { standard } 3 & 6.00 & 272 \\ \text { standard } 4 & 8.00 & 363 \\ \text { standard } 5 & 10.00 & 448 \\ \text { sample } & & 238 \end{array} $$ Report the concentration of sodium in the sample of oat bran as \mug Na/g sample.

Jones and Thatcher developed a spectrophotometric method for analyzing analgesic tablets that contain aspirin, phenacetin, and caffeine. \(^{24}\) The sample is dissolved in \(\mathrm{CHCl}_{3}\) and extracted with an aqueous solution of \(\mathrm{NaHCO}_{3}\) to remove the aspirin. After the extraction is complete, the chloroform is transferred to a \(250-\mathrm{mL}\) volumetric flask and diluted to volume with \(\mathrm{CHCl}_{3} .\) A \(2.00-\mathrm{mL}\) portion of this solution is then diluted to volume in a \(200-\mathrm{mL}\) volumetric flask with \(\mathrm{CHCl}_{3}\). The absorbance of the final solution is measured at wavelengths of \(250 \mathrm{nm}\) and \(275 \mathrm{nm}\), at which the absorptivities, in \(\mathrm{ppm}^{-1} \mathrm{~cm}^{-1},\) for caffeine and phenacetin are $$ \begin{array}{lcc} & \mathrm{a}_{250} & \mathrm{a}_{275} \\ \hline \text { caffeine } & 0.0131 & 0.0485 \\ \text { phenacetin } & 0.0702 & 0.0159 \end{array} $$ Aspirin is determined by neutralizing the \(\mathrm{NaHCO}_{3}\) in the aqueous solution and extracting the aspirin into \(\mathrm{CHCl}_{3}\). The combined extracts are diluted to \(500 \mathrm{~mL}\) in a volumetric flask. A 20.00 -mL portion of the solution is placed in a 100 -mL volumetric flask and diluted to volume with \(\mathrm{CHCl}_{3}\). The absorbance of this solution is measured at \(277 \mathrm{nm}\), where the absorptivity of aspirin is \(0.00682 \mathrm{ppm}^{-1} \mathrm{~cm}^{-1}\). An analgesic tablet treated by this procedure is found to have absorbances of 0.466 at \(250 \mathrm{nm}, 0.164\) at \(275 \mathrm{nm}\), and 0.600 at \(277 \mathrm{nm}\) when using a cell with a \(1.00 \mathrm{~cm}\) pathlength. Report the milligrams of aspirin, caffeine, and phenacetin in the analgesic tablet.

A spectrophotometric method for the analysis of iron has a linear calibration curve for standards of \(0.00,5.00,10.00,15.00,\) and 20.00 \(\mathrm{mg} \mathrm{Fe} / \mathrm{L}\). An iron ore sample that is \(40-60 \% \mathrm{w} / \mathrm{w}\) is analyzed by this method. An approximately \(0.5-\mathrm{g}\) sample is taken, dissolved in a minimum of concentrated HCl, and diluted to \(1 \mathrm{~L}\) in a volumetric flask using distilled water. A \(5.00 \mathrm{~mL}\) aliquot is removed with a pipet. To what volume- \(10,25,50,100,250,500,\) or \(1000 \mathrm{~mL}\) - should it be diluted to minimize the uncertainty in the analysis? Explain.

A solution of \(5.00 \times 10^{-5} \mathrm{M} 1,3\) -dihydroxynaphthelene in \(2 \mathrm{M} \mathrm{NaOH}\) has a fluorescence intensity of 4.85 at a wavelength of \(459 \mathrm{nm}\). What is the concentration of 1,3 -dihydroxynaphthelene in a solution that has a fluorescence intensity of 3.74 under identical conditions?

The equilibrium constant for an acid-base indicator is determined by preparing three solutions, each of which has a total indicator concentration of \(1.35 \times 10^{-5} \mathrm{M}\). The \(\mathrm{pH}\) of the first solution is adjusted until it is acidic enough to ensure that only the acid form of the indicator is present, yielding an absorbance of \(0.673 .\) The absorbance of the second solution, whose \(\mathrm{pH}\) is adjusted to give only the base form of the indicator, is 0.118 . The \(\mathrm{pH}\) of the third solution is adjusted to 4.17 and has an absorbance of 0.439 . What is the acidity constant for the acid-base indicator?
See all solutions

Recommended explanations on Chemistry Textbooks

Chemical Reactions

Read Explanation

Physical Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Kinetics

Read Explanation

Nuclear Chemistry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.