Materials Included In Kit

Additional Materials Required

Safety Precautions

Concentrated nitric acid is severely corrosive, a strong oxidizer and toxic by ingestion and inhalation. Keep sodium carbonate acid neutralizer on hand to clean up acid spills. Reactions of nitric acid with metals generate nitrogen dioxide, a toxic, reddish-brown gas. Work with nitric acid only in a fume hood. Copper(II) sulfate, copper(II) nitrate and zinc nitrate solutions are toxic and irritating to skin and body tissue. Iron(III) chloride and iron(III) nitrate solutions may be skin and body tissue irritants. Zinc sulfate is a skin irritant. Wear chemical splash goggles, chemical-resistant gloves and a chemical-resistant apron or lab coat. Please follow all normal laboratory safety guidelines. Remind students to wash their hands thoroughly with soap and water before leaving the laboratory. Please review current Safety Data Sheets for additional safety, handling and disposal information.

Disposal

Please consult your current Flinn Scientific Catalog/Reference Manual for general guidelines and specific procedures, and review all federal, state and local regulations that may apply, before proceeding. Excess concentrated acid may be neutralized according to Flinn Suggested Disposal Method #24b. Copper(II) sulfate, copper(II) nitrate, iron(III) chloride, iron(III) nitrate, zinc nitrate, and zinc sulfate solutions may be rinsed down the drain with excess water according to Flinn Suggested Disposal Method #26b. All spectroscopic sample solutions may be combined and neutralized according to Flinn Suggested Disposal Method #24b.

Lab Hints

• This laboratory activity can be completed in three 50-minute class periods. It is important to allow time between the Introductory Activity and the Guided-Inquiry Activity for students to discuss and design the guided-inquiry procedures. Also, all student-designed procedures must be approved for safety before students are allowed to implement them in the lab. Prelab Questions may be completed before lab begins the first day.
• Ideally, each lab group should have a spectrophotometer or colorimeter to measure the absorbance of their solutions. However, by first preparing all solutions, then measuring the absorbance values of one solution after another, it is possible to complete this lab with only a few of these instruments in the classroom.
• Before attempting this laboratory, students should be able to:
  • Use basic laboratory instruments, including a balance, volumetric pipets and a spectrophotometer and colorimeter.
  • Display data in graphic form.
  • Balance an oxidation–reduction reaction.
• Assign the lab groups one of the following sets of salt solutions for the Introductory Activity.

Set A
0.1 M copper(II) nitrate
0.1 M iron(III) nitrate
0.1 M zinc sulfate

Set B
0.1 M copper(II) sulfate
0.1 M iron(III) sulfate
0.1 M zinc nitrate

• Volumetric flasks ensure the greatest accuracy in this quantitative analysis lab. If you do not have enough flasks for each student group, students may share by transferring their prepared solutions to a labeled bottle and then passing on their flasks (rinsed clean!) to the next group.
• Students are asked to determine a suitable wavelength at which to measure the absorbance of the brass solution. Discuss the factors that influence this choice (e.g., no interferences, adequate range of concentrations and sensitivity, highest absorbance value ≤1.00). Students will use the 0.1 M Cu²⁺(aq) ion spectrum to determine the wavelength, but the calibration curve should extend to 0.4 M. They should select a top absorbance value near 1.000 (say 0.800) and divide this value by 4 (since 0.4 M is four times 0.1 M). The students should look at the spectrum to find the wavelength where the 0.1 M Cu²⁺(aq) ion has an absorbance value of 0.200.
• The reaction of brass with concentrated nitric acid completely dissolves the zinc and copper metals in the brass.
• In Part B, students are to add 15.8 M nitric acid to their samples to dissolve the brass. Ask students to calculate the amount needed to dissolve the brass, assuming the sample is all copper.
• The volume of HNO₃(aq) that is required is calculated from the overall balanced equation:

Cu + 4HNO₃ → Cu(NO₃)₂ + 2NO₂ + 2H₂O

The setup for the calculated minimum volume is:

mL HNO₃ needed = 1.0 g brass(as Cu) (1 mol Cu/63.55 g Cu) (4 mol HNO₃/1 mol Cu) (1000 mL HNO₃/15.8 mol HNO₃). The minimum volume is 4 mL.

• The nitric acid–brass reaction is quite vigorous and produces both NO and NO₂ gases. The NO also reacts with the oxygen in the air to produce the toxic reddish-brown gas NO₂. A fume hood is required for this reaction. While the reaction is very rapid at first, as the acid is consumed, the rate slows. If the beaker is placed on a hot plate at a low setting, the metal should quickly dissolve within 20 minutes. Do not allow the students to remove their beakers from the fume hood until they have added the 30 mL of distilled water, washing down any drops on the bottom of the watch glass into the beaker. We recommend that concentrated nitric acid (15.8 M) only be handled and dispensed by the instructor!
• The brass supplied in this kit has a nominal copper content of 70%, with a range, depending on the particular production lot, of 68.5% to 71.5%.

Teacher Tips

• Visible spectroscopy is an important tool in determining the composition of metals, either in industry, such as alloying amounts or as a forensics tool when analyzing bullets taken from a crime scene.
• Spectrophotometry is also used to determine the rate law for the dye-fading reactions with sodium hydroxide in the Advanced Inquiry Lab series for AP Chemistry from Flinn Scientific. See “Kinetics of Crystal Violet Fading,” Investigation 11, Flinn Catalog No. AP7644.

Further Extensions

Alignment to Curriculum Framework for AP^® Chemistry

Enduring Understanding and Essential Knowledge
Atoms are so small that they are difficult to study directly, atomic models are constructed to explain experimental data on collections of atoms. (ID)

1D3: The interaction of electromagnetic waves or light with matter is a powerful means to probe the structure of atoms and molecules, and to measure their concentration.

Learning Objectives
1.15 The student can justify the selection of a particular type of spectroscopy to measure properties associated with vibrational or electronic motions of molecules.
1.16 The student can design and/or interpret the results of an experiment regarding the absorption of light to determine the concentration of an absorbing species in a solution.

Science Practices
4.1 The student can justify the selection of the kind of data needed to answer a particular scientific question.
4.2 The student can design a plan for collecting data to answer a particular scientific question.
5.1 The student can analyze data to identify patterns or relationships.
6.4 The student can make claims and predictions about natural phenomena based on scientific theories and models.

Answers to Prelab Questions

1. Dissolving brass requires an oxidizing acid, such as concentrated nitric acid. Nitrogen dioxide is produced as a byproduct in this reaction. Write a balanced chemical equation for the reaction of copper metal with concentrated nitric acid to produce copper(II) nitrate, nitrogen dioxide and water.

   Cu(s) + 4HNO₃(aq) → Cu(NO₃)₂(aq) + 2NO₂(g) + 2H₂O(l)

2. Nitrogen dioxide is a toxic, reddish-brown gas. What safety precautions are needed in this inquiry lab to protect against this hazard, as well as the hazards due to the use of concentrated acid?

   Work with concentrated nitric acid in a fume hood only. Wear chemical splash goggles, heavy-duty nitrile gloves and a chemical-resistant apron or lab coat, especially when measuring or transferring nitric acid. (The instructor may do this step for all student groups.) When the brown fumes have dissipated, carefully add about 30 mL of distilled or deionized water to dilute the acid prior to removing the beaker from the hood.

3. Copper(II) ions appear blue in aqueous solution. This is the transmitted color. The wavelengths of light that are NOT absorbed give rise to the perceived or transmitted color of a substance. Based on the principle of complementary colors, which colors or wavelengths of light would you expect to be most strongly absorbed by Cu²⁺ ions?

Solutions of copper(II) ions have a strong absorbance at visible wavelengths above 600 nm, corresponding to absorption of yellow, orange and red light. Yellow is the complementary color of blue light. The maximum absorbance is actually in the near infrared region, that is, at wavelengths > 700 nm.

4. Spectroscopy measurements may be made in either percent transmittance (%T) or absorbance (A). Based on the mathematical relationship between absorbance and transmittance, A = −log T, explain why calibration curves of absorbance versus concentration may deviate from a straight line when A < 0.1 and when A > 1.

   Absorbance measurements are most accurate in the range 0.1–1.0. Recall that a spectrophotometer detects the difference or ratio between the power of the incident and transmitted light. At very low absorbance values, the percent transmittance is high (for A = 0.1, % T = 80%). Since almost all of the incident light goes through, there is not a great difference between the incident power and the transmitted light for the instrument to detect. At very high absorbance values the percent transmittance is very low (for A = 1.0, % T = 10%). The instrument does not accurately detect low power of transmitted light.

Sample Data

Introductory Activity

Sample calculation:

Mass of brass sample: 0.973 g
Volume of dissolved brass solution: 100 mL
Sample absorbance: 0.251 at 630 nm
Slope of Cu²⁺ calibration curve: 2.3459 M^–1

1. Copper concentration in brass solution
   = (Sample absorbance)/(Slope of Cu²⁺ calibration curve)
   = 0.251/2.3459 M^–1
   = 0.107 M
2. Moles copper in brass sample
   = (Volume of dissolved brass sample)(Copper concentration in brass solution)
   = (0.107 M)(0.1 L)
   = 0.0107 moles
3. Percent copper in brass sample
   = [(moles Cu)(Molar mass Cu)/grams sample)] x 100

= 69.9%

Answers to Questions

Guided-Inquiry

1. For each salt solution, determine the species (cation and/or anion) that is responsible for the absorbance spectrum. Explain your reasoning.

   Only copper(II) and iron(III) salt solutions absorbed visible light. The spectra of copper(II) nitrate and copper(II) sulfate were almost identical, suggesting that the copper(II) cation is responsible for the visible light absorption at wavelengths > 600 nm. Similarly, the spectra of iron(III) chloride and iron(III) nitrate showed very close overlap as well, consistent with the iron(III) cation absorbing visible light from 400−550 nm. Chloride, sulfate and nitrate anions do not contribute to the visible spectra of the transition metal salts.

2. Do Zn²⁺ ions absorb visible light? Discuss the answer in terms of (a) the color and appearance of Zn²⁺ aqueous solutions and (b) the electronic structure of Zn²⁺ ions. Hint: See the Background section for information on the electronic transitions of transition metal ions.

   The zinc salt solutions were colorless and did not show any absorption of visible light in the 400−700 nm region. Zinc ions have a valence electron configuration of 3d¹⁰ after loss of their 4s² electrons. Absorption of visible light by colored transition metal ions is associated with d–d transitions. Since Zn²⁺ ions have filled d-orbitals, electrons cannot be promoted or excited to a higher energy, unfilled d-orbital.

3. Identify a suitable wavelength for analysis of Cu²⁺ ions in aqueous solution. The radiant intensity of light is highest in the middle of the visible range and falls off dramatically at long wavelengths (700 nm).

   Copper(II) ions absorb all wavelengths of visible light from 580 to 700 nm. The maximum absorbance occurs at >700 nm, as evidenced by the steeply rising curve. It’s best to choose a wavelength for analysis in the 600–650 nm range because the spectrophotometer may not give accurate measurements at the 700 nm cutoff for visible light.

4. If Cu²⁺ ions and Fe³⁺ ions are both suspected of being present in the solution, will Fe³⁺ ions interfere with the analysis of Cu²⁺ at the wavelength selected? Why or why not? Revise your answer to Question 3, if needed.

   Iron(III) ions absorb visible light below 550 nm. Choosing a wavelength above this value for analyzing the concentration of Cu²⁺ ions in solution will not show any interference due to Fe³⁺ ions.

5. In order to accommodate the range of possible [Cu²⁺] concentrations that may be obtained by dissolving brass, it’s recommended that the calibration curve extend from approximately 0.05 M to 0.4 M. Estimate the absorbance of standard solutions containing 0.05 M, 0.1 M, 0.2 M and 0.4 M copper(II) nitrate at the selected wavelength. Revise your answer to Question 3, if needed.

   We selected 630 nm as the optimum wavelength for analysis of copper ion solutions. The absorbance of the 0.1 M Cu(NO₃)₂ solution was approximately 0.20 at this wavelength. According to Beer’s law, absorbance is directly proportional to concentration. The estimated absorbance for various Cu²⁺ standard solutions is shown.

   {12452_Answers_Table_2}
6. Calculate the volumes of 0.4 M Cu(NO₃)₂ stock solution and water required to prepare 8.0 mL of each standard solution for your calibration curve.

   The results of dilution calculations to prepare 8.0 mL of each standard solution are included in the table.

1. Quality control samples were taken of a batch of nickel–iron alloy produced by Ironic Steel, Inc., in the furnace at their Springfield plant. The alloy must contain 43% nickel, ±0.5%, with the remaining percent iron. A 1.200 g sample was dissolved in hydrochloric acid and diluted to 100 mL in a volumetric flask. The Beer’s law plot for the absorbance of Fe(NO₃)₃(aq) versus its concentration is listed.

Calculate the percent iron contained in the alloy sample. Based on your results, is the batch of 43% nickel–iron alloy acceptable?

Mass of alloy sample: 1.200 g
Volume of dissolved alloy solution: 100 mL
Sample absorbance: 0.3730
Slope of Fe³⁺ calibration curve: 3.4919 M^–1

a. Iron concentration in alloy solution

= (Sample absorbance)/(Slope of Fe³⁺ calibration curve)
= 0.3730/3.4919 M^–1
= 0.107 M

b. Moles iron in alloy sample

= (Volume of dissolved alloy sample)(Iron concentration of dissolved alloy sample)
= (0.107 M)(0.1 L)
= 0.0107 moles

c. Percent iron in alloy sample

= [(moles Fe)(Molar mass Fe)/grams sample)] x 100
= [(0.0107 moles)(55.85 g/mol)/1.200 g] x 100
= 49.8%

Therefore, the amount of nickel is (100 – 49.8)% or 50.2% Ni; the batch is not acceptable.

2. The characteristic flame test colors of metal ions are due to atomic emission spectra. Discuss the relationship between the absorption and emission of light and the factors responsible for flame test colors. Include quantization of electron energy levels and Planck’s law in your answer.

When a substance is heated in a flame, the atoms absorb energy from the flame. This absorbed energy allows the electrons to be promoted to excited energy levels. From these excited energy levels, there is a natural tendency for the electrons to make a transition or drop back down to the ground state. When an electron makes a transition from a higher energy level to a lower energy level, a particle of light called a photon is emitted. Both the absorption and emission of energy are quantized—only certain energy levels are allowed.

The energy of each emitted photon is equal to the difference in energy between the excited state and the state to which the electron relaxes. The energy of the emitted photon determines the color of light observed in the flame. The flame color may be described in terms of its wavelength, and Planck’s law may be used to calculate the energy of the emitted photon.

ΔE is the difference in energy between the two energy levels in joules (J).
h is Planck’s constant (h = 6.626 x 10^–34 J•sec).
c is the speed of light (c = 2.998 x 10⁸ m/sec).
λ (lambda) is the wavelength of light in meters.

3. The wavelength of the characteristic, bright yellow-orange flame test color of sodium is 590 nm. Calculate the average energy (ΔE) associated with this atomic emission line.

Use Planck’s law to calculate the energy difference associated with the atomic emission D line of sodium (590 nm).
Planck’s law E = hc/λ
h = Planck’s constant, 6.626 x 10⁻³⁴ J•sec
c = speed of light, 2.998 x 10⁸ m/sec
λ = wavelength in m

References

AP^® Chemistry Guided-Inquiry Experiments: Applying the Science Practices; The College Board: New York, NY, 2013.

Introduction

The relative proportions of copper, zinc and iron in brass influence its properties and uses. How can the percent composition of brass be determined to verify these properties?

Concepts

• Spectroscopy
• Absorbance
• Beer’s law
• Calibration curve
• Concentration
• Electron transitions

Background

Brass is a generic term for alloys of copper and zinc. In addition to these metals, brass may also contain small amounts of iron, lead, aluminum and tin. More than 300 different brass alloys are known, with uses ranging from decorative hardware to architectural construction, musical instruments, and electrical switches. The amount of copper in brass affects its color, hardness, ductility, mechanical strength, electrical conductivity, corrosion resistance, etc. Visible spectroscopy provides a simple tool for determining the percent copper in brass.

Spectroscopy involves the interaction of electromagnetic radiation and matter. The absorption of electromagnetic radiation results in different types of transitions in a substance, depending on the energy of the radiation. Low energy microwave radiation, for instance, is converted to energy of molecular rotation. Absorption of infrared radiation excites vibrational frequencies associated with covalent bonds in a molecule. Visible and ultraviolet light cause electron transitions between different electron energy levels in a substance. The energy of electromagnetic radiation is quantized, as are the energy levels associated with various transitions, whether rotational, vibrational or electronic. Furthermore, the energies of these transitions are characteristic of an atom, molecule or compound. As a result, the absorption spectra of substances generally consist of specific lines or bands that can be used as a type of fingerprint to identify a substance.

Transition metal ions have filled or partially filled d orbitals. The presence of water molecules or other ligands surrounding a transition metal ion in solution leads to energy differences among the d orbitals. Depending on the metal ion involved, the energy difference may correspond to different wavelengths and energies of visible light. This property of transition metal ions gives many their characteristic—and beautiful—colors.

The concentration of a colored transition metal ion solution can be determined by measuring the color intensity. A visible spectrophotometer is used to measure the absorption of visible light. In general, absorbance is proportional to concentration. The higher the concentration of a transition metal ion solution, the more intense its color will be, and the greater its absorbance. The linear relationship between absorbance (A) and concentration (c) is expressed in Beer’s law, Equation 1, where b is the path length in cm and a is a proportionality constant. If c is given in units of molarity (M = moles/L), then a is known as the molar absorptivity coefficient, with units M⁻¹ cm⁻¹.

Experiment Overview

The purpose of this advanced inquiry lab is to design a procedure to analyze the amount of copper in brass using visible spectroscopy. Brass can be dissolved by reacting it with concentrated nitric acid, which oxidizes the possible metal components of the alloy to their most common ions, Cu²⁺, Zn²⁺ and Fe³⁺. The lab begins with an introductory activity to distinguish among these metal ions using visible spectroscopy. Students measure the absorbance of metal ion solutions at regular wavelength intervals from 400 to 700 nm and investigate the influence of the anion on the absorption spectra. The results provide a model for guided-inquiry design of an experiment to construct a calibration curve and determine the concentration of copper ions in a solution prepared by dissolving brass in nitric acid. Students must investigate the concentration range over which Beer’s law is valid and identify the optimum wavelength for analysis.

Materials

Prelab Questions

1. Dissolving brass requires an oxidizing acid, such as concentrated nitric acid. Nitrogen dioxide is produced as a byproduct in this reaction. Write a balanced chemical equation for the reaction of copper metal with concentrated nitric acid to produce copper(II) nitrate, nitrogen dioxide and water.
2. Nitrogen dioxide is a toxic, reddish-brown gas. What safety precautions are needed in this inquiry lab to protect against this hazard as well as the hazards due to the use of concentrated acid?
3. Copper(II) ions appear blue in aqueous solution. This is the transmitted color. The wavelengths of light that are not absorbed give rise to the perceived or transmitted color of a substance. Based on the principle of complementary colors, which colors or wavelengths of light would you expect to be most strongly absorbed by Cu²⁺ ions?

4. Spectroscopy measurements may be made in either percent transmittance (%T) or absorbance (A). Based on the mathematical relationship between absorbance and transmittance, A = −log T, explain why calibration curves of absorbance versus concentration may deviate from a straight line when A < 0.1 and when A >1.

Safety Precautions

Concentrated nitric acid is severely corrosive, a strong oxidizer and toxic by ingestion and inhalation. Reactions of nitric acid with metals generate nitrogen dioxide, a toxic, reddish-brown gas. Work with nitric acid only in a fume hood. Copper(II) sulfate, copper(II) nitrate and zinc nitrate solutions are toxic and irritating to skin and body tissue. Iron(III) chloride and iron(III) nitrate solutions may be skin and body tissue irritants. Zinc sulfate is a mild body tissue irritant. Wear chemical splash goggles, chemical-resistant gloves and a chemical-resistant apron or lab coat. Please follow all normal laboratory safety guidelines. Wash hands thoroughly with soap and water before leaving the laboratory.

Procedure

Introductory Activity

Visible Spectra of Transition Metal Ions

1. Read the entire procedure before beginning the activity.
2. Each group will be given 2–3 transition metal salt solutions to analyze. The possible solutions are Cu(NO₃)₂ and CuSO₄; FeCl₃ and Fe(NO₃)₃; and Zn(NO₃)₂ and ZnSO₄. All solutions are 0.1 Molar.
3. Follow the procedure for colorimetric measurements of the solutions as directed by the instructor. Generally, spectrophotometers are used as follows: Turn the instrument on and allow it to warm up for 15–20 minutes. Set the wavelength to the desired value. Handle cuvets at the top so no fingerprints are in the light path. Polish cuvets with a lens tissue. Place a cuvet that is about ⅔ full of distilled or deionized water into the sample holder and set the percent transmittance to 100% with the appropriate control. Fill a cuvet about ⅔ full of a test solution, place it in the spectrophotometer and record both wavelength and the absorbance.
4. Measure the absorbance of the first metal salt solution to be tested at 400 nm.
5. Increase the wavelength to 420 nm and reset the 100% T using the blank (distilled or deionized water). Replace the metal salt solution in the instrument and measure the absorbance.
6. Repeat step 5 every 20 nm to 700 nm.
7. Repeat steps 4–6 for each metal salt solution you have been assigned.
8. Plot the visible spectrum (absorbance versus wavelength) from 400 to 700 nm for each salt solution.
9. Report all spectra to the class for analysis in the guided-inquiry investigation.

1. For each salt solution, determine the species (cation and/or anion) that is responsible for the absorbance spectrum. Explain your reasoning.
2. Do Zn²⁺ ions absorb visible light? Discuss the answer in terms of (a) the color and appearance of Zn²⁺ aqueous solutions and (b) the electronic structure of Zn²⁺ ions. Hint: See the Background section for information on the electronic transitions of transition metal ions.

Consider the information provided in Questions 3−5 together. Recall that absorbance measurements are most accurate in the range 0.1 to 1.0.

3. Identify a suitable wavelength for analysis of Cu²⁺ ions in aqueous solution. The radiant intensity of light is highest in the middle of the visible range, and falls off dramatically at long wavelengths (700 nm).
4. If Cu²⁺ ions and Fe³⁺ ions are both suspected of being present in the solution, will Fe³⁺ ions interfere with the analysis of Cu²⁺ at the wavelength selected? Why or why not? Revise your answer to Question 3, if needed.
5. In order to accommodate the range of possible [Cu²⁺] concentrations that may be obtained by dissolving brass, it’s recommended that the calibration curve extend from approximately 0.05 M to 0.4 M. Estimate the absorbance of standard solutions containing 0.05 M, 0.1 M, 0.2 M and 0.4 M copper(II) nitrate at the selected wavelength. Revise your answer to Question 3, if needed.
6. Calculate the volumes of 0.4 M Cu(NO3)2 stock solution and water required to prepare 8.0 mL of each standard solution for your calibration curve.
7. Write detailed, step-by-step procedures for preparing the standard solutions and obtaining the calibration curve data. Include the glassware (e.g., pipets) and laboratory techniques needed to ensure precision and accuracy in transferring liquids, as well as the procedure for spectroscopic measurements.
8. Carry out the experiment, record the data, and graph the results. Analyze the best-fit straight line for the data and determine the slope of the Beer’s law curve.

Concentrated nitric acid is severely corrosive and the reaction of nitric acid with brass generates a toxic gas. The following procedure is provided to ensure laboratory safety.

1. Add two (2) brass shot pieces (about 1 g) to a small tared weighing dish and determine the mass to ±0.001 g. Place the shot samples in a small beaker.
2. Concentrated nitric acid is required to dissolve the brass shot. You will need a slight excess of acid to completely dissolve the shot. Since the reaction generates the toxic gas nitrogen dioxide, NO₂(g), a fume hood is required for this step in the procedure. Place your beaker in the fume hood and have your instructor dispense 6 mL of 15.8 M nitric acid into your beaker. Cover the beaker with a watch glass.
3. Place the beaker on a hot plate dialed to a low setting (1–2). The shot should be completely dissolved within 10–20 minutes. Once the brass has dissolved, carefully remove the beaker from the hot plate and let it cool.
4. With the beaker still in the hood, use a graduated cylinder to carefully add 30 mL of deionized water to the beaker and stir the solution with a glass rod. The beaker can now be removed from the hood.
5. Using a stirring rod, quantitatively transfer the solution in the beaker to a 100-mL volumetric flask. Use a wash bottle to rinse the beaker with a small portion of deionized water. Transfer the rinse water to the flask. Repeat this washing two more times. Fill the flask to the mark with deionized water. Cap and mix well.
6. Analyze the absorbance of the brass solution and calculate the percent copper in the original brass sample.
7. Dispose of all solutions as directed by your instructor.

Student Worksheet PDF

12452_Student1.pdf