5.6: Ludwig van Beethoven
Along with Mozart, Beethoven is likely one of the best known composers, and his 5th symphony is undoubtedly one of the best known works of classical music. Of the three composers we study in the Classical era, Beethoven exerted the greatest influence on the composers of the Romantic period that followed. Beethoven’s music represents a transition from Classical to Romantic style.
Biography
Background and early life
Beethoven was the grandson of Lodewijk van Beethoven (1712–73), a musician from Mechelen in the Southern Netherlands (now part of Belgium), who at the age of twenty moved to Bonn. Lodewijk (the Dutch cognate of German Ludwig ) was employed as a bass singer at the court of the Elector of Cologne, eventually rising to become Kapellmeister (music director). Lodewijk had one son, Johann (1740–1792), who worked as a tenor in the same musical establishment and gave lessons on piano and violin to supplement his income. Johann married Maria Magdalena Keverich in 1767; she was the daughter of Johann Heinrich Keverich, who had been the head chef at the court of the Archbishopric of Trier.
Beethoven was born of this marriage in Bonn. There is no authentic record of the date of his birth; however, the registry of his baptism, in a Roman Catholic service at the Parish of St. Regius on 17 December 1770, survives. As children of that era were traditionally baptised the day after birth in the Catholic Rhine country, and it is known that Beethoven’s family and his teacher Johann Albrechtsberger celebrated his birthday on 16 December, most scholars accept 16 December 1770 as Beethoven’s date of birth. Of the seven children born to Johann van Beethoven, only Ludwig, the second-born, and two younger brothers survived infancy.Caspar Anton Carl was born on 8 April 1774, and Nikolaus Johann, the youngest, was born on 2 October 1776.
Beethoven’s first music teacher was his father. Although tradition has it that Johann van Beethoven was a harsh instructor, and that the child Beethoven, “made to stand at the keyboard, was often in tears,” the Grove Dictionary of Music and Musicians claimed that no solid documentation supported this, and asserted that “speculation and myth-making have both been productive.” Beethoven had other local teachers: the court organist Gilles van den Eeden (d. 1782), Tobias Friedrich Pfeiffer (a family friend, who taught Beethoven the piano), and Franz Rovantini (a relative, who instructed him in playing the violin and viola). Beethoven’s musical talent was obvious at a young age. Johann, aware of Leopold Mozart’s successes in this area (with son Wolfgang and daughter Nannerl), attempted to exploit his son as a child prodigy, claiming that Beethoven was six (he was seven) on the posters for Beethoven’s first public performance in March 1778.
Some time after 1779, Beethoven began his studies with his most important teacher in Bonn, Christian Gottlob Neefe, who was appointed the Court’s Organist in that year. Neefe taught Beethoven composition, and by March 1783 had helped him write his first published composition: a set of keyboard variations (WoO 63). Beethoven soon began working with Neefe as assistant organist, at first unpaid (1781), and then as a paid employee (1784) of the court chapel conducted by the Kapellmeister Andrea Luchesi. His first three piano sonatas, named “Kurfürst” (“Elector”) for their dedication to the Elector Maximilian Friedrich (1708–1784), were published in 1783. Maximilian Frederick noticed Beethoven’s talent early, and subsidised and encouraged the young man’s musical studies.
Maximilian Frederick’s successor as the Elector of Bonn was Maximilian Franz, the youngest son of Empress Maria Theresa of Austria, and he brought notable changes to Bonn. Echoing changes made in Vienna by his brother Joseph, he introduced reforms based on Enlightenment philosophy, with increased support for education and the arts. The teenage Beethoven was almost certainly influenced by these changes. He may also have been influenced at this time by ideas prominent in freemasonry, as Neefe and others around Beethoven were members of the local chapter of the Order of the Illuminati.
In March 1787 Beethoven traveled to Vienna (possibly at another’s expense) for the first time, apparently in the hope of studying with Mozart. The details of their relationship are uncertain, including whether they actually met. Having learned that his mother was ill, Beethoven returned about two weeks after his arrival. His mother died shortly thereafter, and his father lapsed deeper into alcoholism. As a result, Beethoven became responsible for the care of his two younger brothers, and spent the next five years in Bonn.
Beethoven was introduced in these years to several people who became important in his life. Franz Wegeler, a young medical student, introduced him to the von Breuning family (one of whose daughters Wegeler eventually married). Beethoven often visited the von Breuning household, where he taught piano to some of the children. Here he encountered German and classical literature. The von Breuning family environment was less stressful than his own, which was increasingly dominated by his father’s decline. Beethoven also came to the attention of Count Ferdinand von Waldstein, who became a lifelong friend and financial supporter.
In 1789 Beethoven obtained a legal order by which half of his father’s salary was paid directly to him for support of the family. He also contributed further to the family’s income by playing viola in the court orchestra. This familiarised Beethoven with a variety of operas, including three by Mozart that were performed at court in this period. He also befriended Anton Reicha, a flautist and violinist of about his own age who was a nephew of the court orchestra’s conductor, Josef Reicha.
Establishing his career in Vienna
From 1790 to 1792, Beethoven composed a significant number of works (none were published at the time, and most are now listed as works without opus) that demonstrated his growing range and maturity. Musicologists have identified a theme similar to those of his Third Symphony in a set of variations written in 1791. Beethoven was probably first introduced to Joseph Haydn in late 1790, when the latter was traveling to London and stopped in Bonn around Christmas time. A year and a half later they met in Bonn on Haydn’s return trip from London to Vienna in July 1792, and it is likely that arrangements were made at that time for Beethoven to study with the old master. With the Elector’s help, Beethoven left Bonn for Vienna in November 1792, amid rumors of war spilling out of France; he learned shortly after his arrival that his father had died. Mozart had also recently died. Count Waldstein, in his farewell note to Beethoven, wrote: “Through uninterrupted diligence you will receive Mozart’s spirit through Haydn’s hands.” Over the next few years, Beethoven responded to the widespread feeling that he was a successor to the recently deceased Mozart by studying that master’s work and writing works with a distinctly Mozartean flavor.
Beethoven did not immediately set out to establish himself as a composer, but rather devoted himself to study and performance. Working under Haydn’s direction, he sought to master counterpoint. He also studied violin under Ignaz Schuppanzigh. Early in this period, he also began receiving occasional instruction from Antonio Salieri, primarily in Italian vocal composition style; this relationship persisted until at least 1802, and possibly 1809. With Haydn’s departure for England in 1794, Beethoven was expected by the Elector to return home. He chose instead to remain in Vienna, continuing his instruction in counterpoint with Johann Albrechtsberger and other teachers. Although his stipend from the Elector expired, a number of Viennese noblemen had already recognised his ability and offered him financial support, among them Prince Joseph Franz Lobkowitz, Prince Karl Lichnowsky, and Baron Gottfried van Swieten.
By 1793, Beethoven had established a reputation as an improviser in the salons of the nobility, often playing the preludes and fugues of J. S. Bach’s Well-Tempered Clavier. His friend Nikolaus Simrock had begun publishing his compositions; the first are believed to be a set of variations (WoO 66). By 1793, he had established a reputation in Vienna as a piano virtuoso, but he apparently withheld works from publication so that their publication in 1795 would have greater impact. Beethoven’s first public performance in Vienna was in March 1795, a concert in which he first performed one of his piano concertos. It is uncertain whether this was the First or Second. Documentary evidence is unclear, and both concertos were in a similar state of near-completion (neither was completed or published for several years). Shortly after this performance, he arranged for the publication of the first of his compositions to which he assigned an opus number, the three piano trios, Opus 1. These works were dedicated to his patron Prince Lichnowsky, and were a financial success; Beethoven’s profits were nearly sufficient to cover his living expenses for a year.
Musical maturity
Beethoven composed his first six string quartets (Op. 18) between 1798 and 1800 (commissioned by, and dedicated to, Prince Lobkowitz). They were published in 1801. With premieres of his First and Second Symphonies in 1800 and 1803, Beethoven became regarded as one of the most important of a generation of young composers following Haydn and Mozart. He also continued to write in other forms, turning out widely known piano sonatas like the “Pathétique” sonata (Op. 13), which Cooper describes as “surpass[ing] any of his previous compositions, in strength of character, depth of emotion, level of originality, and ingenuity of motivic and tonal manipulation.” He also completed his Septet (Op. 20) in 1799, which was one of his most popular works during his lifetime.
For the premiere of his First Symphony , Beethoven hired the Burgtheater on 2 April 1800, and staged an extensive program of music, including works by Haydn and Mozart, as well as his Septet, the First Symphony, and one of his piano concertos (the latter three works all then unpublished). The concert, which the Allgemeine musikalische Zeitung described as “the most interesting concert in a long time,” was not without difficulties; among the criticisms was that “the players did not bother to pay any attention to the soloist.”
Mozart and Haydn were undeniable influences. For example, Beethoven’s quintet for piano and winds is said to bear a strong resemblance to Mozart’s work for the same configuration, albeit with his own distinctive touches. But Beethoven’s melodies, musical development, use of modulation and texture, and characterization of emotion all set him apart from his influences, and heightened the impact some of his early works made when they were first published. By the end of 1800 Beethoven and his music were already much in demand from patrons and publishers.
In May 1799, Beethoven taught piano to the daughters of Hungarian Countess Anna Brunsvik. During this time, Beethoven fell in love with the younger daughter Josephine who has therefore been identified as one of the more likely candidates for the addressee of his letter to the “Immortal Beloved” (in 1812). Shortly after these lessons, Josephine was married to Count Josef Deym. Beethoven was a regular visitor at their house, continuing to teach Josephine, and playing at parties and concerts. Her marriage was by all accounts happy (despite initial financial problems), and the couple had four children. Her relationship with Beethoven intensified after Deym died suddenly in 1804.
Beethoven had few other students. From 1801 to 1805, he tutored Ferdinand Ries, who went on to become a composer and later wrote Beethoven remembered , a book about their encounters. The young Carl Czerny studied with Beethoven from 1801 to 1803. Czerny went on to become a renowned music teacher himself, instructing Franz Liszt, and gave on 11 February 1812 the Vienna premiere of Beethoven’s fifth piano concerto (the “Emperor”).
Beethoven’s compositions between 1800 and 1802 were dominated by two large-scale orchestral works, although he continued to produce other important works such as the piano sonata Sonata quasi una fantasia known as the “Moonlight Sonata.” In the spring of 1801 he completed The Creatures of Prometheus , a ballet. The work received numerous performances in 1801 and 1802, and Beethoven rushed to publish a piano arrangement to capitalise on its early popularity. In the spring of 1802 he completed the Second Symphony, intended for performance at a concert that was canceled. The symphony received its premiere instead at a subscription concert in April 1803 at the Theater an der Wien, where Beethoven had been appointed composer in residence. In addition to the Second Symphony, the concert also featured the First Symphony, the Third Piano Concerto, and the oratorio Christ on the Mount of Olives . Reviews were mixed, but the concert was a financial success; Beethoven was able to charge three times the cost of a typical concert ticket.
Beethoven’s business dealings with publishers also began to improve in 1802 when his brother Carl, who had previously assisted him casually, began to assume a larger role in the management of his affairs. In addition to negotiating higher prices for recently composed works, Carl also began selling some of Beethoven’s earlier unpublished works, and encouraged Beethoven (against the latter’s preference) to also make arrangements and transcriptions of his more popular works for other instrument combinations. Beethoven acceded to these requests, as he could not prevent publishers from hiring others to do similar arrangements of his works.
Loss of Hearing
Around 1796, by the age of 26, Beethoven began to lose his hearing. He suffered from a severe form of tinnitus, a “ringing” in his ears that made it hard for him to hear music; he also tried to avoid conversations. The cause of Beethoven’s deafness is unknown, but it has variously been attributed to typhus, auto-immune disorders (such as systemic lupus erythematosus), and even his habit of immersing his head in cold water to stay awake. The explanation from Beethoven’s autopsy was that he had a “distended inner ear,” which developed lesions over time.
As early as 1801, Beethoven wrote to friends describing his symptoms and the difficulties they caused in both professional and social settings (although it is likely some of his close friends were already aware of the problems). Beethoven, on the advice of his doctor, lived in the small Austrian town ofHeiligenstadt, just outside Vienna, from April to October 1802 in an attempt to come to terms with his condition. There he wrote his Heiligenstadt Testament, a letter to his brothers which records his thoughts of suicide due to his growing deafness and records his resolution to continue living for and through his art. Over time, his hearing loss became profound: at the end of the premiere of his Ninth Symphony in 1824, he had to be turned around to see the tumultuous applause of the audience because he could hear neither it nor the orchestra. Beethoven’s hearing loss did not prevent him from composing music, but it made playing at concerts—a lucrative source of income—increasingly difficult. After a failed attempt in 1811 to perform his ownPiano Concerto No. 5 (the “Emperor”), which was premiered by his student Carl Czerny, he never performed in public again until he conducted the Ninth Symphony in 1824.
A large collection of Beethoven’s hearing aids, such as a special ear horn, can be viewed at the Beethoven House Museum in Bonn, Germany. Despite his obvious distress, Czerny remarked that Beethoven could still hear speech and music normally until 1812. Around 1814 however, by the age of 44, Beethoven was almost totally deaf, and when a group of visitors saw him play a loud arpeggio of thundering bass notes at his piano remarking, “Ist es nicht schön?” (Is it not beautiful?), they felt deep sympathy considering his courage and sense of humor (he lost the ability to hear higher frequencies first).
As a result of Beethoven’s hearing loss, his conversation books are an unusually rich written resource. Used primarily in the last ten or so years of his life, his friends wrote in these books so that he could know what they were saying, and he then responded either orally or in the book. The books contain discussions about music and other matters, and give insights into Beethoven’s thinking; they are a source for investigations into how he intended his music should be performed, and also his perception of his relationship to art. Out of a total of 400 conversation books, it has been suggested that 264 were destroyed (and others were altered) after Beethoven’s death by Anton Schindler, who wished only an idealised biography of the composer to survive. However, Theodore Albrecht contests the verity of Schindler’s destruction of a large number of conversation books.
Patronage
While Beethoven earned income from publication of his works and from public performances, he also depended on the generosity of patrons for income, for whom he gave private performances and copies of works they commissioned for an exclusive period prior to their publication. Some of his early patrons, including Prince Lobkowitz and Prince Lichnowsky, gave him annual stipends in addition to commissioning works and purchasing published works.
Perhaps Beethoven’s most important aristocratic patron was Archduke Rudolph, the youngest son of Emperor Leopold II, who in 1803 or 1804 began to study piano and composition with Beethoven. The cleric (Cardinal-Priest) and the composer became friends, and their meetings continued until 1824. Beethoven dedicated 14 compositions to Rudolph, including the Archduke Trio (1811) and his great Missa Solemnis (1823). Rudolph, in turn, dedicated one of his own compositions to Beethoven. The letters Beethoven wrote to Rudolph are today kept at the Gesellschaft der Musikfreunde in Vienna. Another patron was Count (later Prince) Andreas Razumovsky, for whom the String Quartets Nos. 7–9, Op. 59, Rasumovsky were named.
In the Autumn of 1808, after having been rejected for a position at the royal theatre, Beethoven received an offer from Napoleon’s brother Jérôme Bonaparte, then king of Westphalia, for a well-paid position as Kapellmeister at the court in Cassel. To persuade him to stay in Vienna, the Archduke Rudolph, Prince Kinsky and Prince Lobkowitz, after receiving representations from the composer’s friends, pledged to pay Beethoven a pension of 4000 florins a year. Only Archduke Rudolph paid his share of the pension on the agreed date. Kinsky, immediately called to military duty, did not contribute and soon died after falling from his horse. Lobkowitz stopped paying in September 1811. No successors came forward to continue the patronage, and Beethoven relied mostly on selling composition rights and a small pension after 1815. The effects of these financial arrangements were undermined to some extent by war with France, which caused significant inflation when the government printed money to fund its war efforts.
The Middle Period
Beethoven’s return to Vienna from Heiligenstadt was marked by a change in musical style, and is now designated as the start of his middle or “heroic” period. According to Carl Czerny, Beethoven said, “I am not satisfied with the work I have done so far. From now on I intend to take a new way.” This “heroic” phase was characterised by a large number of original works composed on a grand scale. The first major work employing this new style was the Third Symphony in E flat, known as the Eroica . This work was longer and larger in scope than any previous symphony. When it premiered in early 1805 it received a mixed reception. Some listeners objected to its length or misunderstood its structure, while others viewed it as a masterpiece.
Listen: Symphony No. 5
Please listen to the following audio file to hear Symphony No. 5, Op. 67 (1st movement), composed during Beethoven’s middle period.
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=246
The “middle period” is sometimes associated with a “heroic” manner of composing, but the use of the term “heroic” has become increasingly controversial in Beethoven scholarship. The term is more frequently used as an alternative name for the middle period. The appropriateness of the term “heroic” to describe the whole middle period has been questioned as well: while some works, like the Third and Fifth Symphonies, are easy to describe as “heroic”, many others, like his Symphony No. 6, Pastoral , are not.
Some of the middle period works extend the musical language Beethoven had inherited from Haydn and Mozart. The middle period work includes the Third through Eighth Symphonies, the Rasumovsky, Harp and Serioso string quartets, the “Waldstein” and “Appassionata” piano sonatas, Christ on the Mount of Olives , the opera Fidelio , the Violin Concerto and many other compositions. During this time Beethoven’s income came from publishing his works, from performances of them, and from his patrons. His position at the Theater an der Wien was terminated when the theater changed management in early 1804, and he was forced to move temporarily to the suburbs of Vienna with his friend Stephan von Breuning. This slowed work on Fidelio , his largest work to date, for a time. It was delayed again by the Austrian censor, and finally premiered in November 1805 to houses that were nearly empty because of the French occupation of the city. In addition to being a financial failure, this version of Fidelio was also a critical failure, and Beethoven began revising it.
During May 1809, when the attacking forces of Napoleon bombarded Vienna, according to Ferdinand Ries, Beethoven, very worried that the noise would destroy what remained of his hearing, hid in the basement of his brother’s house, covering his ears with pillows.
The work of the middle period established Beethoven as a master. In a review from 1810, he was enshrined by E. T. A. Hoffmann as one of the three great “Romantic” composers; Hoffman called Beethoven’s Fifth Symphony “one of the most important works of the age.”
Personal and family difficulties
Beethoven’s love life was hampered by class issues. In late 1801 he met a young countess, Julie (“Giulietta”) Guicciardi through the Brunsvik family, at a time when he was giving regular piano lessons to Josephine Brunsvik. Beethoven mentions his love for Julie in a November 1801 letter to his boyhood friend, Franz Wegeler, but he could not consider marrying her, due to the class difference. Beethoven later dedicated to her his Sonata No. 14 , now commonly known as the “Moonlight sonata” or “Mondscheinsonate” (in German).
His relationship with Josephine Brunsvik deepened after the death in 1804 of her aristocratic first husband, the Count Joseph Deym. Beethoven wrote Josephine 15 passionate love letters from late 1804 to around 1809/10. Although his feelings were obviously reciprocated, Josephine was forced by her family to withdraw from him in 1807. She cited her “duty” and the fact that she would have lost the custodianship of her aristocratic children had she married a commoner. After Josephine married Baron von Stackelberg in 1810, Beethoven may have proposed unsuccessfully to Therese Malfatti, the supposed dedicatee of “Für Elise”; his status as a commoner may again have interfered with those plans.
In the spring of 1811 Beethoven became seriously ill, suffering headaches and high fever. On the advice of his doctor, he spent six weeks in theBohemian spa town of Teplitz. The following winter, which was dominated by work on the Seventh symphony, he was again ill, and his doctor ordered him to spend the summer of 1812 at the spa Teplitz. It is certain that he was at Teplitz when he wrote a love letter to his “Immortal Beloved.” The identity of the intended recipient has long been a subject of debate; candidates include Julie Guicciardi, Therese Malfatti, Josephine Brunsvik, and Antonie Brentano.
Beethoven visited his brother Johann at the end of October 1812.  He was unable to convince Johann to end the relationship and appealed to the local civic and religious authorities. Johann and Therese married on 9 November.
In early 1813 Beethoven apparently went through a difficult emotional period, and his compositional output dropped. His personal appearance degraded—it had generally been neat—as did his manners in public, especially when dining. Beethoven took care of his brother (who was suffering from tuberculosis) and his family, an expense that he claimed left him penniless.
Beethoven was finally motivated to begin significant composition again in June 1813, when news arrived of the defeat of one of Napoleon’s armies atVitoria, Spain, by a coalition of forces under the Duke of Wellington. This news stimulated him to write the battle symphony known as Wellington’s Victory . It was first performed on 8 December, along with his Seventh Symphony, at a charity concert for victims of the war. The work was a popular hit, probably because of its programmatic style, which was entertaining and easy to understand. It received repeat performances at concerts Beethoven staged in January and February 1814. Beethoven’s renewed popularity led to demands for a revival of Fidelio , which, in its third revised version, was also well received at its July opening. That summer he composed a piano sonata for the first time in five years ( No. 27, Opus 90 ). This work was in a markedly more Romantic style than his earlier sonatas. He was also one of many composers who produced music in a patriotic vein to entertain the many heads of state and diplomats who came to the Congress of Vienna that began in November 1814.  Compared to its first setting in 1805 (a gift for Josephine Brunsvik), it was “far more dramatic … The entire spirit is that of an operatic scena.”
Custody Struggle and Illness
Between 1815 and 1817 Beethoven’s output dropped again. Beethoven attributed part of this to a lengthy illness (he called it an “inflammatory fever”) that afflicted him for more than a year, starting in October 1816. Biographers have speculated on a variety of other reasons that also contributed to the decline, including the difficulties in the personal lives of his would-be paramours and the harsh censorship policies of the Austrian government. The illness and death of his brother Carl from tuberculosis may also have played a role.
Carl had been ill for some time, and Beethoven spent a small fortune in 1815 on his care. After Carl died on 15 November 1815, Beethoven immediately became embroiled in a protracted legal dispute with Carl’s wife Johanna over custody of their son Karl, then nine years old. Beethoven, who considered Johanna an unfit parent because of her morals (she had an illegitimate child by a different father before marrying Carl and had been convicted of theft) and financial management, had successfully applied to Carl to have himself named sole guardian of the boy. A late codicil to Carl’s will gave him and Johanna joint guardianship. While Beethoven was successful at having his nephew removed from her custody in February 1816, the case was not fully resolved until 1820, and he was frequently preoccupied by the demands of the litigation and seeing to Karl’s welfare, whom he first placed in a private school.
The Austrian court system had one court for the nobility and members of the Landtafel, the Landrechte, and many other courts for commoners, among them the Civil Court of the Vienna Magistrate. Beethoven disguised the fact that the Dutch “van” in his name did not denote nobility as does the German “von” and his case was tried in the Landrechte. Owing to his influence with the court, Beethoven felt assured of the favorable outcome of being awarded sole guardianship. While giving evidence to the Landrechte, however, Beethoven inadvertently admitted that he was not nobly born. On 18 December 1818 the case was transferred to the Magistracy, where he lost sole guardianship.
Beethoven appealed and regained custody. Johanna’s appeal to the Emperor was not successful: the Emperor “washed his hands of the matter.” During the years of custody that followed, Beethoven attempted to ensure that Karl lived to the highest moral standards. Beethoven had an overbearing manner and frequently interfered in his nephew’s life. Karl attempted suicide on 31 July 1826 by shooting himself in the head. He survived and was brought to his mother’s house, where he recuperated. He and Beethoven were reconciled, but Karl insisted on joining the army and last saw Beethoven in early 1827.
Late works
Beethoven began a renewed study of older music, including works by J. S. Bach and Handel, that were then being published in the first attempts at complete editions. He composed the overture The Consecration of the House , which was the first work to attempt to incorporate these influences. A new style emerged, now called his “late period”. He returned to the keyboard to compose his first piano sonatas in almost a decade: the works of the late period are commonly held to include the last five piano sonatas and the Diabelli Variations , the last two sonatas for cello and piano, the late string quartets (see below), and two works for very large forces: the Missa Solemnis and the Ninth Symphony.
By early 1818 Beethoven’s health had improved, and his nephew moved in with him in January. On the downside, his hearing had deteriorated to the point that conversation became difficult, necessitating the use of conversation books. His household management had also improved somewhat; Nanette Streicher, who had assisted in his care during his illness, continued to provide some support, and he finally found a skilled cook. His musical output in 1818 was still somewhat reduced, but included song collections and the “Hammerklavier” Sonata, as well as sketches for two symphonies that eventually coalesced into the epic Ninth. In 1819 he was again preoccupied by the legal processes around Karl, and began work on the Diabelli Variations and the Missa Solemnis .
For the next few years he continued to work on the Missa, composing piano sonatas and bagatelles to satisfy the demands of publishers and the need for income, and completing the Diabelli Variations. He was ill again for an extended time in 1821, and completed the Missa in 1823, three years after its original due date. He also opened discussions with his publishers over the possibility of producing a complete edition of his work, an idea that was arguably not fully realised until 1971. Beethoven’s brother Johann began to take a hand in his business affairs, much in the way Carl had earlier, locating older unpublished works to offer for publication and offering the Missa to multiple publishers with the goal of getting a higher price for it.
Two commissions in 1822 improved Beethoven’s financial prospects. The Philharmonic Society of London offered a commission for a symphony, and Prince Nikolas Golitsin of St. Petersburg offered to pay Beethoven’s price for three string quartets. The first of these commissions spurred Beethoven to finish the Ninth Symphony, which was first performed, along with the Missa Solemnis, on 7 May 1824, to great acclaim at the Kärntnertortheater. The Allgemeine musikalische Zeitung gushed, “inexhaustible genius had shown us a new world,” and Carl Czerny wrote that his symphony “breathes such a fresh, lively, indeed youthful spirit … so much power, innovation, and beauty as ever [came] from the head of this original man, although he certainly sometimes led the old wigs to shake their heads.” Unlike his more lucrative earlier concerts, this did not make Beethoven much money, as the expenses of mounting it were significantly higher. A second concert on 24 May, in which the producer guaranteed Beethoven a minimum fee, was poorly attended; nephew Karl noted that “many people [had] already gone into the country.” It was Beethoven’s last public concert.
Listen: Piano Sonata No. 32
Please listen to the following audio file to hear Piano Sonata No. 32 in C minor, Op. 111 (1st movement), written between 1821 and 1822, during Beethoven’s late period.
An audio element has been excluded from this version of the text. You can listen to it online here: http://pb.libretexts.org/map/?p=246
Beethoven then turned to writing the string quartets for Golitsin. This series of quartets, known as the “Late Quartets,” went far beyond what musicians or audiences were ready for at that time. One musician commented that “we know there is something there, but we do not know what it is.” Composer Louis Spohr called them “indecipherable, uncorrected horrors.” Opinion has changed considerably from the time of their first bewildered reception: their forms and ideas inspired musicians and composers including Richard Wagner and Béla Bartók, and continue to do so. Of the late quartets, Beethoven’s favorite was the Fourteenth Quartet, op. 131 in C ♯ minor, which he rated as his most perfect single work. The last musical wish of Schubert was to hear the Op. 131 quartet, which he did on 14 November 1828, five days before his death.
Beethoven wrote the last quartets amidst failing health. In April 1825 he was bedridden, and remained ill for about a month. The illness—or more precisely, his recovery from it—is remembered for having given rise to the deeply felt slow movement of the Fifteenth Quartet, which Beethoven called “Holy song of thanks (‘Heiliger Dankgesang’) to the divinity, from one made well.” He went on to complete the quartets now numberedThirteenth, Fourteenth, and Sixteenth. The last work completed by Beethoven was the substitute final movement of the Thirteenth Quartet, which replaced the difficult Große Fuge . Shortly thereafter, in December 1826, illness struck again, with episodes of vomiting and diarrhea that nearly ended his life.
In 1825, his nine symphonies were performed in a cycle for the first time, by the Leipzig Gewandhaus Orchestra under Johann Philipp Christian Schulz. This was repeated in 1826.
Illness and death
Beethoven was bedridden for most of his remaining months, and many friends came to visit. He died on 26 March 1827 at the age of 56 during a thunderstorm. His friend Anselm Hüttenbrenner, who was present at the time, said that there was a peal of thunder at the moment of death. An autopsy revealed significant liver damage, which may have been due to heavy alcohol consumption. It also revealed considerable dilation of the auditory and other related nerves.
Beethoven’s funeral procession on 29 March 1827 was attended by an estimated 20,000 Viennese citizens. Franz Schubert, who died the following year and was buried next to Beethoven, was one of the torchbearers. Beethoven was buried in a dedicated grave in the Währing cemetery, north-west of Vienna, after a requiem mass at the church of the Holy Trinity (Dreifaltigkeitskirche). His remains were exhumed for study in 1862, and moved in 1888 to Vienna’s Zentralfriedhof. In 2012, his crypt was checked to see if his teeth had been stolen during a series of grave robberies of other famous Viennese composers.
There is dispute about the cause of Beethoven’s death: alcoholic cirrhosis, syphilis, infectious hepatitis, lead poisoning,sarcoidosis and Whipple’s disease have all been proposed. Friends and visitors before and after his death clipped locks of his hair, some of which have been preserved and subjected to additional analysis, as have skull fragments removed during the 1862 exhumation. Some of these analyses have led to controversial assertions that Beethoven was accidentally poisoned to death by excessive doses of lead-based treatments administered under instruction from his doctor.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Ludwig van Beethoven. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Ludwig_van_Beethoven . License : CC BY-SA: Attribution-ShareAlike


9.1: Transcluding (Reusing) a Page
Our transcluding video tutorial on YouTube: https://youtu.be/Ef5HjZ6ogGc
One of the powers of the LibreTexts Bookshelves/Course Shell system is to share content without having to copy it. Content can be added to new pages in two ways: copying (also called "forking") and transcluding (also called "reusing"). The former is the typical copy and paste approach and effectively creates two copies of content that can be independently curated. This is useful for deviating the content from the original source. However, often the content may not need to be changed at all, and in this case, transcluding the content is ideal.
Transcluding content keeps the content from the source page, and only mirrors this content to the new page. This helps to reduce clutter and facilitates effective curation of the libraries, a key aspect of the LibreTexts project. This is because typos can be changed in the source and these will immediately be updated in all transcluded pages as well.
Note that the section labels (e.g. Figure 11.2.4) will be updated in the transcluded pages based on the number given first in the transcluded page's title. So even though the content is mirrored from the original source, the section appears to fit seamlessly into other LibreTexts.
The remixer transcludes all content automatically by default and if you wish to change/edit the content you will fork it to produce a copy.
How to Transclude
- Navigate to the page on which you want to reuse content and open the page in Edit mode.
- Select Elements > Content reuse from the Editor toolbar.
- Use the Browse or Search tab to select your source page to appear in the Article to reuse field.
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Specify how much content you want to reuse:
- To reuse the entire page , leave the Reuse section unselected.
- To reuse only a section of a page , select the appropriate section from the Reuse section drop-down list. If reusing a section, the entire section will be reused, including any subsections. To include the title of a section, select the Show title checkbox.
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Click
Insert
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- Save the page to see your published content reuse.
- After you create a transcluded page in this way, it is a good idea to flip the "Transcluded" meta-tag under the page settings at the top of the page (often collapsed) to "yes".
From the perspective of the student reader, the difference between traditional copying (forking) and transclusion (reusing) is impossible to tell.
It is best to transclude all content when a new text is created by remixing existing content UNLESS a specific need arises to edit the content. And this customization can be done one section at a time.
How to switch Transcluded content into Forked Content
A transcluded page can be swtiched to a forked page by clicking on the Forker option under page options. Alternatively, if the meta-tag is set on a trancluded page, there is a forker icon next to the title that will do the same thing.


14.3: The Milky Way — Our Home Galaxy
It is estimated that the Milky Way Galaxy has a population of 100 billion to 400 billion stars. Initially, astronomers did not associate the Milky Way Galaxy with other galaxies they saw through their telescopes. Recall, that they called these other galaxies “nebulae. ”
Characteristics of the Milky Way Galaxy
The Milky Way Galaxy is a Spiral Galaxy and most likely a barred spiral. This characteristic is difficult to confirm due to our location within the Milky Way Galaxy and all of the nebulosity we have to look through. Recall, that the Milky Way Galaxy is part of the Local Group of galaxies, and that group is a part of the Virgo supercluster of galaxies.
This large barred spiral galaxy is believed to be similar in shape to the Milky Way Galaxy. Dust lanes, spiral arms, and a bar at the galaxy’s center are visible.
Some have referred to this image as a ‘picture postcard’ of the Milky Way Galaxy.


1.8: The Island of Skeletons
THE ISLAND OF SKELETONS.
IG Wave and his little nephew, Red Shell, lived together in a deep forest. The boy was the only relative that the old man had, and he was very fond of him. He had brought Red Shell and his sister, Wild Sage, to his home some years before, just after the great plague had killed most of his tribe, among them the father and mother of the children. But they had not been many months in the forest before Wild Sage was stolen by a giant who lived on the Island of Skeletons.
Big Wave warned the boy never to go towards the east; for, if by any chance, he should cross a certain magic line of sacred meal that Big Wave had drawn, he would be at the mercy of the giant.
The boy obeyed for a time; but by and by he grew tired of playing in one place, so he went towards the east, not noticing when he crossed the magic line, till he came to the shore of a great lake.
He amused himself for a while, throwing pebbles into the water, and shooting arrows. A man came up to him, and said, “Well, boy, where is your lodge?”
Red Shell told him. Then the man proposed shooting arrows to see who could shoot the higher. Red Shell had had much practice, and though he was only a boy, his arm was strong, and he drew the bow far back and sent the arrow much higher than the man did.
The man laughed and said, “You are a brave boy; now let us see whether you can swim as well as you can shoot.”
They jumped into the water and tried holding their breath while swimming. Again the boy proved himself the victor.
 I am on my way to an island where there are pretty birds, and you can shoot as many as you please.”
Red Shell said he would go, and looked about for a canoe. The man began singing, and presently there appeared a canoe drawn by six white swans, three on either side. The boy and his companion stepped in and the man guided the swans by singing.
The island was so long that he could not see the end of it, but it was not very wide. It was thickly wooded and there was so much undergrowth that the ground could hardly be seen, but Red Shell noticed heaps of bones under the bushes, and asked what they were. He was told that the island had once been a famous hunting-ground and these were the bones of the animals that had been killed.
After wandering about for some time, the man proposed another swim. They had been in the water but a few minutes when the boy heard singing, and looking around he saw the man going off in the canoe and taking his own and Red Shell’s clothes with him. He shouted, but neither the man nor the birds paid any attention to him.
Thus he was left alone and naked, and it was fast growing dark. Then he remembered his uncle’s warnings, and was so miserable from cold, hunger and fear, that at last he sat down and cried.
By-and-by he heard a voice calling to him, “Hist! keep still.”
He looked round and saw a skeleton lying on the ground not far from him. It beckoned to him and said, “Poor boy, it was the same with me, but I will help you if you will do me a service. Go to that tree” (pointing to one close by) “dig on the west side of it, and you will find a pouch of smoking mixture and a pipe. Bring them to me. You can get a flint on the shore. Bring that also.”
The boy was terribly frightened, but the skeleton spoke kindly, and not as though he meant to do harm. Red Shell therefore went to the tree, and brought the pipe and smoking mixture. Then he found a flint and on being asked to do so struck fire, lit the pipe and handed the same to the skeleton.
It smoked quickly, drawing the smoke into the mouth and letting it escape between the ribs. Red Shell watched and saw mice run out from between the bones. When the skeleton was rid of them it said: “Now I feel better, and can tell you what to do to escape my fate.  You must lose the trail for them by jumping into the water many times on your way to a hollow tree, which you will find on the other side of the island. In the morning after they have gone, come to me.”
Red Shell thanked the skeleton and started at once to find the tree. It was quite dark, so he could see nothing, but he ran from tree to tree, climbing hallway up each one, and running into the water many times before he found the place where he had been told to sleep.
Towards morning he heard the splash of a canoe in the water, and soon a giant followed by three large dogs, strode into the forest.
“You must hunt this animal,” the giant said to the dogs.
They scented the trail and dashed through the bushes. They rushed up one tree and then another, and at last came back to the giant with their tails between their legs, for they had found nothing.
 He skinned it and ate it raw. Then he drove the two others down to the canoe, jumped in and went away.
When they were out of sight of the island, Red Shell crept from his hiding place and went back to the skeleton.
“You are still alive?” it asked in surprise.   You must go down to the shore before the darkness comes and dig a pit in the sand. Lie down in it and cover yourself with sand. When he leaves his canoe, get into it and say ‘Come swans, let us go home.’ If the man calls you, you must not turn round or look at him. When you are free, do not forget the skeleton.”
Red Shell promised to come back to the island and to do all that he could for the poor bones. He went down to the shore and dug the pit deep enough so that when he stood in it his head was on a level with the water. When he heard the song in the distance he knew the swans were coming; so he covered his head with sand and waited till he heard a footstep on the dry leaves.
Then he crept out stealthily, stepped into the canoe and whispered to the swans, “Come, let us go home.” He began the song that he had heard their master sing to them, and the canoe glided from the shore.
The swans carried him down the lake to a large cleft rock in the center. They drew the canoe through the opening and through the cave till they came to a stone door. Red Shell tried to open it, but could not. Then he turned the canoe around and struck the door with the stern.
The door flew open and Red Shell found himself in a fine lodge. He saw his own clothes and many others heaped in a corner near the fire which was burning brightly. A kettle of soup was steaming over it and there were some potatoes in the ashes on the hearth.
Seeing no one, the boy ate supper and then lay down to sleep on a couch of wild-cat skins.
In the morning he went out and stepping into the canoe, said, “Come, swans, let us go to the island.”
He saw the two dogs lying asleep in the sun and, on landing, found that then had killed their master.
The skeleton was delighted to see him and praised him for his courage and for being true to his word. But he said to him, “You must not go home yet. Travel toward the east three days and you will come to some huge rocks. There you will see a young girl drawing water from a spring.  You will be able to get her away. When you have done so, come back to me.”
Red Shell at once set out for the east and in three days he found the rocks of which he had been told. As he came near them he saw a lovely girl drawing water. “Sister,” he said, going up to her, “you must come home with me.”
She was frightened and tried to run away. Looking back, she saw that it was really her brother, when she was even more afraid, though she turned and spoke to him. “Hist,” said she, “a giant keeps me here.
Red Shell did not move.
“Go,” said Wild Sage.
“No,” he answered, “not till you go with me. Take me to your lodge.”
The giant had gone to a cranberry swamp, and Wild Sage knew that he would not return until the evening; so she ventured to take her brother home with her. She dug a pit in one corner of the lodge, told him to get into it, and then covered it with her bed of buffalo skins.
Just before the darkness came the giant’s dogs rushed in, barking furiously. “Who?” said the giant, “is hidden here?”
“No one,” said Wild Sage.
“There is, there is,” said the giant, “or the dogs would not bark like that.”
They did not discover Red Shell, however, so the giant sat down to his supper.
“This boy is not tender, he is not cooked enough, get up and cook him more,” said the giant.
“Cook it yourself, if it doesn’t suit you,” she answered.
The giant took no notice of her answer, but called to her to come and take off his moccasins.
“Take them off yourself,” she said.
“Kaw,” thought the giant, “now I know she has some one hidden.
Early the next day the giant said he was going to the cranberry swamp to get some children for his dinner. He did not go far from the lodge, but hid himself in some bushes close to the shore.
He saw Wild Sage and her brother get into a canoe, and threw a hook after them, which caught the boat and drew it towards the shore. But Red Shell took up a stone and broke the hook, and they floated off once more.
The giant was in a terrible rage. He lay down flat on the ground, and, putting his mouth to the water, drank so fast that the canoe was drawn close to the shore He began to swell from drinking such a quantity, and could not move. Red Shell took another stone and threw it at him. It struck him and he snapped in two, and the water he had swallowed flowed back into the lake.
Red Shell and his sister then sailed to the island, where the two dogs who had eaten their master rushed down to meet them. The boy raised his hand threateningly, and said: “Off to the woods as wolves. You no longer deserve to be dogs.”
The animals slunk away growling, and as they disappeared were seen to change into lean and hungry wolves.
Red Shell went to the skeleton, who commanded him to gather all the bones that he could find on the island and to lay them side by side in one place. Then he was to say to them, “Dead folk, arise!”
It took him and his sister many days, for there were bones everywhere. When all had been arranged in one place, Red Shell stood off at a little distance and called loudly, “Dead folk, arise!” The bones raised themselves and took human form. All the men had bows and arrows,but some had only one arm, and others only one leg. The skeleton whom Red Shell had first met became a tall, handsome warrior, perfect in every limb. He saluted Red Shell as Chief, and the others did the same.
Then the boy and his sister crossed the lake and traveled westward till they came to their uncles’ lodge. He was very old, his fire was out and he was still mourning for his nephew. But as he listened to the story of the lad’s adventures, and realized that he had come back unhurt, some of his years left him.
They built a long lodge with many fireplaces; then Red Shell returned to the island and brought back those who had been skeletons. The handsome brave, who was known as White Eagle, married Wild Sage, and they all dwelt together in peace to the end of their lives.


7.9: Rebirth of the Ku Klux Klan (KKK)
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Suspicions of immigrants, Catholics, and modernists contributed to a string of reactionary organizations.
Two events in 1915 are widely credited with inspiring the rebirth of the Klan: the lynching of Leo Frank and the release of The Birth of a Nation , a popular and groundbreaking film that valorized the Reconstruction Era Klan as a protector of feminine virtue and white racial purity. Taking advantage of this sudden surge of popularity, Colonel William Joseph Simmons organized what is often called the “second” Ku Klux Klan in Georgia in late 1915. This new Klan, modeled after other fraternal organizations with elaborate rituals and a hierarchy, remained largely confined to Georgia and Alabama until 1920, when Simmons began a professional recruiting effort that resulted in individual chapters being formed across the country and membership rising to an estimated five million. 38
Partly in response to the migration of southern blacks to northern cities during World War I, the KKK expanded above the Mason-Dixon Line. Membership soared in Philadelphia, Detroit, Chicago, and Portland, while Klan-endorsed mayoral candidates won in Indianapolis, Denver, and Atlanta. 39 The Klan often recruited through fraternal organizations such as the Freemasons and through various Protestant churches. In many areas, local Klansmen visited churches of which they approved and bestowed a gift of money on the presiding minister, often during services. The Klan also enticed people to join through large picnics, parades, rallies, and ceremonies. The Klan established a women’s auxiliary in 1923 headquartered in Little Rock, Arkansas. The Women of the Ku Klux Klan mirrored the KKK in practice and ideology and soon had chapters in all forty-eight states, often attracting women who were already part of the Prohibition movement, the defense of which was a centerpiece of Klan activism. 40
Contrary to its perception of as a primarily southern and lower-class phenomenon, the second Klan had a national reach composed largely of middle-class people. Sociologist Rory McVeigh surveyed the KKK newspaper Imperial Night-Hawk for the years 1923 and 1924, at the organization’s peak, and found the largest number of Klan-related activities to have occurred in Texas, Pennsylvania, Indiana, Illinois, and Georgia. The Klan was even present in Canada, where it was a powerful force within Saskatchewan’s Conservative Party. In many states and localities, the Klan dominated politics to such a level that one could not be elected without the support of the KKK. For example, in 1924, the Klan supported William Lee Cazort for governor of Arkansas, leading his opponent in the Democratic Party primary, Thomas Terral, to seek honorary membership through a Louisiana klavern so as not to be tagged as the anti-Klan candidate. In 1922, Texans elected Earle B. Mayfield, an avowed Klansman who ran openly as that year’s “klandidate,” to the U.S. Senate. At its peak the Klan claimed between four and five million members. 41
Despite the breadth of its political activism, the Klan is today remembered largely as a violent vigilante group—and not without reason. Members of the Klan and affiliated organizations often carried out acts of lynching and “nightriding”—the physical harassment of bootleggers, union activists, civil rights workers, or any others deemed “immoral” (such as suspected adulterers) under the cover of darkness or while wearing their hoods and robes. In fact, Klan violence was extensive enough in Oklahoma that Governor John C. Walton placed the entire state under martial law in 1923. Witnesses testifying before the military court disclosed accounts of Klan violence ranging from the flogging of clandestine brewers to the disfiguring of a prominent black Tulsan for registering African Americans to vote. In Houston, Texas, the Klan maintained an extensive system of surveillance that included tapping telephone lines and putting spies in the local post office in order to root out “undesirables.” A mob organized and led by Klan members in Aiken, South Carolina, lynched Bertha Lowman and her two brothers in 1926, but no one was ever prosecuted: the sheriff, deputies, city attorney, and state representative all belonged to the Klan. 42
The Klan dwindled in the face of scandal and diminished energy over the last years of the 1920s. By 1930, the Klan only had about thirty thousand members and it was largely spent as a national force, only to appear again as a much diminished force during the civil rights movement in the 1950s and 1960s.


2.5: Sets are not ordered pairs (or tuples)
You’ll remember from high school algebra the notion of an ordered pair \((x,y)\) . We dealt with those when we wanted to specify a point to plot on a graph: the first coordinate gave the distance from the origin on the x-axis, and the second coordinate on the y-axis. Clearly an ordered pair is not a set, because as the name implies it is ordered: \((3,-4) \neq (-4,3)\) . For this reason, we’ll be very careful to use curly braces to denote sets, and parentheses to denote ordered pairs.
By the way, although the word “coordinate" is often used to describe the elements of an ordered pair, that’s really a geometry-centric word that implies a visual plot of some kind. Normally we won’t be plotting elements like that, but we will still have use to deal with ordered pairs. I’ll just call the constituent parts “elements" to make it more general.
Three-dimensional points need ordered triple s \((x,y,z)\) , and it doesn’t take a rocket scientist to deduce that we could extend this to any number of elements. The question is what to call them, and you do sort of sound like a rocket scientist (or other generic nerd) when you say tuple . (Some people rhyme this word with “Drupal," and others with “couple," by the way, and there seems to be no consensus). If you have an ordered-pair-type thing with 5 elements, therefore, it’s a 5-tuple (or a quintuple). If it has 117 elements, it’s a 117-tuple, and there’s really nothing else to call it. The general term (if we don’t know or want to specify how many elements) is n-tuple . In any case, it’s an ordered sequence of elements that may contain duplicates, so it’s very different than a set.


16.1: Balancing Oxidation-Reduction Reactions
- Define electrochemistry and a number of important associated terms
- Split oxidation-reduction reactions into their oxidation half-reactions and reduction half-reactions
- Produce balanced oxidation-reduction equations for reactions in acidic or basic solution
- Identify oxidizing agents and reducing agents
Electricity refers to a number of phenomena associated with the presence and flow of electric charge. Electricity includes such diverse things as lightning, static electricity, the current generated by a battery as it discharges, and many other influences on our daily lives. The flow or movement of charge is an electric current (Figure \(\PageIndex{1}\)). Electrons or ions may carry the charge. The elementary unit of charge is the charge of a proton, which is equal in magnitude to the charge of an electron. The SI unit of charge is the coulomb (C) and the charge of a proton is 1.602 × 10 −19 C. The presence of an electric charge generates an electric field. Electric current is the rate of flow of charge.
The SI unit for electrical current is the SI base unit called the ampere (A), which is a flow rate of 1 coulomb of charge per second (1 A = 1 C/s). An electric current flows in a path, called an electric circuit . In most chemical systems, it is necessary to maintain a closed path for current to flow. The flow of charge is generated by an electrical potential difference, or potential, between two points in the circuit. Electrical potential is the ability of the electric field to do work on the charge. The SI unit of electrical potential is the volt (V). When 1 coulomb of charge moves through a potential difference of 1 volt, it gains or loses 1 joule (J) of energy. Table \(\PageIndex{1}\) summarizes some of this information about electricity.
| Quantity | Definition | Measure or Unit |
|---|---|---|
| Electric charge | Charge on a proton | 1.602 × 10 −19 C |
| Electric current | The movement of charge | ampere = A = 1 C/s |
| Electric potential | The force trying to move the charge | volt = V = J/C |
| Electric field | The force acting upon other charges in the vicinity |
Electrochemistry studies oxidation-reduction reactions, which were first discussed in an earlier chapter, where we learned that oxidation was the loss of electrons and reduction was the gain of electrons. The reactions discussed tended to be rather simple, and conservation of mass (atom counting by type) and deriving a correctly balanced chemical equation were relatively simple. In this section, we will concentrate on the half-reaction method for balancing oxidation-reduction reactions. The use of half-reactions is important partly for balancing more complicated reactions and partly because many aspects of electrochemistry are easier to discuss in terms of half-reactions. There are alternate methods of balancing these reactions; however, there are no good alternatives to half-reactions for discussing what is occurring in many systems. The half-reaction method splits oxidation-reduction reactions into their oxidation “half” and reduction “half” to make finding the overall equation easier.
Electrochemical reactions frequently occur in solutions, which could be acidic, basic, or neutral. When balancing oxidation-reduction reactions, the nature of the solution may be important. It helps to see this in an actual problem. Consider the following unbalanced oxidation-reduction reaction in acidic solution:
\[\ce{MnO4-}(aq)+\ce{Fe^2+}(aq)⟶\ce{Mn^2+}(aq)+\ce{Fe^3+}(aq) \nonumber \]
We can start by collecting the species we have so far into an unbalanced oxidation half-reaction and an unbalanced reduction half-reaction . Each of these half-reactions contain the same element in two different oxidation states. The Fe 2 + has lost an electron to become Fe 3 + ; therefore, the iron underwent oxidation. The reduction is not as obvious; however, the manganese gained five electrons to change from Mn 7 + to Mn 2 + .
\[\begin{align*}
&\textrm{oxidation (unbalanced): }\ce{Fe^2+}(aq)⟶\ce{Fe^3+}(aq)\\
&\textrm{reduction (unbalanced): }\ce{MnO4-}(aq)⟶\ce{Mn^2+}(aq)
\end{align*} \nonumber \]
In acidic solution, there are hydrogen ions present, which are often useful in balancing half-reactions. It may be necessary to use the hydrogen ions directly or as a reactant that may react with oxygen to generate water. Hydrogen ions are very important in acidic solutions where the reactants or products contain hydrogen and/or oxygen. In this example, the oxidation half-reaction involves neither hydrogen nor oxygen, so hydrogen ions are not necessary to the balancing. However, the reduction half-reaction does involve oxygen. It is necessary to use hydrogen ions to convert this oxygen to water.
\[\textrm{charge not balanced: }\ce{MnO4-}(aq)+\ce{8H+}(aq)⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l) \nonumber \]
The situation is different in basic solution because the hydrogen ion concentration is lower and the hydroxide ion concentration is higher. After finishing this example, we will examine how basic solutions differ from acidic solutions. A neutral solution may be treated as acidic or basic, though treating it as acidic is usually easier.
The iron atoms in the oxidation half-reaction are balanced (mass balance); however, the charge is unbalanced, since the charges on the ions are not equal. It is necessary to use electrons to balance the charge. The way to balance the charge is by adding electrons to one side of the equation. Adding a single electron on the right side gives a balanced oxidation half-reaction:
\[\textrm{oxidation (balanced): }\ce{Fe^2+}(aq)⟶\ce{Fe^3+}(aq)+\ce{e-} \nonumber \]
You should check the half-reaction for the number of each atom type and the total charge on each side of the equation. The charges include the actual charges of the ions times the number of ions and the charge on an electron times the number of electrons.
\[\begin{align*}
&\textrm{Fe: } \mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{Charge: } \mathrm{Does\:[1×(+2)]=[1×(+3)+1×(−1)]?\:Yes.}
\end{align*} \nonumber \]
If the atoms and charges balance, the half-reaction is balanced. In oxidation half-reactions, electrons appear as products (on the right). As discussed in the earlier chapter, since iron underwent oxidation, iron is the reducing agent.
Now return to the reduction half-reaction equation:
\[\textrm{reduction (unbalanced): }\ce{MnO4-}(aq)+\ce{8H+}(aq)⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l) \nonumber \]
The atoms are balanced (mass balance), so it is now necessary to check for charge balance. The total charge on the left of the reaction arrow is [(−1) × (1) + (8) × (+1)], or +7, while the total charge on the right side is [(1) × (+2) + (4) × (0)], or +2. The difference between +7 and +2 is five; therefore, it is necessary to add five electrons to the left side to achieve charge balance.
\[\textrm{Reduction (balanced): }\ce{MnO4-}(aq)+\ce{8H+}(aq)+\ce{5e-}⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l) \nonumber \]
You should check this half-reaction for each atom type and for the charge, as well:
\[\begin{align*}
&\textrm{Mn: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(8×1) =(4×2)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(1×4) =(4×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[1×(−1)+8×(+1)+5×(−1)]=[1×(+2)]?\:Yes.}
\end{align*} \nonumber \]
Now that this half-reaction is balanced, it is easy to see it involves reduction because electrons were gained when \(\ce{MnO4-}\) was reduced to Mn 2 + . In all reduction half-reactions, electrons appear as reactants (on the left side). As discussed in the earlier chapter, the species that was reduced, \(\ce{MnO4-}\) in this case, is also called the oxidizing agent. We now have two balanced half-reactions.
\[\begin{align*}
&\textrm{oxidation: }\ce{Fe^2+}(aq)⟶\ce{Fe^3+}(aq)+\ce{e-}\\
&\textrm{reduction: }\ce{MnO4-}(aq)+\ce{8H+}(aq)+\ce{5e-}⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l)
\end{align*} \nonumber \]
It is now necessary to combine the two halves to produce a whole reaction. The key to combining the half-reactions is the electrons. The electrons lost during oxidation must go somewhere. These electrons go to cause reduction. The number of electrons transferred from the oxidation half-reaction to the reduction half-reaction must be equal. There can be no missing or excess electrons. In this example, the oxidation half-reaction generates one electron, while the reduction half-reaction requires five. The lowest common multiple of one and five is five; therefore, it is necessary to multiply every term in the oxidation half-reaction by five and every term in the reduction half-reaction by one. (In this case, the multiplication of the reduction half-reaction generates no change; however, this will not always be the case.) The multiplication of the two half-reactions by the appropriate factor followed by addition of the two halves gives
\[\begin{align*}
&\textrm{oxidation: }5×(\ce{Fe^2+}(aq)⟶\ce{Fe^3+}(aq)+\ce{e-})\\
&\underline{\textrm{reduction: }\ce{MnO4-}(aq)+\ce{8H+}(aq)+\ce{5e-}⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l)}\\
&\textrm{overall: }\ce{5Fe^2+}(aq)+\ce{MnO4-}(aq)+\ce{8H+}(aq)⟶\ce{5Fe^3+}(aq)+\ce{Mn^2+}(aq)+\ce{4H2O}(l)
\end{align*} \nonumber \]
The electrons do not appear in the final answer because the oxidation electrons are the same electrons as the reduction electrons and they “cancel.” Carefully check each side of the overall equation to verify everything was combined correctly:
\[\begin{align*}
&\textrm{Fe: }\mathrm{Does\:(5×1)=(5×1)?\:Yes.}\\
&\textrm{Mn: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(8×1)=(4×2)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(1×4)=(4×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[5×(+2)+1×(−1)+8×(+1)]=[5×(+3)+1×(+2)]?\:Yes.}
\end{align*} \nonumber \]
Everything checks, so this is the overall equation in acidic solution. If something does not check, the most common error occurs during the multiplication of the individual half-reactions.
Now suppose we wanted the solution to be basic. Recall that basic solutions have excess hydroxide ions. Some of these hydroxide ions will react with hydrogen ions to produce water. The simplest way to generate the balanced overall equation in basic solution is to start with the balanced equation in acidic solution, then “convert” it to the equation for basic solution. However, it is necessary to exercise caution when doing this, as many reactants behave differently under basic conditions and many metal ions will precipitate as the metal hydroxide. We just produced the following reaction, which we want to change to a basic reaction:
\[\ce{5Fe^2+}(aq)+\ce{MnO4-}(aq)+\ce{8H+}(aq)⟶\ce{5Fe^3+}(aq)+\ce{Mn^2+}(aq)+\ce{4H2O}(l) \nonumber \]
However, under basic conditions, \(\ce{MnO4-}\) normally reduces to MnO 2 and iron will be present as either Fe(OH) 2 or Fe(OH) 3 . For these reasons, under basic conditions, this reaction will be
\[\ce{3Fe(OH)2}(s)+\ce{MnO4-}(aq)+\ce{2H2O}(l)⟶\ce{3Fe(OH)3}(s)+\ce{MnO2}(s)+\ce{OH-}(aq) \nonumber \]
(Under very basic conditions \(\ce{MnO4-}\) will reduce to \(\ce{MnO4^2-}\), instead of MnO 2 .)
It is still possible to balance any oxidation-reduction reaction as an acidic reaction and then, when necessary, convert the equation to a basic reaction. This will work if the acidic and basic reactants and products are the same or if the basic reactants and products are used before the conversion from acidic or basic. There are very few examples in which the acidic and basic reactions will involve the same reactants and products. However, balancing a basic reaction as acidic and then converting to basic will work. To convert to a basic reaction, it is necessary to add the same number of hydroxide ions to each side of the equation so that all the hydrogen ions (H + ) are removed and mass balance is maintained. Hydrogen ion combines with hydroxide ion (OH − ) to produce water.
Let us now try a basic equation. We will start with the following basic reaction:
\[\ce{Cl-}(aq)+\ce{MnO4-}(aq)⟶\ce{ClO3-}(aq)+\ce{MnO2}(s) \nonumber \]
Balancing this as acid gives
\[\ce{Cl-}(aq)+\ce{2MnO4-}(aq)+\ce{2H+}(aq)⟶\ce{ClO3-}(aq)+\ce{2MnO2}(s)+\ce{H2O}(l) \nonumber \]
In this case, it is necessary to add two hydroxide ions to each side of the equation to convert the two hydrogen ions on the left into water:
\[\ce{Cl-}(aq)+\ce{2MnO4-}(aq)+\ce{(2H+ + 2OH- )}(aq)⟶\ce{ClO3-}(aq)+\ce{2MnO2}(s)+\ce{H2O}(l)+\ce{2OH-}(aq) \nonumber \]
\[\ce{Cl-}(aq)+\ce{2MnO4-}(aq)+\ce{(2H2O)}(l)⟶\ce{ClO3-}(aq)+\ce{2MnO2}(s)+\ce{H2O}(l)+\ce{2OH-}(aq) \nonumber \]
Note that both sides of the equation show water. Simplifying should be done when necessary, and gives the desired equation. In this case, it is necessary to remove one H 2 O from each side of the reaction arrows.
\[\ce{Cl-}(aq)+\ce{2MnO4-}(aq)+\ce{H2O}(l)⟶\ce{ClO3-}(aq)+\ce{2MnO2}(s)+\ce{2OH-}(aq) \nonumber \]
Again, check each side of the overall equation to make sure there are no errors:
\[\begin{align*}
&\textrm{Cl: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{Mn: }\mathrm{Does\:(2×1)=(2×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(1×2)=(2×1)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(2×4+1×1)=(3×1+2×2+2×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[1×(−1)+2×(−1)]=[1×(−1)+2×(−1)]?\:Yes.}
\end{align*} \nonumber \]
Everything checks, so this is the overall equation in basic solution.
Balance the following reaction equation in acidic solution:
\[\ce{MnO4-}(aq)+\ce{Cr^3+}(aq)⟶\ce{Mn^2+}(aq)+\ce{Cr2O7^2-}(aq) \nonumber \]
Sol ution
This is an oxidation-reduction reaction, so start by collecting the species given into an unbalanced oxidation half-reaction and an unbalanced reduction half-reaction.
&\textrm{oxidation (unbalanced): }\ce{Cr^3+}(aq)⟶\ce{Cr2O7^2-}(aq)\\
&\textrm{reduction (unbalanced): }\ce{MnO4-}(aq)⟶\ce{Mn^2+}(aq)
\end{align*} \nonumber \]
Starting with the oxidation half-reaction, we can balance the chromium
\[\textrm{oxidation (unbalanced): }\ce{2Cr^3+}(aq)⟶\ce{Cr2O7^2-}(aq) \nonumber \]
In acidic solution, we can use or generate hydrogen ions (H + ). Adding seven water molecules to the left side provides the necessary oxygen; the “left over” hydrogen appears as 14 H + on the right:
\[\textrm{oxidation (unbalanced): }\ce{2Cr^3+}(aq)+\ce{7H2O}(l)⟶\ce{Cr2O7^2-}(aq)+\ce{14H+}(aq) \nonumber \]
The left side of the equation has a total charge of [2 × (+3) = +6], and the right side a total charge of [−2 + 14 × (+1) = +12]. The difference is six; adding six electrons to the right side produces a mass- and charge-balanced oxidation half-reaction (in acidic solution):
\[\textrm{oxidation (balanced): }\ce{2Cr^3+}(aq)+\ce{7H2O}(l)⟶\ce{Cr2O7^2-}(aq)+\ce{14H+}(aq)+\ce{6e-} \nonumber \]
Checking the half-reaction:
\[\begin{align*}
&\textrm{Cr: }\mathrm{Does\:(2×1)=(1×2)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(7×2)=(14×1)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(7×1) =(1×7)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[2×(+3)]=[1×(−2)+14×(+1)+6×(−1)]?\:Yes.}
\end{align*} \nonumber \]
Now work on the reduction. It is necessary to convert the four oxygen atoms in the permanganate into four water molecules. To do this, add eight H + to convert the oxygen into four water molecules:
\[\textrm{reduction (unbalanced): }\ce{MnO4-}(aq)+\ce{8H+}(aq)⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l) \nonumber \]
Then add five electrons to the left side to balance the charge:
\[\textrm{reduction (balanced): }\ce{MnO4-}(aq)+\ce{8H+}(aq)+\ce{5e-}⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l) \nonumber \]
Make sure to check the half-reaction:
\[\begin{align*}
&\textrm{Mn: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(8×1)=(4×2)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(1×4)=(4×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[1×(−1)+8×(+1)+5×(−1)]=[1×(+2)]?\:Yes.}
\end{align*} \nonumber \]
Collecting what we have so far:
\[\begin{align*}
&\textrm{oxidation: }\ce{2Cr^3+}(aq)+\ce{7H2O}(l)⟶\ce{Cr2O7^2-}(aq)+\ce{14H+}(aq)+\ce{6e-}\\
&\textrm{reduction: }\ce{MnO4-}(aq)+\ce{8H+}(aq)+\ce{5e-}⟶\ce{Mn^2+}(aq)+\ce{4H2O}(l)
\end{align*} \nonumber \]
The least common multiple for the electrons is 30, so multiply the oxidation half-reaction by five, the reduction half-reaction by six, combine, and simplify:
\[\ce{10Cr^3+}(aq)+\ce{35H2O}(l)+\ce{6MnO4-}(aq)+\ce{48H+}(aq)⟶\ce{5Cr2O7^2-}(aq)+\ce{70H+}(aq)+\ce{6Mn^2+}(aq)+\ce{24H2O}(l) \nonumber \]
\[\ce{10Cr^3+}(aq)+\ce{11H2O}(l)+\ce{6MnO4-}(aq)⟶\ce{5Cr2O7^2-}(aq)+\ce{22H+}(aq)+\ce{6Mn^2+}(aq) \nonumber \]
Checking each side of the equation:
\[\begin{align*}
&\textrm{Mn: }\mathrm{Does\:(6×1)=(6×1)?\:Yes.}\\
&\textrm{Cr: }\mathrm{Does\:(10×1)=(5×2)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(11×2)=(22×1)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(11×1+6×4)=(5×7)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[10×(+3)+6×(−1)]=[5×(−2)+22×(+1)+6×(+2)]?\:Yes.}
\end{align*} \nonumber \]
This is the balanced equation in acidic solution.
Balance the following equation in acidic solution:
\[\ce{Hg2^2+ + Ag ⟶ Hg + Ag+} \nonumber \]
- Answer
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\[\ce{Hg2^2+}(aq)+\ce{2Ag}(s)⟶\ce{2Hg}(l)+\ce{2Ag+}(aq) \nonumber \]
Balance the following reaction equation in basic solution:
\[\ce{MnO4-}(aq)+\ce{Cr(OH)3}(s)⟶\ce{MnO2}(s)+\ce{CrO4^2-}(aq) \nonumber \]
Soluti on
This is an oxidation-reduction reaction, so start by collecting the species given into an unbalanced oxidation half-reaction and an unbalanced reduction half-reaction
\[\begin{align*}
&\textrm{oxidation (unbalanced): }\ce{Cr(OH)3}(s)⟶\ce{CrO4^2-}(aq)\\
&\textrm{reduction (unbalanced): }\ce{MnO4-}(aq)⟶\ce{MnO2}(s)
\end{align*} \nonumber \]
Starting with the oxidation half-reaction, we can balance the chromium
\[\textrm{oxidation (unbalanced): }\ce{Cr(OH)3}(s)⟶\ce{CrO4^2-}(aq) \nonumber \]
In acidic solution, we can use or generate hydrogen ions (H + ). Adding one water molecule to the left side provides the necessary oxygen; the “left over” hydrogen appears as five H + on the right side:
\[\textrm{oxidation (unbalanced): }\ce{Cr(OH)3}(s)+\ce{H2O}(l)⟶\ce{CrO4^2-}(aq)+\ce{5H+}(aq) \nonumber \]
The left side of the equation has a total charge of [0], and the right side a total charge of [−2 + 5 × (+1) = +3]. The difference is three, adding three electrons to the right side produces a mass- and charge-balanced oxidation half-reaction (in acidic solution):
\[\textrm{oxidation (balanced): }\ce{Cr(OH)3}(s)+\ce{H2O}(l)⟶\ce{CrO4^2-}(aq)+\ce{5H+}(aq)+\ce{3e-} \nonumber \]
Checking the half-reaction:
\[\begin{align*}
&\textrm{Cr: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(1×3+1×2)=(5×1)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(1×3+1×1)=(4×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does \:[0=[1×(−2)+5×(+1)+3×(−1)]?\:Yes.}
\end{align*} \nonumber \]
Now work on the reduction. It is necessary to convert the four O atoms in the MnO 4 − minus the two O atoms in MnO 2 into two water molecules. To do this, add four H + to convert the oxygen into two water molecules:
\[\textrm{reduction (unbalanced): }\ce{MnO4-}(aq)+\ce{4H+}(aq)⟶\ce{MnO2}(s)+\ce{2H2O}(l) \nonumber \]
Then add three electrons to the left side to balance the charge:
\[\textrm{reduction (balanced): }\ce{MnO4-}(aq)+\ce{4H+}(aq)+\ce{3e-}⟶\ce{MnO2}(s)+\ce{2H2O}(l) \nonumber \]
Make sure to check the half-reaction:
\[\begin{align*}
&\textrm{Mn: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(4×1)=(2×2)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(1×4)=(1×2+2×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[1×(−1)+4×(+1)+3×(−1)]=[0]?\:Yes.}
\end{align*} \nonumber \]
Collecting what we have so far:
\[\begin{align*}
&\textrm{oxidation: }\ce{Cr(OH)3}(s)+\ce{H2O}(l)⟶\ce{CrO4^2-}(aq)+\ce{5H+}(aq)+\ce{3e-}\\
&\textrm{reduction: }\ce{MnO4-}(aq)+\ce{4H+}(aq)+\ce{3e-}⟶\ce{MnO2}(s)+\ce{2H2O}(l)
\end{align*} \nonumber \]
In this case, both half reactions involve the same number of electrons; therefore, simply add the two half-reactions together.
\[\ce{MnO4-}(aq)+\ce{4H+}(aq)+\ce{Cr(OH)3}(s)+\ce{H2O}(l)⟶\ce{CrO4^2-}(aq)+\ce{MnO2}(s)+\ce{2H2O}(l)+\ce{5H+}(aq) \nonumber \]
\[\ce{MnO4-}(aq)+\ce{Cr(OH)3}(s)⟶\ce{CrO4^2-}(aq)+\ce{MnO2}(s)+\ce{H2O}(l)+\ce{H+}(aq) \nonumber \]
Checking each side of the equation:
\[\begin{align*}
&\textrm{Mn: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{Cr: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(1×3)=(2×1+1×1)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(1×4+1×3)=(1×4+1×2+1×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[1×(−1)]=[1×(−2)+1×(+1)]?\:Yes.}
\end{align*} \nonumber \]
This is the balanced equation in acidic solution. For a basic solution, add one hydroxide ion to each side and simplify:
\[\ce{OH-}(aq)+\ce{MnO4-}(aq)+\ce{Cr(OH)3}(s)⟶\ce{CrO4^2-}(aq)+\ce{MnO2}(s)+\ce{H2O}(l)+\ce{(H+ + OH- )}(aq) \nonumber \]
\[\ce{OH-}(aq)+\ce{MnO4-}(aq)+\ce{Cr(OH)3}(s)⟶\ce{CrO4^2-}(aq)+\ce{MnO2}(s)+\ce{2H2O}(l) \nonumber \]
Checking each side of the equation:
\[\begin{align*}
&\textrm{Mn: }\mathrm{Does\:(1×1)=(1×1)?\: Yes.}\\
&\textrm{Cr: }\mathrm{Does\:(1×1)=(1×1)?\:Yes.}\\
&\textrm{H: }\mathrm{Does\:(1×1+1×3)=(2×2)?\:Yes.}\\
&\textrm{O: }\mathrm{Does\:(1×1+1×4+1×3)=(1×4+1×2+2×1)?\:Yes.}\\
&\textrm{Charge: }\mathrm{Does\:[1×(−1)+1×(−1)]=[1×(−2)]?\:Yes.}
\end{align*} \nonumber \]
This is the balanced equation in basic solution.
Balance the following in the type of solution indicated.
- \(\ce{H2 + Cu^2+ ⟶ Cu \:\:\:(acidic\: solution)}\)
- \(\ce{H2 + Cu(OH)2 ⟶ Cu\:\:\:(basic\: solution)}\)
- \(\ce{Fe + Ag+ ⟶ Fe^2+ + Ag}\)
- Identify the oxidizing agents in reactions (a), (b), and (c).
- Identify the reducing agents in reactions (a), (b), and (c).
- Answer a
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\(\ce{H2}(g)+\ce{Cu^2+}(aq)⟶\ce{2H+}(aq)+\ce{Cu}(s)\)
- Answer b
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\(\ce{H2}(g)+\ce{Cu(OH)2}(s)⟶\ce{2H2O}(l)+\ce{Cu}(s)\)
- Answer c
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\(\ce{Fe}(s)+\ce{2Ag+}(aq)⟶\ce{Fe^2+}(aq)+\ce{2Ag}(s)\)
- Answer d
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oxidizing agent = species reduced: Cu 2 + , Cu(OH) 2 , Ag +
- Answer e
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reducing agent = species oxidized: H 2 , H 2 , Fe.
Summary
An electric current consists of moving charge. The charge may be in the form of electrons or ions. Current flows through an unbroken or closed circular path called a circuit. The current flows through a conducting medium as a result of a difference in electrical potential between two points in a circuit. Electrical potential has the units of energy per charge. In SI units, charge is measured in coulombs (C), current in amperes \(\mathrm{\left(A=\dfrac{C}{s}\right)}\), and electrical potential in volts \(\mathrm{\left(V=\dfrac{J}{C}\right)}\).
Oxidation is the loss of electrons, and the species that is oxidized is also called the reducing agent. Reduction is the gain of electrons, and the species that is reduced is also called the oxidizing agent. Oxidation-reduction reactions can be balanced using the half-reaction method. In this method, the oxidation-reduction reaction is split into an oxidation half-reaction and a reduction half-reaction. The oxidation half-reaction and reduction half-reaction are then balanced separately. Each of the half-reactions must have the same number of each type of atom on both sides of the equation and show the same total charge on each side of the equation. Charge is balanced in oxidation half-reactions by adding electrons as products; in reduction half-reactions, charge is balanced by adding electrons as reactants. The total number of electrons gained by reduction must exactly equal the number of electrons lost by oxidation when combining the two half-reactions to give the overall balanced equation. Balancing oxidation-reduction reaction equations in aqueous solutions frequently requires that oxygen or hydrogen be added or removed from a reactant. In acidic solution, hydrogen is added by adding hydrogen ion (H + ) and removed by producing hydrogen ion; oxygen is removed by adding hydrogen ion and producing water, and added by adding water and producing hydrogen ion. A balanced equation in basic solution can be obtained by first balancing the equation in acidic solution, and then adding hydroxide ion to each side of the balanced equation in such numbers that all the hydrogen ions are converted to water.
Glossary
- circuit
- path taken by a current as it flows because of an electrical potential difference
- current
- flow of electrical charge; the SI unit of charge is the coulomb (C) and current is measured in amperes \(\mathrm{\left(1\: A=1\:\dfrac{C}{s}\right)}\)
- electrical potential
- energy per charge; in electrochemical systems, it depends on the way the charges are distributed within the system; the SI unit of electrical potential is the volt \(\mathrm{\left(1\: V=1\:\dfrac{J}{C}\right)}\)
- half-reaction method
- method that produces a balanced overall oxidation-reduction reaction by splitting the reaction into an oxidation “half” and reduction “half,” balancing the two half-reactions, and then combining the oxidation half-reaction and reduction half-reaction in such a way that the number of electrons generated by the oxidation is exactly canceled by the number of electrons required by the reduction
- oxidation half-reaction
- the “half” of an oxidation-reduction reaction involving oxidation; the half-reaction in which electrons appear as products; balanced when each atom type, as well as the charge, is balanced
- reduction half-reaction
- the “half” of an oxidation-reduction reaction involving reduction; the half-reaction in which electrons appear as reactants; balanced when each atom type, as well as the charge, is balanced


4.E: Permutations with Similar Elements (Optional Exercises)
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Do the following problems using the techniques learned in this section.
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In how many different ways can five children hold hands to play "Ring Around the Rosy"?
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In how many ways can three people be made to sit at a round table?
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In how many different ways can six children ride a "Merry Go Around" with six horses?
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In how many ways can three couples be seated at a round table, so that men and women sit alternately?
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In how many ways can six trinkets be arranged on a chain?
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In how many ways can five keys be put on a key ring?
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Find the number of different permutations of the letters of the word MASSACHUSETTS.
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Find the number of different permutations of the letters of the word MATHEMATICS.
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Seven flags are to be flown on seven poles: 3 flags are red, 2 are white, and 2 are blue,. How many different arrangements are possible?
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How many different ways can 3 pennies, 2 nickels and 5 dimes be arranged in a row?
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How many four-digit numbers can be made using two 2's and two 3's?
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How many five-digit numbers can be made using two 6's and three 7's?
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If a coin is tossed 5 times, how many different outcomes of 3 heads and 2 tails are possible?
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If a coin is tossed 10 times, how many different outcomes of 7 heads and 3 tails are possible?
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If a team plays ten games, how many different outcomes of 6 wins and 4 losses are possible?
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If a team plays ten games, how many different ways can the team have a winning season?
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1.1: Scope of Practice Introduction
- Discuss nursing scope of practice and standards of care
- Compare various settings in which nurses work
- Describe contributions of interprofessional health care team members
- Describe levels of nursing education and the NCLEX
- Discuss basic legal considerations and ethics
- Outline professional nursing organizations
- Examine quality and evidence-based practice in nursing
You are probably wondering, “What is scope of practice? What does it mean for me and my nursing practice?” Scope of practice is defined as services that a trained health professional is deemed competent to perform and permitted to undertake according to the terms of their professional nursing license. [1] Nursing scope of practice provides a framework and structured guidance for activities one can perform based on their nursing license. As a nurse and a nursing student, is always important to consider: Just because your employer asks you to do a task…can you perform this task according to your scope of practice – or are you putting your nursing license at risk?
Nurses must also follow legal standards in when providing nursing care. Standards are set by several organizations, including the American Nurses Association (ANA), your state’s Nurse Practice Act, agency policies and procedures, and federal regulators. These standards assure safe, competent care is provided to the public.
This chapter will provide an overview of basic concepts related to nursing scope of practice and standards of care.
- American Nurses Association. (n.d.). Scope of practice. https://www.nursingworld.org/practice-policy/scope-of-practice/ ↵


6.2: Mindfulness Practice
Breathe and Smile Receptive – put away all distractions, focusing on the context of RS 3 and RS 4. Construct 10 ASL sentences, 5 with RS Things Around the House as the topic of conversation and 5 with RS 4 Foods and Preparation as the topic of conversation. Include a related Fingerspelled loan sign in each appropriate sentence. Practice your sentences throughout the week until they are fluid and smooth without any hitches, embodying all fingerspelling habits learned in Lessons 1-5, then proceed to the following:
- Find a partner who has done the same Lesson 6 Mindfulness Practice 1
- Put aside all distractions and focus on maintaining good fingerspelling habits learned in Lessons 1-5
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Remember your context of conversation – Things Around the House and Foods and
Preparation. Take turns signing one ASL sentence at a time, incorporating the appropriate loan sign with clarity or speed and concentrating on fluidity and ease. Your partner fingerspells only the loan sign back to you as confirmation that they understood your fingerspelling.
- When your mind wonders, because it will, bring it back to the topic of conversation, eye contact, breathing and smiling.
When you complete the above practice, answer the following questions:
- What distractions did you notice?
- How present were you on a scale of 1 – 10 (1 being not very present, 10 being fully present)
- Did you maintain eye contact or fall back on the habit of looking at the hand?
- What prevented you from being present?
- How many of the 10 loan signs were you able to catch?