Solution Definition in Chemistry

A solution is a homogeneous mixture of two or more substances. A solution may exist in any phase. A solution consists of a solute and a solvent. The solute is the substance that is dissolved in the solvent. The amount of solute that can be dissolved in solvent is called its solubility. For example, in a saline solution, salt is the solute dissolved in water as the solvent. For solutions with components in the same phase, the substances present in lower concentration are solutes, while the substance present in highest abundance is the solvent. Using air as an example, oxygen and carbon dioxide gases are solutes, while nitrogen gas is the solvent. Characteristics of a Solution A chemical solution exhibits several properties: A solution consists of a homogeneous mixture.A solution is composed of one phase (e.g., solid, liquid, gas).Particles in a solution are not visible to the naked eye.A solution does not scatter a light beam.Components of a solution cannot be separated using simple mechanical filtration. Solution Examples Any two substances which can be evenly mixed may form a solution. Even though materials of different phases may combine to form a solution, the end result always exists of a single phase. An example of a solid solution is brass. An example of a liquid solution is aqueous hydrochloric acid (HCl in water). An example of a gaseous solution is air. Solution Type Example gas-gas air gas-liquid carbon dioxide in soda gas-solid hydrogen gas in palladium metal liquid-liquid gasoline solid-liquid sugar in water liquid-solid mercury dental amalgam solid-solid sterling silver

The Changes In The Narrators View Of Sonny Essay - 1269 Words

The Changes in the Narrators View of Sonny Can one know anothers thoughts? Through dialogue, actions, and events, the thoughts and views of a man of whom we know not even a name are shown. The man is the narrator of quot;Sonnys Bluesquot; and his thoughts we are shown are those directed towards his brother. Over the course of the story, there are three major stages or phases that the narrator goes through, in which his thoughts about his brother change. We see that those stages of thought vary greatly over the narrators life, from confusion about his brother to understanding. Each phase brings different views of his own responsibility toward his brother, his brothers manhood, and his brothers sense of reality. Through out the†¦show more content†¦Therefore, whenever he did something for Sonny it was because his mother had wanted him to, not because he cared about Sonny. As soon as taking care of Sonny stopped working with his schedule, he sent him to his mother-in-laws house. During the story, however, a long separation brought the narrator into his second stage of thinking, and changed his views of Sonny. The narrator recognized that Sonny wasnt just a kid any more. Sonny had been in the Navy and had been living on his own for some time. Yet he didnt see him as a man either. quot;He was a man by then, of course, but I wasnt willing to see it.quot;(52) He saw Sonny as a teenager of sorts. Sonny dressed strangely, became family with strange friends, and listened to still stranger music.quot; In the narrators eyes, Sonny foolishly thought he knew everything. Even though the narrators views on Sonnys manhood changed, during the second stage his feelings about Sonnys sense of reality didnt. When he saw Sonny after Sonnys stay in the Navy, the narrator still viewed Sonny as if he were on drugs. quot;He carried himself, loose and dreamlike all the time, ...and his music seemed to be merely an excuse for the life he led. It sounded just that weird and disordered.quot;(52) He thought that Sonny had been driven even farther from reality than before. He thought that Sonnys view of reality was so distorted that he might as well have beenShow MoreRelatedThe Sounds of Sonnys Blues1247 Words   |  5 Pagesstory focused around the narration of Sonny’s brother. The narrator in the case of Sonny’s Blues is the most important character in a cast of characters not only because he is the narrator, but due to the dynamic change of his character we see at the end of the story. Baldwin effectively uses the first-person narration of Sonny’s brother in order to convey the theme of communication. Throughout the entire story of Sonny’s Blues, the narrator and his brother interact through exchanging words countlessRead MoreEssay on Sonnys Blues Character Analysis843 Words   |  4 PagesSonny’s brother who is the narrator and goes through his life and how he reacts to the many problems his younger brother has come into. The brothers grew up in the poverty stricken city of Harlem where the brothers had to avoid drugs and violence constantly. Growing up, Sonny struggled to stay out of trouble and ended up making some bad decisions throughout his life and ends up landing him in jail and addicted to heroin. The un-named brother of Sonny who is the narrator of the story begins to realizeRead MoreAnalysis Of James Baldwin s Sonny s Blues 1578 Words   |  7 Pagesin Europe, at young age of 24. Finally, after 10 years of working all over Europe, Baldwin returned home to New York and began writing there. Sonny, just like Baldwin, had to leave Harlem at a very young age to chase what he truly wanted to do, which was music. Though it worked out fo r Baldwin and he was successful by the time he returned home, Sonny and Baldwin both eventually did end up back in Harlem, the place they so badly wanted to escape in their earlier years. â€Å"Sonny’s Blues† is a shortRead MoreAnalysis Of The Story Sonny s Blues By James Baldwin1481 Words   |  6 PagesIn reading the story Sonny s Blues by James Baldwin, we learn of two brothers and their lives growing up in Harlem. The narrator, who is the older brother in the story, narrates the trials and tribulations he and his younger brother (Sonny) had to endure growing up in such a harsh environment in Harlem (due to the drugs, violence, and Black s being looked down upon in general in the mid-1950s). We start in the future (present), with the narrator having a somewhat successful future being a teacherRead MoreSetting Analysis : Sonny s Blues 921 Words   |  4 Pagesbrotherhood between narrator and his young brother, Sonny. After many conflicts and arguments about the ir different ideals and lifestyles, Sonny tries to open his heart to let his brother understand him by inviting the narrator to come to his jazz music performance at a small nightclub in Greenwich Village. At this place, he meets friends of Sonny, acquaint himself with jazz music and tries to get into Sonny’s world. He carefully observes any changes of his brother on the stage. Sonny is nervous and hasRead MoreBrotherly Love in Sonny’s Blues by James Baldwin1196 Words   |  5 Pagesevent that had a great impact on his relationship with his brother, Sonny. Having to deal with the life-style of poverty, his relationship with his brother becomes affected and rivalry develops. Conclusively, brotherly love is the theme of the story. Despite the narrator’s and his brother’s differences, this theme is revealed throughout the characters’ thoughts, feelings, actions, and dialogue. Therefore, the change in the narrator throughout the text is significant in understanding the theme of theRead MoreJames Baldwin s Reality Through Sonny1253 Words   |  6 Pages James Baldwin’s Reality through Sonny Sonny’s Blues digs deeply into the â€Å"Negro American† during Civil Rights and Jim Crow Era’s. Set in Harlem, New York in the 1950’s. James Baldwin’s stories give insight based on events of his culture and this becomes apparent through the analysis of the characters in Sonny’s Blues. James Baldwin uses his talents to paint a vivid picture of African American life through a fictional story of two brothers who choseRead MorePoint of View and Symbolism in Sonnys Blues1558 Words   |  7 PagesPoint of View and Symbolism in â€Å"Sonny’s Blues† The story â€Å"Sonny’s Blues† by James Baldwin makes excellent use of multiple literary elements. Namely, I think the writer utilizes symbolism and the nuances of point of view to give the story a deeper connotation that could not be said plainly. The meat of the story is about an unnamed older brother’s relationship and differences with his younger brother, Sonny. Sonny’s aspiration to become a jazz pianist leads him in an opposite direction than hisRead MoreDrugs and Musicians in â€Å"Sonny’s Blues† by James Baldwin Essay840 Words   |  4 Pages The last statement is what the narrator believes to be true. However, by delving deeper and examining the theme of music in the story, it is nothing but beneficial for Sonny and the other figures involved. Sonny’s drug use and his music are completely free of one another. Sonny views his jazz playing as a ray of light to lead him away from the dim and dismal future that Harlem has to off er. The first moment music is introduced in the story is while the narrator is teaching at school. He has justRead MoreEssay on Sonnys Blues by James Baldwin1316 Words   |  6 Pagesâ€Å"Sonny’s Blues† revolves around the narrator as he learns who his drug-hooked, piano-playing baby brother, Sonny, really is. The author, James Baldwin, paints views on racism, misery and art and suffering in this story. His written canvas portrays a dark and continual scene pertaining to each topic. As the story unfolds, similarities in each generation can be observed. The two African American brothers share a life similar to that of their father and his brother. The father’s brother had a thirst

Lacsap’s Fractions Free Essays

Lacsap’s Fractions IB Math 20 Portfolio By: Lorenzo Ravani Lacsap’s Fractions Lacsap is backward for Pascal. If we use Pascal’s triangle we can identify patterns in Lacsap’s fractions. The goal of this portfolio is to ? nd an equation that describes the pattern presented in Lacsap’s fraction. We will write a custom essay sample on Lacsap’s Fractions or any similar topic only for you Order Now This equation must determine the numerator and the denominator for every row possible. Numerator Elements of the Pascal’s triangle form multiple horizontal rows (n) and diagonal rows (r). The elements of the ? rst diagonal row (r = 1) are a linear function of the row number n. For every other row, each element is a parabolic function of n. Where r represents the element number and n represents the row number. The row numbers that represents the same sets of numbers as the numerators in Lacsap’s triangle, are the second row (r = 2) and the seventh row (r = 7). These rows are respectively the third element in the triangle, and equal to each other because the triangle is symmetrical. In this portfolio we will formulate an equation for only these two rows to ? nd Lacsap’s pattern. The equation for the numerator of the second and seventh row can be represented by the equation: (1/2)n * (n+1) = Nn (r) When n represents the row number. And Nn(r) represents the numerator Therefore the numerator of the sixth row is Nn(r) = (1/2)n * (n+1) Nn(r) = (1/2)6 * (6+1) Nn(r) = (3) * (7) Nn(r) = 21 Figure 2: Lacsap’s fractions. The numbers that are underlined are the numerators. Which are the same as the elements in the second and seventh row of Pascal’s triangle. Figure 1: Pascal’s triangle. The circled sets of numbers are the same as the numerators in Lacsap’s fractions. Graphical Representation The plot of the pattern represents the relationship between numerator and row number. The graph goes up to the ninth row. The rows are represented on the x-axis, and the numerator on the y-axis. The plot forms a parabolic curve, representing an exponential increase of the numerator compared to the row number. Let Nn be the numerator of the interior fraction of the nth row. The graph takes the shape of a parabola. The graph is parabolical and the equation is in the form: Nn = an2 + bn + c The parabola passes through the points (0,0) (1,1) and (5,15) At (0,0): 0 = 0 + 0 + c ! ! At (1,1): 1 = a + b ! ! ! At (5,15): 15 = 25a + 5b ! ! ! 15 = 25a + 5(1 – a) ! 15 = 25a + 5 – 5a ! 15 = 20a + 5 ! 10 = 20a! ! ! ! ! ! ! therefore c = 0 therefore b = 1 – a Check with other row numbers At (2,3): 3 = (1/2)n * (n+1) ! (1/2)(2) * (2+1) ! (1) * (3) ! N3 = (3) therefore a = (1/2) Hence b = (1/2) as well The equation for this graph therefore is Nn = (1/2)n2 + (1/2)n ! which simpli? es into ! Nn = (1/2)n * (n+1) Denominator The difference between the numerator and the denominator of the same fraction w ill be the difference between the denominator of the current fraction and the previous fraction. Ex. If you take (6/4) the difference is 2. Therefore the difference between the previous denominator of (3/2) and (6/4) is 2. ! Figure 3: Lacsap’s fractions showing differences between denominators Therefore the general statement for ? nding the denominator of the (r+1)th element in the nth row is: Dn (r) = (1/2)n * (n+1) – r ( n – r ) Where n represents the row number, r represents the the element number and Dn (r) represents the denominator. Let us use the formula we have obtained to ?nd the interior fractions in the 6th row. Finding the 6th row – First denominator ! ! ! ! ! ! ! ! ! ! ! ! – Second denominator ! ! ! ! ! ! ! ! ! ! ! ! ! denominator = 6 ( 6/2 + 1/2 ) – 1 ( 6 – 1 ) ! = 6 ( 3. 5 ) – 1 ( 5 ) ! 21 – 5 = 16 denominator = 6 ( 6/2 + 1/2 ) – 2 ( 6 – 2 ) ! = 6 ( 3. 5 ) – 2 ( 4 ) ! = 21 – 8 = 13 ! ! -Third denominator ! ! ! ! ! ! ! ! ! ! ! ! – Fourth denominator ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! – Fifth denominator ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! denominator = 6 ( 6/2 + 1/2 ) – 3 ( 6 – 3 ) ! = 6 ( 3. 5 ) – 3 ( 3 ) ! = 21 – 9 = 12 denominator = 6 ( 6/2 + 1/2 ) – 2 ( 6 – 2 ) ! = 6 ( 3. 5 ) – 2 ( 4 ) ! = 21 – 8 = 13 denominator = 6 ( 6/2 + 1/2 ) – 1 ( 6 – 1 ) ! = 6 ( 3. 5 ) – 1 ( 5 ) ! = 21 – 5 = 16 ! ! We already know from the previous investigation that the numerator is 21 for all interior fractions of the sixth row. Using these patterns, the elements of the 6th row are 1! (21/16)! (21/13)! (21/12)! (21/13)! (21/16)! 1 Finding the 7th row – First denominator ! ! ! ! ! ! ! ! ! ! ! ! – Second denominator ! ! ! ! ! ! ! ! ! ! ! ! – Third denominator ! ! ! ! ! ! ! ! ! ! ! ! – Fourth denominator ! ! ! ! ! ! ! ! ! ! ! ! ! ! denominator = 7 ( 7/2 + 1/2 ) – 1 ( 7 – 1 ) ! =7(4)–1(6) ! = 28 – 6 = 22 denominator = 7 ( 7/2 + 1/2 ) – 2 ( 7 – 2 ) ! =7(4)–2(5) ! = 28 – 10 = 18 denominator = 7 ( 7/2 + 1/2 ) – 3 ( 7 – 3 ) ! =7(4)–3(4) ! = 28 – 12 = 16 denominator = 7 ( 7/2 + 1/2 ) – 4 ( 7 – 3 ) ! =7(4)–3(4) ! = 28 – 12 = 16 ! ! ! ! ! ! Fifth denominator ! ! ! ! ! ! ! ! ! ! ! ! – Sixth denominator ! ! ! ! ! ! ! ! ! ! ! ! denominator = 7 ( 7/2 + 1/2 ) – 2 ( 7 – 2 ) ! ! =7(4)–2(5) ! ! = 28 – 10 = 18 ! ! denominator = 7 ( 7/2 + 1/2 ) – 1 ( 7 – 1 ) ! =7(4)–1(6) ! = 28 – 6 = 22 We already know from the previous investigation that the numerator is 28 for all interior fractions of the seventh row. Using these patterns, the elements of the 7th row are 1 (28/22) (28/18) (28/16) (28/16) (28/18) (28/22) 1 General Statement To ? nd a general statement we combined the two equations needed to ? nd the numerator and to ? nd the denominator. Which are (1/2)n * (n+1) to ? d the numerator and (1/2)n * (n+1) – n( r – n) to ? nd the denominator. By letting En(r) be the ( r + 1 )th element in the nth row, the general statement is: En(r) = {[ (1/2)n * (n+1) ] / [ (1/2)n * (n+1) – r( n – r) ]} Where n represents the row number and r represents the the element number. Limitations The ‘1’ at the beginning and end of each row is taken out before making calculations. Therefore the second element in each equation is now regarded as the ? rst element. Secondly, the r in the general statement should be greater than 0. Thirdly the very ? rst row of the given pattern is counted as the 1st row. Lacsap’s triangle is symmetrical like Pascal’s, therefore the elements on the left side of the line of symmetry are the same as the elements on the right side of the line of symmetry, as shown in Figure 4. Fourthly, we only formulated equations based on the second and the seventh rows in Pascal’s triangle. These rows are the only ones that have the same pattern as Lacsap’s fractions. Every other row creates either a linear equation or a different parabolic equation which doesn’t match Lacsap’s pattern. Lastly, all fractions should be kept when reduced; provided that no fractions common to the numerator and the denominator are to be cancelled. ex. 6/4 cannot be reduced to 3/2 ) Figure 4: The triangle has the same fractions on both sides. The only fractions that occur only once are the ones crossed by this line of symmetry. 1 Validity With this statement you can ? nd any fraction is Lacsap’s pattern and to prove this I will use this equa tion to ? nd the elements of the 9th row. The subscript represents the 9th row, and the number in parentheses represents the element number. – E9(1)!! ! – First element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(2)!! ! – Second element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(3)!! ! – Third element! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 1( 9 – 1) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 1( 8 ) ]} {[ 45 ] / [ 45 – 8 ]} {[ 45 ] / [ 37 ]} 45/37 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 2( 9 – 2) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 2 ( 7 ) ]} {[ 45 ] / [ 45 – 14 ]} {[ 45 ] / [ 31 ]} 45/31 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 3 ( 9 – 3) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 3( 6 ) ]} {[ 45 ] / [ 45 – 18 ]} {[ 45 ] / [ 27 ]} 45/27 E9(4)!! ! – Fourth element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(4)!! ! – Fifth element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(3)!! ! – Sixth element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(2)!! ! – Seventh element! ! ! ! ! ! ! ! ! ! ! ! ! – E9(1)!! ! – Eighth element! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! [ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 4( 9 – 4) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 4( 5 ) ]} {[ 45 ] / [ 45 – 20 ]} {[ 45 ] / [ 25 ]} 45/25 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 4( 9 – 4) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 4( 5 ) ]} {[ 45 ] / [ 45 – 20 ]} {[ 45 ] / [ 25 ]} 45/25 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 3 ( 9 – 3) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 3( 6 ) ]} {[ 45 ] / [ 45 – 18 ]} {[ 45 ] / [ 27 ]} 45/27 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 2( 9 – 2) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 2 ( 7 ) ]} {[ 45 ] / [ 45 – 14 ]} {[ 45 ] / [ 31 ]} 45/31 {[ n( n/2 + 1/2 ) ] / [ n( n/2 + 1/2 ) – r( n – r) ]} {[ 9( 9/2 + 1/2 ) ] / [ 9( 9/2 + 1/2 ) – 1( 9 – 1) ]} {[ 9( 5 ) ] / [ 9( 5 ) – 1( 8 ) ]} {[ 45 ] / [ 45 – 8 ]} {[ 45 ] / [ 37 ]} 45/37 From these calculations, derived from the general statement the 9th row is 1 (45/37)! ! (45/31)! ! (45/27)! (45/25)! (45/25)! (45/27) (45/31)! (45/37)! ! 1 Using the general statement we have obtained from Pascal’s triangle, and following the limitations stated, we will be able to produce the elements of any given row in Lacsap’s pattern. This equatio n determines the numerator and the denominator for every row possible. How to cite Lacsap’s Fractions, Essay examples

A Journey free essay sample

At nine months I learned to walk. At the age of five I learned how to ride a bike without training wheels. In kindergarten I learned that I’m allergic to peanut butter. In second grade, on September 11th, 2001, I learned that there is a lot of evil in the world. In fourth grade I learned how to do long division after having several mental breakdowns. In fifth grade, I learned that I sing like a dying animal while trying out for the school play, â€Å"The Little Mermaid.† In sixth grade I learned that change can be a good thing as I entered middle school and made new friends. In seventh grade I learned that I love the city of Boston after a school trip there. In ninth grade I learned that I hate geometry. In tenth grade I learned how to light a Bunsen burner. In eleventh grade I learned about rhetorical strategies in AP English. We will write a custom essay sample on A Journey or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page I’ve been alive for 17 and a half years and I’ve learned a lot. But I’m still learning. I’m learning new formulas, new words, new theories, new lessons and I’m learning about myself. I’m only 17 years old, and I still have a lot of things that I need to learn about myself. I don’t know how I feel about abortions. I don’t know if I’m a Republican or a Democrat. I don’t know if I’ll ever get married. I don’t know how I feel about the death penalty. I don’t know if I want to be a forensic scientist or a writer. When looking at the big picture, I don’t know much, but I do know that I love learning. I love knowing that I don’t know everything. I love knowing that there are tons of books that I haven’t read. I love knowing that there are hundreds of countries that I’ve never been to. I love knowing that there is a whole world out there that I have yet to learn about. I want to learn about it all. I hope to travel the world one day. I think that traveling is the best way to learn. Learn about other people, learn about other cultures, learn about life, learn about yourself. I’ve always wanted to travel to go someplace new and explore. Someplace where no one knows me. In sixth grade I made a pact with a group of my friends we would travel to Australia after we graduated high school, and we would raise dingos and koalas. We all signed a piece of paper while sitting in art class. We were determined to make it happen. Looking back, that was obviously an unrealistic plan. We’re not all friends anymore. No parents would let a group of teenagers travel to Australia by themselves. We didn’t know the first thing about dingos or koalas. And we totally disregarded the fact that we would be preparing for college after high school. Although our initial plan did not work out, my desire to go to Australia still lives on. In my life time I hope to go to Australia and swim in the Great Barrier Reef and learn about the animals that live there. I hope to go to Austria and ski the Alps. I hope to go to Spain and attend a soccer game. I hope to go to China and visit the Great Wall. I hope to go to Greece and see the Parthenon. I hope to travel to Ireland and go to the county where my Grandpa grew up. I hope to go to Kenya and go on a safari. I hope to go to India and visit the Taj Mahal. I think that traveling will allow me to learn. Learn all of the lessons that I have yet to uncover. Learn who I truly am. Learn what the world has to offer. Learn the secret to a happy life. Traveling will allow me to learn, and learning is the greatest gift that the world has to offer.