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Just Be a Man About It
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"Just Be a Man About it"
Single by Toni Braxton
from the album The Heat
Released
June 20, 2000 (2000-06-20)
Format
DVD
Recorded
1999
Studio
Noontime Music Studios
(Atlanta)
Genre
R&B, neo soul, pop
Length
4:50
Label
LaFace
Arista
Songwriter(s)
Toni Braxton
Johntá Austin
Teddy Bishop
Bryan-Michael Cox
Producer(s)
Teddy Bishop
Toni Braxton (co.)
Bryan-Michael Cox (co.)
Toni Braxton singles chronology
"He Wasn't Man Enough"
(2000) "Just Be a Man About it"
(2000) "Spanish Guitar"
(2000)
"He Wasn't Man Enough"
(2000) "Just Be a Man About It"
(2000) "Spanish Guitar"
(2000)
"Just Be a Man About It" is a song by American singer Toni Braxton from her third studio album, The Heat (2000). It was released on June 20, 2000, as the album's second single.
Contents
1 Background
2 Critical reception
3 Music video
4 Track listing
5 Charts
5.1 Weekly charts
5.2 Year-end charts
6 Release history
7 References
8 External links
Background[edit]
"Just Be a Man About It" is a telephone breakup song, where Braxton questions the status of a partner's manhood and Dr. Dre plays the wayward lover breaking the news to her.[1] According to producer Teddy Bishop, Martin Lawrence and Will Smith were originally considered to perform the speaking parts of the song. However, due to scheduling conflicts, it never happened. Dr. Dre was asked to appear on the song due to being in the studio next door while the song was being recorded.[2]
Critical reception[edit]
The song received mostly positive reviews from music critics. Stephen Thomas Erlewine from AllMusic called it "an instant classic" and picked it one of the best songs of the album, alongside "He Wasn't Man Enough" and "Spanish Guitar".[1] CD Universe wrote that "[e]ven Dr. Dre's guest appearance on 'Just Be a Man About It' nods to the sensual recitatives and bedside manners of '70s love men like Barry White."[3] Bog Roget wrote for Amazon that Braxton "delivers a tough take on reality with 'Just Be a Man About It,' which pits her against Dr. Dre in a breakup scenario that carries much more force than weightless trifles".[4]
Music video[edit]
The music video for "Just Be a Man About It", directed by Bille Woodruff, opens with Braxton's boyfriend (played by rapper Dr. Dre, who also provides additional vocals to the track) leaving a Hawthorne, California, strip club named Bare Elegance, accompanied by a woman. He stops at a payphone to call up Braxton, who is in their apartment. She looks happy at first, but as he tells her not to wait up for him that night alleging he needs space, her facial expression suddenly changes. The following scenes show Braxton wandering about the apartment, talking on the phone, standing next to the balcony, lying on a couch, and throwing objects such as vases and a picture frame containing a photo of Dre. At one point, Dre hangs up on Braxton, which infuriates her. Toward the end of the video, a man (Q-Tip) shows up to visit Braxton; they hug each other and proceed to cuddle on the couch, much to Dre's annoyance by the time he arrives home. When Dre questions Braxton about the man, she remains indifferent, causing him to give her the finger and leave.
Track listing[edit]
DVD single[5]
"Just Be a Man About It" (music video)
"Spanish Guitar" (music video)
Charts[edit]
Weekly charts[edit]
Chart (2000)
Peak
position
US Billboard Hot 100[6]
32
US Hot R&B/Hip-Hop Songs (Billboard)[7]
6
US Rhythmic (Billboard)[8]
36
Year-end charts[edit]
Chart (2000)
Position
US Hot R&B/Hip-Hop Songs (Billboard)[9]
24
Release history[edit]
Region
Date
Format
Label
Ref.
United States
June 20, 2000 Urban adult contemporary radio
LaFace
Arista
[10]
November 21, 2000 DVD single [5]
France
February 27, 2001 BMG [11]
References[edit]
^ Jump up to: a b Erlewine, Stephen Thomas. "The Heat – Toni Braxton". AllMusic. Retrieved April 26, 2018.
Jump up ^ "Interview: Radio Constantly Changes But Teddy Bishop's Ability To Diversify Allows The Quality Of His Music To Stay The Same". YouKnowIGotSoul.com. February 20, 2012. Retrieved August 17, 2013.
Jump up ^ "Toni Braxton - Heat CD Album". CD Universe. Retrieved February 16, 2012.
Jump up ^ Roget, Bob. "The Heat: Toni Braxton". Amazon (US). Retrieved February 15, 2012.
^ Jump up to: a b "Toni Braxton – Just Be a Man About It/Spanish Guitar (DVD Single)". Amazon (US). Retrieved April 26, 2018.
Jump up ^ "Toni Braxton Chart History (Hot 100)". Billboard. Retrieved April 26, 2018.
Jump up ^ "Toni Braxton Chart History (Hot R&B/Hip-Hop Songs)". Billboard. Retrieved April 26, 2018.
Jump up ^ "Toni Braxton Chart History (Rhythmic)". Billboard. Retrieved April 26, 2018.
Jump up ^ "R&B/Hip-Hop Songs – Year-End 2000". Billboard. Retrieved April 26, 2018.
Jump up ^ "Urban AC: Going For Adds (6/20/00)" (PDF). Radio & Records. June 16, 2000. p. 109. ISSN 0277-4860. Retrieved April 26, 2018 – via American Radio History.
Jump up ^ "Toni Braxton – Just Be a Man About It : Spanish Guitar & interview exclusive [DVD Single]". Amazon (France) (in French). Retrieved April 26, 2018.
External links[edit]
Lyrics of this song at MetroLyrics
show
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Un-Break My Heart: The Remix Collection
Breathe Again: Toni Braxton at Her Best
The Essential Toni Braxton
Playlist: The Very Best of Toni Braxton
Video albums
From Toni with Love... The Video Collection
Tours
Maze Tour
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Related articles
Discography
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Toni Braxton: Unbreak My Heart
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Discography
Toni Braxton
"Another Sad Love Song"
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"You Mean the World to Me"
"I Belong to You/How Many Ways"
Secrets
"You're Makin' Me High"
"Un-Break My Heart"
"How Could an Angel Break My Heart"
"Let It Flow"
"I Don't Want To"
"I Love Me Some Him"
The Heat
"He Wasn't Man Enough"
"Spanish Guitar"
"Just Be a Man About It"
"Maybe"
Snowflakes
"Christmas in Jamaica"
"Snowflakes of Love"
"Christmas Time Is Here"
"Have Yourself a Merry Little Christmas"
"The Christmas Song"
More Than a Woman
"Hit the Freeway"
Libra
"Please"
"Trippin' (That's the Way Love Works)"
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"Suddenly"
"The Time of Our Lives"
Pulse
"Yesterday"
"Hands Tied"
"Make My Heart"
Love, Marriage & Divorce (with Babyface)
"Hurt You"
"Where Did We Go Wrong?"
"Roller Coaster"
Sex & Cigarettes
"Deadwood"
"Long as I Live"
As featured artist
"Give U My Heart"
"Part II"
"We Are the World 25 for Haiti"
Other songs
"I Heart You"
Book:Toni Braxton
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Green Eggs and Ham
From Wikipedia, the free encyclopedia
Jump to: navigation, search
For the television series, see Green Eggs and Ham (TV series). For the video game, see Dr. Seuss: Green Eggs and Ham (video game).
Green Eggs and Ham
Author
Dr. Seuss
Cover artist
Dr. Seuss
Country
United States
Language
English
Genre
Children's literature
Publisher
Random House
The Living Books Company
Publication date
August 12, 1960
ISBN
978-0-394-80016-5
OCLC
184476
Preceded by
One Fish Two Fish Red Fish Blue Fish
Followed by
The Sneetches and Other Stories
Green Eggs and Ham is a children's book by Dr. Seuss, first published on August 12, 1960. As of 2016, the book has sold 8 million copies worldwide.[1] The story has appeared in several adaptations starting with 1973's Dr. Seuss on the Loose starring Paul Winchell as the voice of both Sam and Joey.[2]
Contents
[hide]
1 Plot summary
2 Background
3 Reception and cultural impact
4 Adaptations
5 Temporary ban
6 Selected translations
7 References
8 External links
Plot summary[edit]
A character named "Sam" pesters Joey to try a plate of green eggs and ham. Joey refuses, responding, "I do not like green eggs and ham. I do not like them, Sam-I-am." He continues to repeat this as Sam persistently follows him, asking him to try them in eight locations (house, box, car, tree, train, dark, rain, boat) and with three animals (mouse, fox, and goat). Finally, he gives into Sam's pestering and tries the green eggs and ham, which he does like after all and happily responds, "I do so like green eggs and ham. Thank you. Thank you, Sam-I-am."
Background[edit]
Green Eggs and Ham is one of Seuss's "Beginner Books", written in a very simple vocabulary for beginning readers. The vocabulary of the text consists of just 50 words[3] and was the result of a bet between Seuss and Bennett Cerf, Dr. Seuss's publisher.[3][4][5] that Seuss (after completing The Cat in the Hat using 236 words)[6] could not complete an entire book without exceeding that limit. The 50 words are: a, am, and, anywhere, are, be, boat, box, car, could, dark, do, eat, eggs, fox, goat, good, green, ham, here, house, I, if, in, let, like, may, me, mouse, not, on, or, rain, Sam, say, see, so, thank, that, the, them, there, they, train, tree, try, will, with, would, you.[3]
Reception and cultural impact[edit]
Green Eggs and Ham was published on August 12, 1960.[7][8] By 2018, it had become the fourth-best selling English-language children's hardcover book of all time.[9][10] As of 2014, the book has sold 8 million copies. In 1999 the National Education Association (NEA) conducted an online survey of children and teachers, seeking the 100 most popular children's books. The children ranked Green Eggs and Ham third, just above another Dr. Seuss book, The Cat in the Hat.[11] The teachers ranked it fourth.[12] Teachers ranked it fourth again in a 2007 NEA poll.[13] Scholastic Parent & Child magazine placed it #7 among the "100 Greatest Books for Kids" in 2012.[14] That same year, it was ranked number 12 among the "Top 100 Picture Books" in a survey published by School Library Journal – the first of five Dr. Seuss books on the list.[15]
The book has become sufficiently ingrained in the cultural consciousness that U.S. District Court Judge James Muirhead referenced Green Eggs and Ham in his September 21, 2007 court ruling after receiving an egg in the mail from prisoner Charles Jay Wolff who was protesting against the prison diet. Muirhead ordered the egg destroyed and rendered his judgment in the style of Seuss.[16][17] Senator Ted Cruz read the book on the floor of the United States Senate during his filibuster over the funding over Obamacare.[18] Musician will.i.am has stated that his moniker is inspired by the story.[19]
On September 29, 1991, following Dr. Seuss' death earlier that week, the Reverend Jesse Jackson recited an excerpt of Green Eggs and Ham on Saturday Night Live during a special tribute segment.[20]
Adaptations[edit]
Green Eggs and Ham is the third of the three Geisel stories that were adapted into the television special Dr. Seuss on the Loose, which featured a connecting narration by The Cat In The Hat, in 1973. (The Sneetches and The Zax were the other two.)
The song "Green Eggs and Ham" was recorded by the band Moxy Früvous on their 1992 independent debut album Moxy Früvous and is a rap treatment of the famous story.
The book was also made into a Living Books adaptation for the PC and there were similar differences to reflect the new media such as Sam singing his opening lines.
An upcoming animated television series based on the book, Green Eggs and Ham, will premiere on Netflix in 2018, produced by Warner Bros. Animation, A Very Good Production, A Stern Talking To, Random House Children's Entertainment and Gulfstream Television and distributed by Warner Bros. Television.[21]
The book was also featured as one of the segments brought to life via live-action in a stage-play fashion in the 1994 TV film In Search of Dr. Seuss.
In 1995, Punk Wave band Vader Vader recorded a demo of "Green Eggs And Ham". In 2017, Vader Vader also made a music video for their demo. Unfortunately, they have not been able to secure the rights to release their song or video.
Temporary ban[edit]
Main article: Book censorship in China
In 1965, the book was banned in the People's Republic of China for its "portrayal of early Marxism".[22] The ban was lifted in 1991, following Seuss' death.[23]
Selected translations[edit]
לֹא רָעֵב וְלֹא אוֹהֵב (Lo ra'ev ve-lo ohev, 1960, Hebrew ISBN 9789652294661)
Huevos verdes con jamón (1960, Spanish, ISBN 1880507013)
Groene eieren met ham (1960s, Dutch, ISBN 9024002966)
火腿加綠蛋 (Huǒ tuǐ jiā lǜ dàn, 1992, Chinese, ISBN 9573211254)
Prosciutto e uova verdi (2002, Italian, ISBN 880902446X)
Virent ova! Viret perna! (2003, Latin, ISBN 0865165556)
Kto zje zielone jajka sadzone? (2004, Polish, ISBN 8372781249)
Les œufs verts au jambon (2009, French, ISBN 9781569756881)
Grünes Ei mit Speck (2011, German, ISBN 9783596854417)
References[edit]
Jump up ^ http://entertainment.howstuffworks.com/arts/literature/20-best-selling-childrens-books9.htm
Jump up ^ "Dr. Seuss on the Loose". IMDB. Retrieved December 2, 2013.
^ Jump up to: a b c "10 stories behind Dr. Seuss stories". CNN. March 1, 2018. Retrieved 2009-01-26.
Jump up ^ "Green Eggs and Ham". snopes.com. 2012. Retrieved May 4, 2012.
Jump up ^ "99 Interesting Facts about the world #18". All That is Interesting. Retrieved 22 January 2015.
Jump up ^ [1]
Jump up ^ A 50 -year feast in 50 words Archived 2010-08-15 at the Wayback Machine., Marketplace. Retrieved 12 August 2010.
Jump up ^ "Happy Birthday Sam-I-Am! 50 Years of Green Eggs and Ham". Gnews. 2012. Archived from the original on March 4, 2012. Retrieved May 4, 2012.
Jump up ^ "All-Time Bestselling Children's Books". Publishers Weekly. 17 December 2001. Archived from the original on December 25, 2005.
Jump up ^ Menand, Louis. "A Critic at Large: Cat People: What Dr. Seuss Really Taught Us". The New Yorker. 23 December 2002.
Jump up ^ Kids' top 100 books Archived 2013-02-01 at the Wayback Machine. NEA: National Education Association. Retrieved 26 November 2006.
Jump up ^ "Teachers' Top 100 Books". NEA: National Education Association. Retrieved 26 November 2006.
Jump up ^ National Education Association (2007). "Teachers' Top 100 Books for Children". Retrieved August 19, 2012.
Jump up ^ "Parent & Child 100 Greatest Books for Kids" (PDF). Scholastic Corporation. 2012. Retrieved March 25, 2013.
Jump up ^ Bird, Elizabeth (July 6, 2012). "Top 100 Picture Books Poll Results". A Fuse #8 Production. Blog. School Library Journal (blog.schoollibraryjournal.com). Retrieved August 19, 2012.
Jump up ^ "Judge makes 'Green Eggs and Ham' ruling". MSNBC.
Jump up ^ "ORDER the egg filed by the plaintiff is to be destroyed re: 55 Motion for Contempt, injunction", Wolff v. NH Department of Corrections et al (Case 1:2006cv00321), September 18, 2007, Filing 56
Jump up ^ http://www.cbc.ca/news/world/why-ted-cruz-read-green-eggs-and-ham-in-the-u-s-senate-1.1867499
Jump up ^ https://www.nytimes.com/2011/01/23/magazine/23FOB-Q4-t.html?_r=0
Jump up ^ [2] ABC News 2010
Jump up ^ Andreeva, Nellie (April 29, 2015). "Netflix Picks Up 'Green Eggs and Ham' Animated Series From Ellen DeGeneres". Deadline. Retrieved April 30, 2015.
Jump up ^ http://www.huffingtonpost.co.uk/2012/01/20/surprisingly-banned-books_n_1218905.html
Jump up ^ "Banned Books Week: Green Eggs and Ham". New York Public Library. Retrieved December 2, 2013.
External links[edit]
Claasen, Lynda (16 January 2015). How Dr. Seuss Created Green Eggs and Ham (Video).
[show]
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Characters
The Cat in the Hat
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Horton the Elephant
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Bibliography
And to Think That I Saw It on Mulberry Street
The 500 Hats of Bartholomew Cubbins
The King's Stilts
The Seven Lady Godivas
Horton Hatches the Egg
McElligot's Pool
Thidwick the Big-Hearted Moose
Bartholomew and the Oobleck
If I Ran the Zoo
Scrambled Eggs Super!
Horton Hears a Who!
On Beyond Zebra!
If I Ran the Circus
How the Grinch Stole Christmas!
The Cat in the Hat
The Cat in the Hat Comes Back
Yertle the Turtle and Other Stories
Happy Birthday to You!
Green Eggs and Ham
One Fish, Two Fish, Red Fish, Blue Fish
The Sneetches and Other Stories
Dr. Seuss's Sleep Book
Dr. Seuss's ABC
Fox in Socks
Hop on Pop
I Had Trouble in Getting to Solla Sollew
Come over to My House 1
The Foot Book
I Can Lick 30 Tigers Today! and Other Stories
My Book about ME
Mr. Brown Can Moo! Can You?: Dr. Seuss's Book of Wonderful Noises!
The Lorax
Marvin K. Mooney Will You Please Go Now!
Did I Ever Tell You How Lucky You Are?
The Shape of Me and Other Stuff
Wacky Wednesday 1
Great Day for Up!
Oh, the Thinks You Can Think!
The Cat's Quizzer
I Can Read with My Eyes Shut!
Oh Say Can You Say?
Hunches in Bunches
The Butter Battle Book
You're Only Old Once!
I Am Not Going to Get Up Today!
Oh, the Places You'll Go!
There's a Wocket in My Pocket
I Wish That I Had Duck Feet 1
Daisy-Head Mayzie 2
My Many Colored Days 2
Hooray for Diffendoofer Day! 2
The Bippolo Seed and Other Lost Stories 2
Horton and the Kwuggerbug and More Lost Stories 2
What Pet Should I Get? 2
Adaptations
Television
The Gerald McBoing-Boing Show (1956–57)
Dr. Seuss' How the Grinch Stole Christmas! (1966)
Horton Hears a Who! (1970)
The Cat in the Hat (1971)
The Lorax (1972)
Dr. Seuss on the Loose (1973)
The Hoober-Bloob Highway (1975)
Halloween Is Grinch Night (1977)
Pontoffel Pock, Where Are You? (1980)
The Grinch Grinches the Cat in the Hat (1982)
The Butter Battle Book (1989)
In Search of Dr. Seuss (1994)
Daisy-Head Mayzie (1995)
The Wubbulous World of Dr. Seuss (1996–98) (episodes)
Gerald McBoing-Boing (2005–07)
The Cat in the Hat Knows a Lot About That! (2010–present) (episodes)
Green Eggs and Ham (2018)
Film
Horton Hatches the Egg (short; 1942)
Gerald McBoing-Boing (short; 1950)
How the Grinch Stole Christmas (2000)
The Cat in the Hat (2003)
Horton Hears a Who! (2008)
The Lorax (2012)
The Grinch (2018)
The Cat in the Hat (TBA)
Other media
Welcome (Russian short film)
Seussical (musical)
Dr. Seuss' How the Grinch Stole Christmas! The Musical
The Grinch (video game)
Dr. Seuss: How the Grinch Stole Christmas! (video game)
"You're a Mean One, Mr. Grinch" (song)
The Lorax (play)
Other works
Private Snafu
The Pocket Book of Boners
Your Job in Germany
Our Job in Japan
Design for Death
The 5,000 Fingers of Dr. T.
Hejji
Society of Red Tape Cutters
Flit
Related
A Fish out of Water
Amazing World of Dr. Seuss Museum
Beginner Books
Dr. Seuss Goes to War
The Secret Art of Dr. Seuss
Dr. Seuss National Memorial
Geisel Award
Geisel Library
Geisel School of Medicine
Helen Palmer
PM
Political messages of Dr. Seuss
Read Across America
Seuss Landing
1 as "Theo. LeSieg". 2 Posthumous.
[show]
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List of egg topics
Types
Bird
Fish and amphibian
Monotreme
Fossil record
Cephalopod
Fish
Reptile (dinosaur)
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Biology
Embryo
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List of egg dishes
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Jeepers Creepers (song)
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"Jeepers Creepers" is a popular song and jazz standard. The music was written by Harry Warren and the lyrics by Johnny Mercer for the 1938 movie Going Places. It was premiered by Louis Armstrong and has been covered by many other musicians. The song was nominated for the Academy Award for Best Original Song in 1938 but lost to "Thanks for the Memory".
Contents
1 Overview
2 Other film appearances
3 Popular culture
4 References
Overview[edit]
This song was featured in the 1938 film Going Places starring Dick Powell, Anita Louise, and Ronald Reagan. Louis Armstrong appears in the role of Gabriel, the trainer of a race horse named Jeepers Creepers. Jeepers Creepers is a wild horse and can only be soothed enough to let someone ride him when Gabriel plays the song "Jeepers Creepers" on his trumpet or sings it to him. Gabriel wrote the song specifically for the horse. The phrase "jeepers creepers", a slang expression and minced oath euphemism for Jesus Christ, predates both the song and film.
The lyrics of the song are under copyright; the most famous pair of lines is:
Jeepers Creepers, where'd ya get those peepers?
Jeepers Creepers, where'd ya get those eyes?
There were three popular versions of the song in 1939 by Al Donahue, Louis Armstrong, and Larry Clinton.[1]
Other film appearances[edit]
The song is featured in a 1939 Warner Bros. cartoon short of the same name sung with substitute lyrics by the ghost. Variations played when Porky runs away from the ghost
In 1939, a Warner Bros. short, Symphony of Swing, the song was performed by Artie Shaw and His Orchestra with Shaw on clarinet and a vocal by Tony Pastor.
In the 1942 film Yankee Doodle Dandy, "Jeepers Creepers" is sung by a group of kids who pass by the house of George M. Cohan (played by James Cagney).
The 1949 film My Dream Is Yours includes the song and it is sung by an unidentified female singer during Doug's trip to New York.
The 1957 cartoon short Show Biz Bugs has Daffy Duck performing a tap dance number to the song.
In the 1975 movie The Day of the Locust, the character Faye Greener (Karen Black) sings the song whenever she wants to disturb her father. Louis Armstrong's recording of the song plays over the film's closing credits.
The 1975 documentary film Brother, Can You Spare a Dime? includes Louis Armstrong's version.
In the American satirical comedy film The Cheap Detective, the song is sung by Scatman Crothers at Nix Place. It is also played when Marlene and Peckinpaugh are drinking champagne and during the closing credits.
The song and title was featured prominently in the 2001 horror movie Jeepers Creepers when The Creeper is nearby.
In the 2016 film Café Society, "Jeepers Creepers" is performed by Vince Giordano and the Night Hawks and Kat Edmonson.
In the 2016 episode "High School English" from Family Guy, "Jeepers Creepers" is briefly performed by Stewie Griffin during the segment parody of Of Mice and Men.
Popular culture[edit]
Al Donahue's version of this song reached No. 1 in Billboard magazine in 1938.
Frank Sinatra included the song in his album Swing Easy! (1954).
Dave Brubeck's rendition of the song appears on the 1955 album Brubeck Time aka Instant Brubeck.
Bing Crosby recorded the song on his 1956 album Bing Sings Whilst Bregman Swings.
Tony Bennett included the song in his 1959 album Strike Up the Band.
Al Caiola recorded an instrumental version of this song.
Hayley Mills recorded this song in 1962 for the American Buena Vista label.
In 1988, "Peek-a-Boo", the first single from Siouxsie and the Banshees's ninth studio album Peepshow, was found to be too similar to the lyrics of "Jeepers Creepers". To remedy the situation and to avoid legal action, Siouxsie and the Banshees gave co-songwriting credit on "Peek-a-Boo" to Warren and Mercer.
Mr. Show features an extended musical sequence involving the Gospel of Jeepers Creepers. Later in the 90's, the Kidsongs Children and the Biggles (Julene Renee, Patty Lloyd, Jeni Lytton and Debbie Lyton) sang this song in their Baby Animal Songs video and DVD.
The song was sung by Christine in a 1988 episode of Night Court ("The Law Club").
In 1994, the song was covered by the Fun Songs Kids in Mickey's Fun Songs: Campout at Walt Disney World.
In the 1997 Friends episode "The One With the Dirty Girl", Mrs. Burkart (Gretchen Wyler) sings the song at her husband's funeral.
A recording of the song was made by The Puppini Sisters on their 2006 debut album Betcha Bottom Dollar.
The Hi-Lo's included the song on their 2006 A Musical Thrill album.
A jazz-pop cover, "Jeepers Creepers 2.0" appeared on the 2014 album Little Secret by Nikki Yanofsky.
The song was played in Season 9, Episode 4 of Family Guy, "Welcome Back Carter," as well as Season 14, Episode 7, "High School English".
Alex Mendham & His Orchestra recorded a version of this song in their 2017 album On with the Show.
References[edit]
Jump up ^ Whitburn, Joel (1986). Joel Whitburn's Pop Memories 1890–1954. Wisconsin: Record Research. p. 533. ISBN 0-89820-083-0.
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Third law of thermodynamics
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Specific heat capacity c = {\displaystyle c=}
T {\displaystyle T} ∂ S {\displaystyle \partial S}
N {\displaystyle N} ∂ T {\displaystyle \partial T}
Compressibility β = − {\displaystyle \beta =-}
1 {\displaystyle 1} ∂ V {\displaystyle \partial V}
V {\displaystyle V} ∂ p {\displaystyle \partial p}
Thermal expansion α = {\displaystyle \alpha =}
1 {\displaystyle 1} ∂ V {\displaystyle \partial V}
V {\displaystyle V} ∂ T {\displaystyle \partial T}
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U ( S , V ) {\displaystyle U(S,V)}
Enthalpy
H ( S , p ) = U + p V {\displaystyle H(S,p)=U+pV}
Helmholtz free energy
A ( T , V ) = U − T S {\displaystyle A(T,V)=U-TS}
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G ( T , p ) = H − T S {\displaystyle G(T,p)=H-TS}
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Book:Thermodynamics
v
t
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The third law of thermodynamics is sometimes stated as follows, regarding the properties of systems in thermodynamic equilibrium:
The entropy of a system approaches a constant value as its temperature approaches absolute zero.
At absolute zero (zero kelvin) the system must be in a state with the minimum possible energy. Entropy is related to the number of accessible microstates, and there is typically one unique state (called the ground state) with minimum energy.[1] In such a case, the entropy at absolute zero will be exactly zero. If the system does not have a well-defined order (if its order is glassy, for example), then there may remain some finite entropy as the system is brought to very low temperatures, either because the system becomes locked into a configuration with non-minimal energy or because the minimum energy state is non-unique. The constant value is called the residual entropy of the system.[2]
The Nernst–Simon statement of the third law of thermodynamics concerns thermodynamic processes at a fixed, low temperature:
The entropy change associated with any condensed system undergoing a reversible isothermal process approaches zero as the temperature at which it is performed approaches 0 K.
Here a condensed system refers to liquids and solids.
A classical formulation by Nernst (actually a consequence of the Third Law) is:
It is impossible for any process, no matter how idealized, to reduce the entropy of a system to its absolute-zero value in a finite number of operations.[3]
There also exists a formulation of the Third Law which approaches the subject by postulating a specific energy behavior:
If the composite of two thermodynamic systems constitutes an isolated system, then any energy exchange in any form between those two systems is bounded.[4]
Contents
[hide]
1 History
2 Explanation
2.1 Example : Entropy change of a crystal lattice heated by an incoming photon
2.2 Systems with non-zero entropy at absolute zero
3 Mathematical formulation
4 Consequences of the third law
4.1 Absolute zero
4.2 Specific heat
4.3 Vapor pressure
4.4 Latent heat of melting
4.5 Thermal expansion coefficient
5 See also
6 References
7 Further reading
History[edit]
The 3rd law was developed by the chemist Walther Nernst during the years 1906–12, and is therefore often referred to as Nernst's theorem or Nernst's postulate. The third law of thermodynamics states that the entropy of a system at absolute zero is a well-defined constant. This is because a system at zero temperature exists in its ground state, so that its entropy is determined only by the degeneracy of the ground state.
In 1912 Nernst stated the law thus: "It is impossible for any procedure to lead to the isotherm T = 0 in a finite number of steps."[5]
An alternative version of the third law of thermodynamics as stated by Gilbert N. Lewis and Merle Randall in 1923:
If the entropy of each element in some (perfect) crystalline state be taken as zero at the absolute zero of temperature, every substance has a finite positive entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.
This version states not only ΔS will reach zero at 0 K, but S itself will also reach zero as long as the crystal has a ground state with only one configuration. Some crystals form defects which cause a residual entropy. This residual entropy disappears when the kinetic barriers to transitioning to one ground state are overcome.[6]
With the development of statistical mechanics, the third law of thermodynamics (like the other laws) changed from a fundamental law (justified by experiments) to a derived law (derived from even more basic laws). The basic law from which it is primarily derived is the statistical-mechanics definition of entropy for a large system:
S − S 0 = k B ln Ω {\displaystyle S-S_{0}=k_{\text{B}}\ln \,\Omega \ }
where S is entropy, kB is the Boltzmann constant, and Ω {\displaystyle \Omega } is the number of microstates consistent with the macroscopic configuration. The counting of states is from the reference state of absolute zero, which corresponds to the entropy of S0.
Explanation[edit]
In simple terms, is used for convenience the third law states that the entropy of a perfect crystal of a pure substance approaches zero as the temperature approaches zero. The alignment of a perfect crystal leaves no ambiguity as to the location and orientation of each part of the crystal. As the energy of the crystal is reduced, the vibrations of the individual atoms are reduced to nothing, and the crystal becomes the same everywhere.
a) Single possible configuration for a system at absolute zero, i.e., only one microstate is accessible. Thus S = k ln W = 0. b) At temperatures greater than absolute zero, multiple microstates are accessible due to atomic vibration (exaggerated in the figure). Since the number of accessible microstates is greater than 1, S = k ln W > 0.
The third law provides an absolute reference point for the determination of entropy at any other temperature. The entropy of a system, determined relative to this zero point, is then the absolute entropy of that system. Mathematically, the absolute entropy of any system at zero temperature is the natural log of the number of ground states times Boltzmann's constant kB = 6977137999999999999♠1.38×10−23 J K−1.
The entropy of a perfect crystal lattice as defined by Nernst's theorem is zero provided that its ground state is unique, because ln(1) = 0. If the system is composed of one-billion atoms, all alike, and lie within the matrix of a perfect crystal, the number of permutations of one-billion identical things taken one-billion at a time is Ω = 1. Hence:
S − S 0 = k B ln  Ω = k B ln  1 = 0 {\displaystyle S-S_{0}=k_{\text{B}}\ln \Omega =k_{\text{B}}\ln {1}=0}
The difference is zero, hence the initial entropy S0 can be any selected value so long as all other such calculations include that as the initial entropy. As a result, the initial entropy value of zero is selected S0 = 0 is used for convenience.
S − S 0 = S − 0 = 0 {\displaystyle S-S_{0}=S-0=0}
S = 0 {\displaystyle S=0}
Example : Entropy change of a crystal lattice heated by an incoming photon[edit]
Suppose a system consisting of a crystal lattice with volume V of N identical atoms at T= 0 K, and an incoming photon of wavelength λ and energy ε.
Initially, there is only one accessible microstate :
S 0 = k B ln  Ω = k B ln  1 = 0 {\displaystyle S_{0}=k_{\text{B}}\ln \Omega =k_{\text{B}}\ln {1}=0} .
Let's assume the crystal lattice absorbs the incoming photon. There is a unique atom in the lattice that interacts and absorbs this photon. So after absorption, there is N possible microstates accessible by the system , each of the microstates corresponding to one excited atom, and the other atoms remaining at ground state.
The entropy, energy, and temperature of the system rises and can be calculated. The entropy change is:
Δ S = S − S 0 = k B ln  Ω {\displaystyle \Delta S=S-S_{0}=k_{\text{B}}\ln {\Omega }}
From the second law of thermodynamics:
Δ S = S − S 0 = δ Q T {\displaystyle \Delta S=S-S_{0}={\frac {\delta Q}{T}}}
Hence:
Δ S = S − S 0 = k B ln  ( Ω ) = δ Q T {\displaystyle \Delta S=S-S_{0}=k_{\text{B}}\ln(\Omega )={\frac {\delta Q}{T}}}
Calculating entropy change:
S − 0 = k B ln  N = 1.38 × 10 − 23 × ln  ( 3 × 10 22 ) = 70 × 10 − 23 J K − 1 {\displaystyle S-0=k_{\text{B}}\ln {N}=1.38\times 10^{-23}\times \ln {(3\times 10^{22})}=70\times 10^{-23}\,\mathrm {J} \,\mathrm {K} ^{-1}}
We assume N = 3.1022 and λ = 1 cm . The energy change of the system as a result of absorbing the single photon whose energy is ε:
δ Q = ϵ = h c λ = 6.62 × 10 − 34 J ⋅ s × 3 × 10 8 m s − 1 0.01 m = 2 × 10 − 23 J {\displaystyle \delta Q=\epsilon ={\frac {hc}{\lambda }}={\frac {6.62\times 10^{-34}\,\mathrm {J} \cdot \mathrm {s} \times 3\times 10^{8}\,\mathrm {m} \,\mathrm {s} ^{-1}}{0.01\,\mathrm {m} }}=2\times 10^{-23}\,\mathrm {J} }
The temperature of the system rises by:
T = ϵ Δ S = 2 × 10 − 23 J 70 × 10 − 23 J K − 1 = 0.02857 K {\displaystyle T={\frac {\epsilon }{\Delta S}}={\frac {2\times 10^{-23}\,\mathrm {J} }{70\times 10^{-23}\,\mathrm {J} \,\mathrm {K} ^{-1}}}=0.02857K}
This can be interpreted as the average temperature of the system over the range from 0 < S < 70 × 10 − 23 J K − 1 {\displaystyle 0<S<70\times 10^{-23}\,\mathrm {J} \,\mathrm {K} ^{-1}} [7]. A single atom was assumed to absorb the photon but the temperature and entropy change characterizes the entire system.
Systems with non-zero entropy at absolute zero[edit]
An example of a system which does not have a unique ground state is one whose net spin is a half-integer, for which time-reversal symmetry gives two degenerate ground states. For such systems, the entropy at zero temperature is at least kB*ln(2) (which is negligible on a macroscopic scale). Some crystalline systems exhibit geometrical frustration, where the structure of the crystal lattice prevents the emergence of a unique ground state. Ground-state helium (unless under pressure) remains liquid.
In addition, glasses and solid solutions retain large entropy at 0 K, because they are large collections of nearly degenerate states, in which they become trapped out of equilibrium. Another example of a solid with many nearly-degenerate ground states, trapped out of equilibrium, is ice Ih, which has "proton disorder".
For the entropy at absolute zero to be zero, the magnetic moments of a perfectly ordered crystal must themselves be perfectly ordered; from an entropic perspective, this can be considered to be part of the definition of a "perfect crystal". Only ferromagnetic, antiferromagnetic, and diamagnetic materials can satisfy this condition. However, ferromagnetic materials do not, in fact, have zero entropy at zero temperature, because the spins of the unpaired electrons are all aligned and this gives a ground-state spin degeneracy. Materials that remain paramagnetic at 0 K, by contrast, may have many nearly-degenerate ground states (for example, in a spin glass), or may retain dynamic disorder (a quantum spin liquid).[citation needed]
Mathematical formulation[edit]
Consider a closed system in internal equilibrium. As the system is in equilibrium, there are no irreversible processes so the entropy production is zero. During slow heating, small temperature gradients are generated in the material, but the associated entropy production can be kept arbitrarily low if the heat is supplied slowly enough. The increase in entropy due to the added heat δQ is then given by the second part of the Second law of thermodynamics which states that the entropy change of a system ΔS is given by
Δ S = δ Q T . {\displaystyle \Delta S={\frac {\delta Q}{T}}.} (1)
The temperature rise dT due to the heat δQ is determined by the heat capacity C(T,X) according to
δ Q = C ( T , X ) d T . {\displaystyle \delta Q=C(T,X)\mathrm {d} T.} (2)
The parameter X is a symbolic notation for all parameters (such as pressure, magnetic field, liquid/solid fraction, etc.) which are kept constant during the heat supply. E.g. if the volume is constant we get the heat capacity at constant volume CV. In the case of a phase transition from liquid to solid, or from gas to liquid the parameter X can be one of the two components.[clarification needed] Combining relations (1) and (2) gives
Δ S = C ( T , X ) d T T . {\displaystyle \Delta S={\frac {C(T,X)\mathrm {d} T}{T}}.} (3)
Integration of Eq.(3) from a reference temperature T0 to an arbitrary temperature T gives the entropy at temperature T
S ( T , X ) = S ( T 0 , X ) + ∫ T 0 T C ( T ′ , X ) T ′ d T ′ . {\displaystyle S(T,X)=S(T_{0},X)+\int _{T_{0}}^{T}{\frac {C(T^{\prime },X)}{T^{\prime }}}\mathrm {d} T^{\prime }.} (4)
We now come to the mathematical formulation of the third law. There are three steps:
1: In the limit T0→0 the integral in Eq.(4) is finite.[clarification needed] So that we may take T0=0 and write
S ( T , X ) = S ( 0 , X ) + ∫ 0 T C ( T ′ , X ) T ′ d T ′ . {\displaystyle S(T,X)=S(0,X)+\int _{0}^{T}{\frac {C(T^{\prime },X)}{T^{\prime }}}\mathrm {d} T^{\prime }.} (5)
2. The value of S(0,X) is independent of X.[clarification needed] In mathematical form
S ( 0 , X ) = S ( 0 ) . {\displaystyle S(0,X)=S(0).} (6)
So Eq.(5) can be further simplified to
S ( T , X ) = S ( 0 ) + ∫ 0 T C ( T ′ , X ) T ′ d T ′ . {\displaystyle S(T,X)=S(0)+\int _{0}^{T}{\frac {C(T^{\prime },X)}{T^{\prime }}}\mathrm {d} T^{\prime }.} (7)
Equation (6) can also be formulated as
lim T → 0 ( ∂ S ( T , X ) ∂ X ) T = 0. {\displaystyle \lim _{T\rightarrow 0}\left({\frac {\partial S(T,X)}{\partial X}}\right)_{T}=0.} (8)
In words: at absolute zero all isothermal processes are isentropic. Eq.(8) is the mathematical formulation of the third law.
3: Classically, one is free to choose the zero of the entropy, and it is convenient to take
S ( 0 ) = 0 {\displaystyle S(0)=0} (9)
so that Eq.(7) reduces to the final form
S ( T , X ) = ∫ 0 T C ( T ′ , X ) T ′ d T ′ . {\displaystyle S(T,X)=\int _{0}^{T}{\frac {C(T^{\prime },X)}{T^{\prime }}}\mathrm {d} T^{\prime }.} (10)
However, reinterpreting Eq. (9) in view of the quantized nature of the lowest-lying energy states, the physical meaning of Eq.(9) goes deeper than just a convenient selection of the zero of the entropy. It is due to the perfect order at zero kelvin as explained above.
Consequences of the third law[edit]
Fig.1 Left side: Absolute zero can be reached in a finite number of steps if S(0,X1)≠S(0, X2). Right: An infinite number of steps is needed since S(0,X1)= S(0,X2).
Absolute zero[edit]
The third law is equivalent to the statement that
"It is impossible by any procedure, no matter how idealized, to reduce the temperature of any system to zero temperature in a finite number of finite operations".[8]
The reason that T=0 cannot be reached according to the third law is explained as follows: Suppose that the temperature of a substance can be reduced in an isentropic process by changing the parameter X from X2 to X1. One can think of a multistage nuclear demagnetization setup where a magnetic field is switched on and off in a controlled way.[9] If there were an entropy difference at absolute zero, T=0 could be reached in a finite number of steps. However, at T=0 there is no entropy difference so an infinite number of steps would be needed. The process is illustrated in Fig.1.
Specific heat[edit]
A non-quantitative description of his third law that Nernst gave at the very beginning was simply that the specific heat can always be made zero by cooling the material down far enough.[10] A modern, quantitative analysis follows.
Supposed that the heat capacity of a sample in the low temperature region has the form of a power law C(T,X)=C0Tα asymptotically as T→0, and we wish to find which values of α are compatible with the third law. We have
∫ T 0 T C ( T ′ , X ) T ′ d T ′ = C 0 α ( T α − T 0 α ) . {\displaystyle \int _{T_{0}}^{T}{\frac {C(T^{\prime },X)}{T^{\prime }}}dT^{\prime }={\frac {C_{0}}{\alpha }}(T^{\alpha }-T_{0}^{\alpha }).} (11)
By the discussion of third law (above), this integral must be bounded as T0→0, which is only possible if α>0. So the heat capacity must go to zero at absolute zero
lim T → 0 C ( T , X ) = 0. {\displaystyle \lim _{T\rightarrow 0}C(T,X)=0.} (12)
if it has the form of a power law. The same argument shows that it cannot be bounded below by a positive constant, even if we drop the power-law assumption.
On the other hand, the molar specific heat at constant volume of a monatomic classical ideal gas, such as helium at room temperature, is given by CV=(3/2)R with R the molar ideal gas constant. But clearly a constant heat capacity does not satisfy Eq. (12). That is, a gas with a constant heat capacity all the way to absolute zero violates the third law of thermodynamics. We can verify this more fundamentally by substituting CV in Eq. (4), which yields
S ( T , V ) = S ( T 0 , V ) + 3 2 R ln  T T 0 . {\displaystyle S(T,V)=S(T_{0},V)+{\frac {3}{2}}R\ln {\frac {T}{T_{0}}}.} (13)
In the limit T0 → 0 this expression diverges, again contradicting the third law of thermodynamics.
The conflict is resolved as follows: At a certain temperature the quantum nature of matter starts to dominate the behavior. Fermi particles follow Fermi–Dirac statistics and Bose particles follow Bose–Einstein statistics. In both cases the heat capacity at low temperatures is no longer temperature independent, even for ideal gases. For Fermi gases
C V = π 2 2 R T T F {\displaystyle C_{V}={\frac {\pi ^{2}}{2}}R{\frac {T}{T_{\text{F}}}}} (14)
with the Fermi temperature TF given by
T F = 1 8 π 2 N A 2 h 2 M R ( 3 π 2 N A V m ) 2 / 3 . {\displaystyle T_{\text{F}}={\frac {1}{8\pi ^{2}}}{\frac {N_{\text{A}}^{2}h^{2}}{MR}}\left({\frac {3\pi ^{2}N_{\text{A}}}{V_{\text{m}}}}\right)^{2/3}.} (15)
Here NA is Avogadro's number, Vm the molar volume, and M the molar mass.
For Bose gases
C V = 1.93.. R ( T T B ) 3 / 2 {\displaystyle C_{V}=1.93..R\left({\frac {T}{T_{\text{B}}}}\right)^{3/2}} (16)
with TB given by
T B = 1 11.9.. N A 2 h 2 M R ( N A V m ) 2 / 3 . {\displaystyle T_{\text{B}}={\frac {1}{11.9..}}{\frac {N_{\text{A}}^{2}h^{2}}{MR}}\left({\frac {N_{\text{A}}}{V_{\text{m}}}}\right)^{2/3}.} (17)
The specific heats given by Eq.(14) and (16) both satisfy Eq.(12). Indeed, they are power laws with α=1 and α=3/2 respectively.
Vapor pressure[edit]
The only liquids near absolute zero are ³He and ⁴He. Their heat of evaporation has a limiting value given by
L = L 0 + C p T {\displaystyle L=L_{0}+C_{p}T} (18)
with L0 and Cp constant. If we consider a container, partly filled with liquid and partly gas, the entropy of the liquid–gas mixture is
S ( T , x ) = S l ( T ) + x ( L 0 T + C p ) {\displaystyle S(T,x)=S_{l}(T)+x({\frac {L_{0}}{T}}+C_{p})} (19)
where Sl(T) is the entropy of the liquid and x is the gas fraction. Clearly the entropy change during the liquid–gas transition (x from 0 to 1) diverges in the limit of T→0. This violates Eq.(8). Nature solves this paradox as follows: at temperatures below about 50 mK the vapor pressure is so low that the gas density is lower than the best vacuum in the universe. In other words: below 50 mK there is simply no gas above the liquid.
Latent heat of melting[edit]
The melting curves of ³He and ⁴He both extend down to absolute zero at finite pressure. At the melting pressure, liquid and solid are in equilibrium. The third law demands that the entropies of the solid and liquid are equal at T=0. As a result, the latent heat of melting is zero and the slope of the melting curve extrapolates to zero as a result of the Clausius–Clapeyron equation.
Thermal expansion coefficient[edit]
The thermal expansion coefficient is defined as
α V = 1 V m ( ∂ V m ∂ T ) p . {\displaystyle \alpha _{V}={\frac {1}{V_{m}}}\left({\frac {\partial V_{m}}{\partial T}}\right)_{p}.} (20)
With the Maxwell relation
( ∂ V m ∂ T ) p = − ( ∂ S m ∂ p ) T {\displaystyle \left({\frac {\partial V_{m}}{\partial T}}\right)_{p}=-\left({\frac {\partial S_{m}}{\partial p}}\right)_{T}} (21)
and Eq.(8) with X=p it is shown that
lim T → 0 α V = 0. {\displaystyle \lim _{T\rightarrow 0}\alpha _{V}=0.} (22)
So the thermal expansion coefficient of all materials must go to zero at zero kelvin.
See also[edit]
Adiabatic process
Ground state
Laws of thermodynamics
Quantum thermodynamics
Residual entropy
Thermodynamic entropy
Timeline of thermodynamics, statistical mechanics, and random processes
Quantum refrigerators
References[edit]
Jump up ^ J. Wilks The Third Law of Thermodynamics Oxford University Press (1961).[page needed]
Jump up ^ Kittel and Kroemer, Thermal Physics (2nd ed.), page 49.
Jump up ^ Wilks, J. (1971). The Third Law of Thermodynamics, Chapter 6 in Thermodynamics, volume 1, ed. W. Jost, of H. Eyring, D. Henderson, W. Jost, Physical Chemistry. An Advanced Treatise, Academic Press, New York, page 477.
Jump up ^ Heidrich, M. (2016). "Bounded energy exchange as an alternative to the third law of thermodynamics". Annals of Physics. 373: 665–681. Bibcode:2016AnPhy.373..665H. doi:10.1016/j.aop.2016.07.031.
Jump up ^ Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics, New York, ISBN 0-88318-797-3, page 342.
Jump up ^ Kozliak, Evguenii; Lambert, Frank L. (2008). "Residual Entropy, the Third Law and Latent Heat". Entropy. 10 (3): 274–84. Bibcode:2008Entrp..10..274K. doi:10.3390/e10030274.
Jump up ^ Reynolds and Perkins (1977). Engineering Thermodynamics. McGraw Hill. p. 438. ISBN 0-07-052046-1.
Jump up ^ Guggenheim, E.A. (1967). Thermodynamics. An Advanced Treatment for Chemists and Physicists, fifth revised edition, North-Holland Publishing Company, Amsterdam, page 157.
Jump up ^ F. Pobell, Matter and Methods at Low Temperatures, (Springer-Verlag, Berlin, 2007)[page needed]
Jump up ^ Einstein and the Quantum, A. Douglas Stone, Princeton University Press, 2013.
J. Wilks The Third Law of Thermodynamic

Further reading[edit]
Goldstein, Martin & Inge F. (1993) The Refrigerator and the Universe. Cambridge MA: Harvard University Press. ISBN 0-674-75324-0. Chpt. 14 is a nontechnical discussion of the Third Law, one including the requisite elementary quantum mechanics.
Braun, S.; Ronzheimer, J. P.; Schreiber, M.; Hodgman, S. S.; Rom, T.; Bloch, I.; Schneider, U. (2013). "Negative Absolute Temperature for Motional Degrees of Freedom". Science. 339 (6115): 52–5. arXiv:1211.0545  . Bibcode:2013Sci...339...52B. doi:10.1126/science.1227831. PMID 23288533. Lay summary – New Scientist (3 January 2013).
Levy, A.; Alicki, R.; Kosloff, R. (2012). "Quantum refrigerators and the third law of thermodynamics". Phys. Rev. E. 85: 061126. arXiv:1205.1347  . Bibcode:2012PhRvE..85f1126L. doi:10.1103/PhysRevE.85.061126.
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values: {\displaystyle S-S_{0}=k_{\text{B}}\ln \,\Omega \ }

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values: Edit section: Example : Entropy change of a crystal lattice heated by an incoming photon

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values: {\displaystyle \Delta S=S-S_{0}={\frac {\delta Q}{T}}}

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values: \delta Q = \epsilon = \frac {hc}{\lambda} =\frac{6.62 \times 10^{-34}\,\mathrm{J}\cdot \mathrm{s} \times 3 \times 10^{8} \,\mathrm{m}\,\mathrm{s}^{-1}}{0.01 \,\mathrm{m}}=2 \times 10^{-23} \,\mathrm{J}

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values: {\displaystyle \delta Q=C(T,X)\mathrm {d} T.}

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values: S(T,X) = S(T_0,X) + \int_{T_0}^{T} \frac {C(T^\prime,X)}{T^\prime}\mathrm{d}T^\prime.

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values: S(T,X)=S(0,X) + \int_0^T \frac {C(T^\prime,X)}{T^\prime}\mathrm{d}T^\prime.

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values: \lim_{T \rightarrow 0}\left( \frac { \part S(T,X)}{ \part X}\right)_T = 0.

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values: S(T,X) = \int_0^T \frac {C(T^\prime,X)}{T^\prime}\mathrm{d}T^\prime.

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values: \int_{T_0}^T \frac {C(T^\prime,X)}{T^\prime}dT^\prime = \frac {C_0}{ \alpha}(T^{ \alpha}-T_0^{ \alpha}).

values: {\displaystyle \lim _{T\rightarrow 0}C(T,X)=0.}

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