Charles’s Law: History and Work Problem

And we’re back for the second installment of our gas laws series – Charles’ Law.

If you need to refresh on the other gas laws in this series you can check out our post on Boyles’ Law, Avogadros’ Law, and the Ideal Gas Law.

Let’s review what a gas is: a gas a is a state a matter in which particles are not bound by its container and are not subject to a certain shape. The gas may consist of one type of particle (homogenous) or multiple (heterogeneous). There are many states of matter, we list gases as one of the four general states. And as per usual, we are assuming that the gases we’re working with today are ideal. As in, they exhibit elastic reactions with other particles or the sides of the container; the particles don’t lose energy upon impact.

Boyles’ Law addresses the relationship of pressure and volume upon gases – Charles’ law addresses the relationship between volume and temperature. But – how did Charles’ Law also get the name Gay-Lussac’s Law? And how did Charles almost blow himself up in the pursuit of science?

Charles’ Law – Pressure and Temperature

Jacques Alexandre César Charles (November 12, 1746 – April 7, 1823) was an interesting figure in science, to say. From the point of a science historian, he is a truly delightful character to research. Not only was he a mathematician and scientist, he was also a wikipedia described “balloonist”, a very real and very legitimate niche of history. 

Not to mention, many mathematical discoveries attributed to Jacques A.C. Charles were actually written by another Jacques Charles who was also an active member of the Paris Academy of Sciences at the time. You can read more about the surveyor Jacques Charles here on Wikipedia.

From now on, we’ll assume when I say Charles, that I mean Jacques A.C. Charles, our balloonist.

One other aspect that historians and journalists like to point out is that Charles didn’t publish much of his work. The scientific world became familiar with his work through scientist Joseph-Luis Gay-Lussac (Gay-Loussac) and his work on thermal expansion, published 1787. Gay- Lussac drew heavily from Charles’ work on balloons and hot-gas physics. He did cite Charles’ unpublished work in his

Charles was born in Beaugency, Loiret, France in 1746. Little is known about his early life – most websites turn immediately to his work with hydrogen and ballooning. What we do know is that he was an only child, and studied at the Conservatoire des Arts et Métiers.

Jacques Charles (x)

Charles worked for the Paris Financial Ministry before turning to a career in science. He married Julie Françoise Bouchaud des Hérettes in 1804. In fact, I found more about Julie Bouchaud’s early life than Jacques Charles with a quick Wikipedia search.

Charles’ drew inspiration from Robert Boyles’ law published almost 100 years earlier, surmising that hydrogen would be a useful lifting agent. And yes, the way that Charles had planned the contraption and reaction is truly ingenious.

Now, around this time, air balloons had become a fad, to say. Pairing air travel with the chance for a birds’ eye view of your world appealed to both scientists and citizens alike in 1700’s France.

Air travel had been revolutionized in the fall of 1783 by the Montgolfier brothers, using hot air to fuel the balloon (History of ballooning). Hot air consisting of hydrogen, carbon, oxygen, and nitrogen, just the stuff we breathe.

It’s difficult to determine exactly when Charles and fellow researchers Anne-Jean and Nicholas Louis Robert, launched their hot gas balloon. Some sources cite the date as August 27, 1783, but this date is earlier than the Montgolfier brothers first manned hot air balloon flight (Nov. 21, 1783). It is known that Charles developed the hydrogen balloon in response to this success, so dates such as December 01, 1873 make more sense. With this information, we will assume that the first hot air flight with animals was September 19, 1783, the first manned flight was November 21, 1873 (both Montgolfier) and the first manned hot-gas balloon was on December 1, 1873 (Charles, Robert).

Nonetheless, Charles’ and Roberts’ balloon used fuel consisted purely of hydrogen gas. And the contraption itself is wild.

In the Charles-Robert balloon, barrels of iron nails were doused with potent sulfuric acid H2SO4 (yikes). The iron and sufur would bind together in an aqueous solution, and hydrogen gas was released. The reaction that underwent looks like this:

Feii(s)+ H2SO4 (aq) → FeSO4 (aq) + H2 (g)

The offset gas was piped up to a filter to ensure any remaining acid was washed out, and then piped into a vat to fuel the main balloon. This reaction is exothermic, and everything soon became hot to touch. Knowing how combustible hydrogen gas is, and considering this reaction and you know, the whole hot gas thing, it’s a wonder that they honestly didn’t blow up a la Hindenburg. This is a core value that is reflected in Charles’ Law – if there were no exothermic reaction, there wouldn’t be enough lift for the balloon to get off the ground. 

Nonetheless, hot gas balloons became a main mode of air transport. In just ten years, the United States of America would launch its own air balloon. President George Washington observed the flight. (History of ballooning).

By 1787, Charles was elected to the Royal Academy of Sciences and had written all his work down but remained unpublished. One source cites an experiment involving 5 balloons filled with the same volume of different gases. Charles raised the balloons’ collective temperature to 80 degrees Celsius and observed that each balloon increased in the same amount of volume.

By the 1790’s the French Revolution was underway. 1793 held the Reign of Terror – a bout of executions in response to the revolutionist ideals, an anti-clerical sympathies (this is a gross simplification, please go check the official Wiki page on the French Revolution and la Terruer). 1793 also held the first balloon flight in

Charles, not much of an outspoken revolutionist, did hide a refractory priest. He risked arrest for this, and became part of the “enlightened” society that emerged from the revolution that accepted new views on God and ruling monarchies.

Nonetheless, Charles’ research was highlighted by John Dalton in 1801 and Gay-Loussac in 1802. Gay-Loussac credited Charles’ own unpublished researched from the 1780’s. Charles went on to further research regarding rudimentary aircraft, including the dirigible balloon in 1784.And now with what some may describe as “too much background”, we can jump into Charles’ Law and how it works. 

Charles’ Law: Work Problem

Charles Law looks like this: 


Where ‘V’ is your volume and ‘T’ is your temperature in kelvin. It is most always in kelvin, please keep this in mind as we proceed. Also note that we can play around with this equation a lot, so it’s good to refresh your algebra skills.

Initial Problem: A volume of gas initially measures as 2.80 L at an unknown temperature. When submerged in ice water, the temperature registers at 0.00° C. The volume decreased to 2.57 L. What was the water temperature in both Kelvin and Celsius? Given:     

 V1 = 2.80 L, V2 = 2.75 L                      T1 = unknown      T2 = 0.00° C                        

 It is also known that 1K = x + 273.15 C Where x is the temperature in C.                      Charles’ Law: V1/T1 = V2/T2

Step 1: Set up your equations with your known and unknown values. We already have three values, we just need one more, our initial temperature. We’re going to work in kelvin right now as for me that’s an easier base metric to use. We’ll convert to Celsius at the end. Remember 0° Celsius is 273.15 Kelvin. 

Step 2: Use cross muliplication to get the right ratio. That transitionary equation will look like: (T1)(2.57L) = (2.80L)(273.15K). You should end up with .009408 Liters/Kelvin = 2.80 L/ T1

Step 3:  Isolate your T1 value on one side of the equation. This is a little confusing, so bear with me. I can the 2.80 L value on the left hand side of the equation because the T1 value is assumed to be in kelvin. Therefore the units cancel out on that side of the equation.

 Step 4: Now you can cancel out the liters values and end up with a numerical value equaling T1. I don’t believe that I did the units correctly on paper, I think where I messed myself up was needing to multiply the equation on both sides by (T1/T1). But you should end up with a temperature unit in your answer, whether its kelvin or celsius. For this, we found 297.59 kelvin. 

Step 5: Convert your kelvin to celsius. If we add 273.15 to the celsius value to obtain our kelvin value, then we can reverse the process and obtain celsius. 297.59 – 273.15 = 24.44° celsius, a bit warmer than room temperature. 

Final: Kelvin: 297.59 K, Celsius: 24.44°C = T1. As always, drop any questions, comments, etc in the comment section below

.Further Reading: Lecons de physique de la faculté des sciences de Paris, Volume 1, J. Gay-Lussac (1828) 

Sources: Charles Law, Wikipedia.

 Jacques Charles, Wikipedia. 

Jacques Charles, D.A. Dopperpuhl, Chemistry Explained. 

Julie Charles, Wikipedia.

 History of Ballooning, National Balloon Museum. Jacques Charles the Surveyor (1752 -1791, Wikipedia.

A Happy Landing for America’s First Hot-Air Balloon FlightFiaschetti, P. New Jersey Monthly, January 8, 2018.

Charles, Jacques Alexandre César, Encyclopedia Britannica, Vol. 5, 1911 ed. 

Eccentric France The Bradt Guide to Mad, Magical and Marvellous France, Letcher, P. Bradt Travel Guides, 2003. p.36.

Photos: Sketch of Balloon Jacques Charles  

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