Charles's Law: volume and temperature are proportional
Charles's Law states that at constant pressure, the volume of a fixed amount of gas is directly proportional to its absolute temperature. In equation form: V₁/T₁ = V₂/T₂. Heat a gas and it expands; cool it and it contracts. The ratio V/T is constant for a given sample of gas at constant pressure. If you double the Kelvin temperature, you double the volume.
Jacques Alexandre Cesar Charles, a French physicist and balloonist, first observed this relationship around 1787. He built and launched the first hydrogen-filled balloon in 1783, just months after the Montgolfier brothers' hot-air balloon. Charles did not publish his findings; they were reported by Joseph Louis Gay-Lussac in 1802, who cited Charles's unpublished work and extended it with precise measurements. The law is called Charles's Law in English-speaking countries and Gay-Lussac's first law in France.
Gay-Lussac's 1802 paper reported that for every degree Celsius increase in temperature, a gas expands by 1/267 of its volume at 0 °C. The modern figure is 1/273.15, close to Gay-Lussac's 1/267 value. The difference reflects the precision of early-nineteenth-century thermometry: Gay-Lussac's value was accurate to within about 2 percent.
Absolute zero and the Kelvin scale
Charles's Law leads directly to the concept of absolute zero. If you plot volume against Celsius temperature for a gas at constant pressure, the data points form a straight line. Extrapolating that line to zero volume gives a temperature of approximately -273 °C — exactly the value that would become absolute zero on the Kelvin scale. A gas cannot have negative volume, so temperatures below absolute zero are physically impossible in this framework.
Lord Kelvin (William Thomson) formalized the absolute temperature scale in 1848, building directly on the work of Charles and Gay-Lussac. The Kelvin is defined today as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. Zero Kelvin represents the complete absence of thermal energy, a state that can be approached but never reached.
The molecular explanation
From kinetic theory, temperature is a measure of the average kinetic energy of gas molecules. When you heat a gas, the molecules move faster. At constant pressure, the faster molecules strike the container walls with greater force. To maintain constant pressure, the container must expand so the molecules have farther to travel between collisions, which reduces the collision frequency and keeps the pressure steady. The volume increases in exact proportion to the absolute temperature because the average kinetic energy scales linearly with absolute temperature.
Using the calculator
Select which variable to solve for. Enter the three known values. The calculator converts all temperatures to Kelvin internally before applying the law, then converts the result back to your preferred temperature unit. This means you can enter temperature in Celsius or Fahrenheit and get the correct answer without manual conversion.
Worked examples
Example 1: Heating a balloon. A balloon contains 2.50 L of air at 25 °C (298.15 K). If it is heated to 80 °C (353.15 K) at constant pressure, what is the new volume? V₂ = V₁T₂/T₁ = 2.50 × 353.15 / 298.15 = 2.96 L.
Example 2: Cooling a gas sample. A gas occupies 1.00 L at 100 °C. If it is cooled to 0 °C at constant pressure, what volume? V₂ = 1.00 × 273.15 / 373.15 = 0.732 L. The volume shrinks by about 27 percent.
Example 3: Finding initial temperature. A gas expanded from 1.50 L to 2.25 L at constant pressure, and the final temperature is 350 K. Initial temperature? T₁ = V₁T₂/V₂ = 1.50 × 350 / 2.25 = 233 K (-40 °C).
Example 4: Hot-air balloon principle. Air at 15 °C (288.15 K) has density 1.225 kg/m³. When heated to 100 °C (373.15 K) at constant pressure, the density drops to 1.225 × 288.15/373.15 = 0.946 kg/m³. The buoyant force on a 2,800 m³ balloon is (1.225 - 0.946) × 2,800 × 9.81 = about 7,650 N, enough to lift roughly 780 kg.
Charles's Law in everyday life
Hot-air balloons are the most visible application. Heating air inside the envelope reduces its density, creating buoyancy. A typical recreational balloon holds about 2,800 m³ of air. The burner heats it to roughly 100 °C above ambient, reducing density by about a quarter and producing enough lift for the basket, passengers, and equipment.
A car tire pressure changes with temperature. On a cold morning at 0 °C, a tire might read 32 psi. After driving, the tire heats to perhaps 40 °C. According to Gay-Lussac's Law (constant volume), the pressure would rise to about 37 psi. This is why tire pressure is specified cold and why underinflated tires run hotter.
Historical context: balloons and thermometry
Jacques Charles was primarily a balloonist before he was a gas law discoverer. In August 1783, he watched the Montgolfier brothers launch the first hot-air balloon. Charles immediately recognized that hydrogen, being lighter than hot air, would provide better lift. He commissioned the construction of a silk balloon coated with rubber, and on August 27, 1783 — just weeks after the Montgolfier flight — launched the world's first hydrogen balloon from the Champ de Mars in Paris. An estimated 400,000 people watched.
Charles's interest in gas expansion was practical: understanding how gas volume changed with temperature was essential to predicting balloon behavior. On December 1, 1783, Charles and his assistant Nicolas-Louis Robert made the first manned hydrogen balloon flight, reaching an altitude of about 1,800 feet. Charles himself never flew again after that single flight — he was reportedly terrified by the experience. But his work on gas expansion, combined with the precise measurements published by Gay-Lussac in 1802, became the law that bears his name.
The difference between Charles's and Gay-Lussac's Laws
Students frequently confuse Charles's Law and Gay-Lussac's Law because both involve temperature. The distinction: Charles's Law relates volume and temperature at constant pressure (the balloon expands when heated). Gay-Lussac's Law relates pressure and temperature at constant volume (the pressure in a sealed container rises when heated). Both are special cases of the Combined Gas Law with different variables held constant.
Think of it this way: if the container is flexible like a balloon, Charles's Law applies — volume changes with temperature. If the container is rigid like a sealed metal can, Gay-Lussac's Law applies — pressure changes with temperature. If both volume and pressure can change, you need the Combined Gas Law.
Charles's Law and the Kelvin scale: a deeper look
The relationship between Charles's Law and absolute zero is more than a historical footnote. Plot volume against Celsius temperature for any gas, and the linear extrapolation to V = 0 consistently yields -273 °C regardless of which gas you use or what pressure you begin with. This universal behavior is strong evidence that -273.15 °C represents a genuine physical limit, not an artifact of a particular measurement.
William Thomson (Lord Kelvin) proposed the absolute temperature scale in 1848 based on the Carnot cycle, not Charles's Law. But the two approaches converge: a temperature scale where the efficiency of an ideal heat engine is proportional to temperature difference is the same scale where gas volume is proportional to temperature. The consistency between thermodynamic and gas-thermometric definitions of temperature is one of the elegant unifications of nineteenth-century physics.
Charles's Law in chemical engineering
In industrial processes, Charles's Law governs the design of gas-handling equipment that operates at varying temperatures. A pipeline carrying natural gas at 5 °C in winter might carry the same gas at 35 °C in summer. The volume flow rate changes by about 10 percent purely from thermal expansion. Compressor stations, gas storage facilities, and LNG plants all account for this effect in their design specifications.
The law also applies to air that enters a building's HVAC system. Air at 5 °C entering a heating system expands by roughly 4 percent for every 10 °C of heating. Duct sizing calculations must account for this, or the system will be oversized at design conditions and underperform at off-design conditions.
Frequently asked questions
Jacques Charles was primarily a balloonist before he was a gas law discoverer. In August 1783, he watched the Montgolfier brothers launch the first hot-air balloon. Charles immediately recognized that hydrogen, being lighter than hot air, would provide better lift. He commissioned the construction of a silk balloon coated with rubber, and on August 27, 1783 — just weeks after the Montgolfier flight — launched the world's first hydrogen balloon from the Champ de Mars in Paris. An estimated 400,000 people watched.
Charles's interest in gas expansion was practical: understanding how gas volume changed with temperature was essential to predicting balloon behavior. On December 1, 1783, Charles and his assistant Nicolas-Louis Robert made the first manned hydrogen balloon flight, reaching an altitude of about 1,800 feet. Charles himself never flew again after that single flight — he was reportedly terrified by the experience. But his work on gas expansion, combined with the precise measurements published by Gay-Lussac in 1802, became the law that bears his name.
Students frequently confuse Charles's Law and Gay-Lussac's Law because both involve temperature. The distinction: Charles's Law relates volume and temperature at constant pressure (the balloon expands when heated). Gay-Lussac's Law relates pressure and temperature at constant volume (the pressure in a sealed container rises when heated). Both are special cases of the Combined Gas Law with different variables held constant.
Think of it this way: if the container is flexible like a balloon, Charles's Law applies — volume changes with temperature. If the container is rigid like a sealed metal can, Gay-Lussac's Law applies — pressure changes with temperature. If both volume and pressure can change, you need the Combined Gas Law.
The relationship between Charles's Law and absolute zero is more than a historical footnote. Plot volume against Celsius temperature for any gas, and the linear extrapolation to V = 0 consistently yields -273 °C regardless of which gas you use or what pressure you begin with. This universal behavior is strong evidence that -273.15 °C represents a genuine physical limit, not an artifact of a particular measurement.
William Thomson (Lord Kelvin) proposed the absolute temperature scale in 1848 based on the Carnot cycle, not Charles's Law. But the two approaches converge: a temperature scale where the efficiency of an ideal heat engine is proportional to temperature difference is the same scale where gas volume is proportional to temperature. The consistency between thermodynamic and gas-thermometric definitions of temperature is one of the elegant unifications of nineteenth-century physics.
In industrial processes, Charles's Law governs the design of gas-handling equipment that operates at varying temperatures. A pipeline carrying natural gas at 5 °C in winter might carry the same gas at 35 °C in summer. The volume flow rate changes by about 10 percent purely from thermal expansion. Compressor stations, gas storage facilities, and LNG plants all account for this effect in their design specifications.
The law also applies to air that enters a building's HVAC system. Air at 5 °C entering a heating system expands by roughly 4 percent for every 10 °C of heating. Duct sizing calculations must account for this, or the system will be oversized at design conditions and underperform at off-design conditions.
What is Charles's Law? At constant pressure, the volume of a gas is directly proportional to its absolute temperature. V₁/T₁ = V₂/T₂. Heat a gas, it expands. Cool it, it contracts.
Why must temperature be in Kelvin? The relationship is proportional to absolute temperature. 100 °C is not twice 50 °C, but 373 K is roughly 1.15 times 323 K, which produces the correct volume ratio.
Who discovered Charles's Law? Jacques Charles observed it around 1787 but did not publish. Joseph Gay-Lussac published the quantitative relationship in 1802, crediting Charles.
What happens at absolute zero? Extrapolating Charles's Law predicts zero volume at -273.15 °C. Physically, all molecular motion would cease. Absolute zero cannot be reached, only approached.