Entropy Calculator

What an entropy change measures

Entropy, written S, counts how many ways the energy and particles in a sample can be arranged. It is measured in joules per kelvin (J/K), and a change in entropy answers one question: did the arrangement get more spread out or less? Melting, boiling and warming all move a substance toward the spread out end.

The formula for a change is short. Delta S equals the heat moved in or out reversibly, divided by the absolute temperature at which the transfer happens. Dividing by temperature is what makes entropy different from plain energy: the same 100 joules added to ice at 273 K changes the arrangement far more than 100 joules added to water at 373 K.

Signs follow the direction. Warming, melting and boiling absorb heat and give a positive delta S; cooling, freezing and condensing give a negative one. Those signs come out of the formulas on this page, not out of taste.

A negative delta S for the sample is allowed, and common. Freezing water lowers its own entropy while raising the entropy of the room by more, and the total still rises. That is the second law: system plus surroundings never loses entropy.

Heating and cooling: mass, specific heat and a logarithm

For a temperature change with no phase change, delta S = m x c x ln(T2 / T1). Here m is the mass in kilograms, c is the specific heat in J per kg per kelvin, and T1 and T2 are the temperatures in kelvin. The natural logarithm appears because each extra degree lands on a warmer sample, where the same heat buys less order.

Worked example. One kilogram of water warms from 20 C to 80 C, which is 293.15 K to 353.15 K. The ratio is 1.2047 and its natural log is 0.1862, so delta S = 1 x 4,184 x 0.1862 = 779.1 J/K. The heat that flowed was 251,040 J, and the entropy gain is that heat divided by a temperature somewhere between the two ends.

Double the mass, double the change. Two kilograms over the same span gains 2 x 4,184 x 0.1862 = 1,558.2 J/K.

A metal pan. A 0.5 kg aluminum pan heats from 20 C to 180 C. Aluminum's specific heat is 897 J per kg per kelvin, the ratio 453.15 / 293.15 is 1.5458 and its natural log is 0.4355, so delta S = 0.5 x 897 x 0.4355 = 195.3 J/K. The pan gains a quarter of the water example's entropy while taking in more heat, because most of that heat arrives at a higher temperature.

Cooling runs the sign backwards. Let that water fall from 80 C back to 20 C and the answer is -1,558.2 J/K, exactly what it gained; the kitchen gains the rest.

Temperatures must be absolute. Feeding 20 and 80 into the ratio as Celsius gives a number that means nothing, because the logarithm is measured from zero kelvin.

Melting, boiling, freezing and condensing

Phase changes happen at a fixed temperature, so the logarithm drops out and the formula becomes delta S = m x L / T. L is the latent heat in joules per kilogram and T is the transition temperature in kelvin.

Melting ice. One kilogram of ice at 0 C takes 334,000 J to melt while the temperature holds at 273.15 K. Delta S = 334,000 / 273.15 = 1,222.8 J/K. The molecules go from a locked lattice to a liquid that slides past its neighbors, and that freedom is worth about 1.2 kJ per kelvin per kilogram.

Boiling water. One kilogram at 100 C takes about 2,260,000 J to turn into steam at the same temperature. Delta S = 2,260,000 / 373.15 = 6,056.5 J/K, roughly five times the melting value, because a gas spreads through the whole room while a liquid still clings to itself.

The stages add up. One kilogram from 20 C to a boil gains 4,184 x 0.2412 = 1,009.4 J/K, and boiling it dry adds another 6,056.5 J/K for a total of 7,065.9 J/K. Almost all of it arrives in the last stage, while the thermometer sits still.

Specific heat and latent heat for common substances

Specific heat tells you how much energy moves the temperature; latent heat tells you how much energy moves the arrangement. Both matter here, and both vary by substance. The table lists typical values near room temperature and at ordinary pressure.

Water sits near the top of the specific heat column, which is why it takes so long to heat a pot. The table also explains why an aluminum pan reaches 180 C in a minute while the water inside is still lukewarm: the metal needs 897 joules per kilogram per degree, the water needs 4,184. Latent heats are quoted at the normal transition points, so ethanol's 855 kJ per kilogram is spent at its 78 C boiling point, not at 100 C. The NIST Chemistry WebBook lists measured values for hundreds of substances.

Gases: expansion, mixing and the R ln term

Gases get their own formula because their entropy depends on how much room they have. For an ideal gas held at constant temperature, delta S = n x R x ln(V2 / V1), with n the number of moles, R = 8.314 J per mole per kelvin, and V1 and V2 the starting and ending volumes.

Worked example. One mole of gas expands from 1 L to 2 L at a steady temperature. The ratio is 2 and its natural log is 0.6931, so delta S = 1 x 8.314 x 0.6931 = 5.76 J/K. Expanding from 1 L to 10 L instead gives 8.314 x 2.3026 = 19.1 J/K. Compression reverses the sign: pushing that mole back to 1 L lowers its entropy by 5.76 J/K, which is why a compressor needs work and the coil behind a refrigerator runs warm.

Mixing works the same way. Two different gases that share one volume each spread into the other's space, and the total entropy rises by roughly the sum of two expansion terms. That is why an opened bottle of perfume scents a whole room, and why the scent never collects back in the bottle on its own. Real gases drift from the ideal formula near their condensation point, where molecules attract each other.

Entropy of a chemical reaction

For a chemical reaction, delta S standard = sum S(products) - sum S(reactants), using tabulated standard molar entropies. Those are per mole, measured at 298 K and standard pressure, in J per mole per kelvin rather than J per kg per kelvin.

Worked example. The Haber reaction, N2 + 3 H2 going to 2 NH3. Products: 2 x 192.8 = 385.6 J/K. Reactants: 191.6 + 3 x 130.7 = 583.7 J/K. Delta S standard = 385.6 - 583.7 = -198.1 J/K. Four gas molecules become two, so the mixture loses freedom and the sign is negative, which is exactly what the count of molecules predicts.

The pattern is worth reading before you use the table. Gases sit far above liquids, and liquids far above solids, because freedom to move through space dominates the count. A reaction that makes more gas molecules almost always gains entropy, and one that consumes gas usually loses it.

Reaction entropies are small next to the surroundings term: tens or hundreds of joules per kelvin, while heat released to a room changes thousands. That is why a reaction with a negative delta S can still run, since the heat it gives off raises the entropy of the room by more than it loses. The OpenStax chapter on the second law works through that term.

Reading the result and the usual mistakes

Four checks catch most wrong answers.

Kelvin, not Celsius. Both formulas divide or take a logarithm of an absolute temperature. A temperature of 20 C is 293.15 K; use 20 in the ratio and the answer is meaningless. The calculator converts before it computes, including Fahrenheit inputs.

Per kilogram or per mole. The heating and phase formulas use J per kg per kelvin, and the reaction formula uses J per mole per kelvin. Mixing the two is the most common unit error on homework, and it can be off by a factor of 18 for water.

Joules or kilojoules. A specific heat of 4.184 is in kJ and 4,184 is in J; the results differ by 1,000, and both look plausible in a sentence.

Sign of the surroundings. A negative delta S for the sample is normal. What must be positive is the total for sample plus surroundings.

One free sanity check: compare the answer with the heat involved. Entropy is heat divided by an absolute temperature between 273 K and 373 K for water, so 779.1 J/K should pair with a heat near 779.1 x 320 = 249,312 J. The example moved 251,040 J, close enough given that rough average temperature. When the two are off by a factor of ten, a unit slipped.

Where entropy shows up at home

Entropy is not only a classroom quantity.

Why ice cools a cooler. One kilogram of ice melting absorbs 334,000 J from the food and drinks around it while its own entropy rises by 1,222.8 J/K. The freezer that made the ice dumped more heat, and more entropy, into the kitchen to do it. The cold is borrowed, not created.

Why a freezer warms the room. The heat rejected at the back coil leaves at a higher temperature than the heat pulled from inside, so the kitchen ends up warmer than the electricity alone explains, which is why a freezer in a closed closet stops working well.

Why a kettle takes so long at the end. Heating 1 kg of water from 20 C to 100 C costs about 335 kJ. Boiling it dry costs another 2,260 kJ, nearly seven times more. The last stage is where the time goes.

Why sweat works. Evaporating a gram of water at skin temperature takes about 2,400 J and raises that gram's entropy by roughly 7.8 J/K. The heat comes out of your skin and the vapor carries the entropy into the air. In humid weather the air already holds vapor, so evaporation slows.

Why a hot drink never re-warms itself. Heat flows from the cup to the room until the two match, and the entropy gained by the room exceeds the entropy lost by the cup. The reverse flow would lower total entropy, and nothing in a kitchen pays that bill. The USGS water science school documents the water properties behind these examples.