Freezing Point Depression Calculator (ΔTf = i · Kf · b)

Background.

Dissolve anything in a liquid and it freezes below the temperature the pure liquid would. That is freezing-point depression, and to a good approximation it depends only on how many solute particles are present, not on what they are: ΔTf = i · Kf · b, where b is the molality in moles per kilogram of solvent, Kf is a constant belonging to the solvent, and i is the number of particles each dissolved formula unit releases. This calculator solves that relation forwards, solves it backwards to find an unknown solute's molar mass, and derives Kf itself from a solvent's enthalpy of fusion.

Before the number, the limits — because they change how the result should be read and belong here rather than in a collapsed FAQ. The equation is the dilute-solution limit of an exact thermodynamic condition, and it assumes four things. The solute must be non-volatile and must not dissolve into the solid phase, so that pure solvent crystallises out. The solution must behave ideally, meaning activity can be replaced by mole fraction. The logarithm in the exact relation is linearised as ln(1 − x) ≈ −x. And the enthalpy of fusion is treated as constant across the small temperature interval. In practice the prediction is good to a percent or so below about 0.1 mol/kg for a non-electrolyte, drifts steadily above that, and is systematically off for electrolytes at any concentration you would actually use. A 5 mol/kg brine will freeze lower than this equation says, not higher, and by a margin that matters.

The concentration unit is molality, not molarity, and that is not arbitrary. This is an experiment that deliberately changes the temperature, and molarity changes with temperature because a litre expands. Molality divides by the mass of solvent, which does not expand, so the concentration on the left of the equation is fixed by how the solution was made and stays fixed all the way down to the freezing point. This is why cryoscopic constants are tabulated in K·kg/mol and never in K·L/mol.

The van 't Hoff factor deserves care. For a non-electrolyte — glucose, urea, ethylene glycol — it is exactly 1, because the molecule that dissolves is the particle that counts. For an ionic compound the ideal value is the number of ions per formula unit, 2 for sodium chloride and 3 for calcium chloride, but the measured value is always lower, because oppositely charged ions spend part of their time paired and a paired ion behaves as one particle rather than two. The effective factor is i = ν·φ, where ν is the ion count and φ the practical osmotic coefficient, and φ is below 1 throughout the dilute range. This calculator leaves i as an editable input defaulted to 1 rather than guessing an integer for you.

Kf is not an independent physical constant either. It follows from the solvent's own thermodynamics as Kf = R·Tf²·M_A ÷ ΔfusH, so the third mode of this calculator will compute it for you from data you can look up in the NIST Chemistry WebBook. That mode exists for a specific reason. Published tables of cryoscopic constants for organic solvents disagree between compilations by several percent, because different sources use enthalpies of fusion measured at different temperatures. Rather than ship a table of numbers whose ancestry it cannot vouch for, this page ships the derivation and one verified default: 1.86 K·kg/mol for water, which the derivation independently reproduces as 1.8605 from a peer-reviewed equation of state for ice.

Below the widget you will find that derivation in full, the hand-computed ethylene glycol example the unit tests are built from, a worked derivation for benzene that lands within 0.2 percent of the published constant, guidance on choosing i, and the domain edges where the calculator refuses rather than returning a number.

What is freezing point depression calculator?

Freezing-point depression is one of the four colligative properties, alongside boiling-point elevation, vapour-pressure lowering and osmotic pressure. Colligative means the effect depends on the number of solute particles rather than their identity: one mole of glucose and one mole of urea depress water's freezing point by the same amount, because both contribute one mole of particles.

The physical mechanism is about the liquid, not the solid. Dissolved solute lowers the chemical potential of the liquid solvent by diluting it, while leaving the chemical potential of the solid solvent untouched — because the solute is excluded from the crystal. Freezing happens when those two potentials are equal, and lowering one of them means the system has to be cooled further before they meet. That exclusion assumption is why the equation fails for solutes that form solid solutions with the solvent.

The working form is ΔTf = i · Kf · b. Kf, the cryoscopic constant, has units of K·kg/mol and is a property of the solvent alone. Its value follows from the solvent's normal freezing point Tf, its molar mass M_A and its molar enthalpy of fusion ΔfusH: Kf = R·Tf²·M_A ÷ ΔfusH, with Tf in kelvin, M_A in kg/mol and ΔfusH in J/mol. For water that gives 1.8605 K·kg/mol using the ice melting enthalpy from Feistel and Wagner's 2006 equation of state, against the 1.86 that handbooks print — agreement to 0.03 percent, and a useful demonstration that the tabulated constant is not folklore.

The reverse calculation is a real analytical technique called cryoscopy. Weigh a known mass of an unknown compound into a known mass of solvent, measure how far the freezing point drops, and the molar mass follows from M = mass ÷ (b × kg of solvent) with b = ΔTf ÷ (i·Kf). It was one of the standard methods for determining molecular weights before mass spectrometry, and it is still taught because the whole chain from measurement to answer is visible.

How to use this calculator.

1. 01Pick the mode. 'Freezing point of the solution' is the forward calculation; 'molar mass' is cryoscopy; 'cryoscopic constant' derives Kf from solvent thermodynamics and then applies it.
2. 02Enter the mass of SOLVENT, not the mass or volume of the finished solution. 1000 g of water is one kilogram but roughly 1003 mL at 20 °C, so weighing and measuring are not interchangeable here.
3. 03Choose the van 't Hoff factor deliberately. Use 1 for anything that does not dissociate. For a salt, the ideal value is the ion count, but expect the real depression to fall short of the ideal prediction — read the FAQ on ion pairing before treating an integer as exact.
4. 04Leave Kf at 1.86 only if your solvent is water. For anything else, switch to the derivation mode and enter your solvent's freezing point, molar mass and enthalpy of fusion from the NIST Chemistry WebBook — do not copy a constant from a table without checking its source.
5. 05For cryoscopy, enter the depression as a positive magnitude in K. If your thermometer read −1.86 °C for a solvent that freezes at 0 °C, the depression is 1.86, not −1.86.
6. 06Sanity-check the molality against the regime. Below 0.1 mol/kg the prediction should be good to about a percent for a non-electrolyte. Above 1 mol/kg treat it as an order-of-magnitude guide.
7. 07Read the depression as a difference and the freezing point as an absolute. A difference has the same number in K and °C; an absolute temperature does not.
8. 08For engine coolant or de-icing decisions, do not stop here. Colligative physics gives the freezing point; corrosion inhibition, pump cavitation and the manufacturer's specification decide the concentration, and none of those is on this page.

The formula.

Two equations, one of which is usually presented as a constant to be looked up.

ΔTf = i · Kf · b the colligative relation b = (mass ÷ M) ÷ kg of solvent molality of the solute Kf = R · Tf² · M_A ÷ ΔfusH the constant, from solvent thermodynamics M = mass ÷ (b × kg of solvent) cryoscopy, with b = ΔTf ÷ (i·Kf)

The first line is a linearisation. The exact condition for freezing is that the chemical potential of the solvent be the same in the liquid and the solid, and writing that out gives a relation involving ln(x_solvent), the logarithm of the solvent's mole fraction. Expanding ln(1 − x_solute) as −x_solute for small x_solute, and converting mole fraction to molality, produces the linear form. Everything that goes wrong with this equation at high concentration goes wrong in that expansion or in the assumption that activity equals mole fraction.

The third line is worth having because it explains where a cryoscopic constant comes from. Kf is large when the solvent's freezing point is high (it enters squared), when the solvent's molecules are heavy, and when the solvent's enthalpy of fusion is small. That is why camphor, with a high melting point and a small enthalpy of fusion, has a cryoscopic constant an order of magnitude above water's, and why cryoscopy was historically done in camphor — a bigger constant means a bigger, more easily measured temperature drop from the same amount of unknown.

Worked for water, using data from a source that never publishes a cryoscopic constant: Feistel and Wagner's 2006 equation of state for ice Ih, published in the Journal of Physical and Chemical Reference Data and adopted by IAPWS, gives a melting enthalpy of 333.427 kJ/kg at the normal-pressure melting point. Multiply by water's molar mass, 0.018015 kg/mol, to get 6006.7 J/mol. Then Kf = 8.314462618 × 273.15² × 0.018015 ÷ 6006.7 = 1.8605 K·kg/mol. Handbooks print 1.86. The two agree to 0.03 percent, which is a genuine independent confirmation rather than a circular one.

The same derivation for benzene, entirely from NIST Chemistry WebBook data — freezing point 278.64 K, enthalpy of fusion 9.8663 kJ/mol from Oliver, Eaton and Huffman (1948), molar mass 78.114 g/mol — gives 5.111 K·kg/mol against the commonly tabulated 5.12. Agreement to 0.18 percent.

Rounding: all arithmetic is arbitrary-precision and rounding happens once, at the return boundary, to ten decimal places. Importantly, a Kf derived in the third mode is applied to the depression unrounded. Rounding it to the three significant figures a textbook prints would shift a 1 mol/kg result by about half a millikelvin — small, but it is precisely the kind of silent intermediate rounding that makes a formula right in the middle of its range and wrong at the edges, so the tests assert the unrounded path.

Invalid domain: a van 't Hoff factor of zero or less is rejected, since nothing releases zero particles. A cryoscopic constant of zero or less is rejected. Solvent mass and solute molar mass must both be positive. In cryoscopy mode a measured depression of exactly zero is rejected, because the molar mass it implies is infinite, and a zero solute mass is rejected for the same reason. In the derivation mode an absolute freezing point at or below 0 K is rejected — the formula squares it — and a zero enthalpy of fusion is rejected because the formula divides by it.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Feistel, R. & Wagner, W. (2006). 'A New Equation of State for H2O Ice Ih', Journal of Physical and Chemical Reference Data 35(2), 1021–1047 — the equation of state adopted by IAPWS in 2006. Melting-curve table at the normal-pressure melting point (T = 273.152519 K, p = 0.1013 MPa) gives Δh_melt = 333.427 kJ/kg; §3.5 compares this with the experimental determinations of Giauque & Stout (1936: 333.49 and 333.42), Osborne (1939: 333.54) and Haida et al. (1974: 333.41). This is the value from which this page derives Kf(water) = 1.8605 K·kg/mol. Peer-reviewed, free PDF; retrieved 2026-07-29.
2. [2]NIST Chemistry WebBook, SRD 69 — phase-change data. For benzene: T_fus = 278.64 ± 0.08 K (average of 57 values) and Δ_fus H = 9.8663 kJ/mol at 278.69 K from Oliver, Eaton & Huffman (1948), compilation DH. Used for the worked non-water derivation on this page, which returns Kf = 5.111 K·kg/mol against the commonly tabulated 5.12. Retrieved 2026-07-29.
3. [3]NIST Special Publication 811, 2008 edition, section 8.6.8 'Molality of solute B' — b_B = n_B/m_A, SI unit mol/kg, defined per mass of SOLVENT. Section 8.5 'Temperature interval and temperature difference' establishes that {Δt}°C = {ΔT}K, which is why the depression on this page needs no unit conversion. Retrieved 2026-07-29.
4. [4]IUPAC, 'Quantities, Units and Symbols in Physical Chemistry' (Green Book), 3rd edition, 2nd printing 2012, section 2.10 composition table: molality m_B, b_B = n_B/m_A with SI unit mol kg⁻¹, and note 14 on the m/b symbol collision. Confirms independently of NIST that the concentration variable in the colligative equations is per kilogram of solvent. Retrieved 2026-07-29.
5. [5]Hamer, W. J. & Wu, Y.-C. (1972). 'Osmotic Coefficients and Mean Activity Coefficients of Uni-univalent Electrolytes in Water at 25 °C', Journal of Physical and Chemical Reference Data 1(4), 1047–1100. Tabulates practical osmotic coefficients φ for 79 uni-univalent electrolytes; the effective van 't Hoff factor is i = ν·φ, and φ is below 1 throughout the dilute range, which is why a real salt depresses less than its ion count predicts. Cited for the direction and origin of the deviation; specific tabulated coefficients were not extracted from the scanned tables and no numeric φ is asserted on this page. Free NIST reprint; retrieved 2026-07-29.
6. [6]IUPAC Commission on Isotopic Abundances and Atomic Weights (CIAAW), Standard Atomic Weights 2021, abridged values — the source of every molar mass on this page: H 1.008, C 12.011, O 15.999, giving M(H2O) = 18.015, M(C2H6O2) = 62.068, M(C6H6) = 78.114 g/mol. Retrieved 2026-07-29.