Dimensional Formula of Thermal Conductivity: [MLT⁻³K⁻¹]

The dimensional formula of thermal conductivity is [M¹ L¹ T⁻³ K⁻¹], where K stands for the temperature dimension (some books write Θ instead). Thermal conductivity k measures how easily heat flows through a material, and it is one of the few Class 11 quantities whose bracket needs temperature.

The result comes from the law of heat conduction: the rate of heat flow through a slab is proportional to its area and the temperature difference, and inversely proportional to its thickness. Solve that for k and the dimensions follow. See the thermal conductivity formula guide for values of k and slab problems.

Quick facts

Dimensional formula
[M¹ L¹ T⁻³ K⁻¹]
Defining formula
$\frac{Q}{t} = \frac{kA\,\Delta T}{L}$
SI unit
W m⁻¹ K⁻¹
Base units
kg m s⁻³ K⁻¹
Type
Scalar property of a material
Typical values
Copper about 400, glass about 1, air about 0.025 W m⁻¹ K⁻¹

Derivation: dimensional formula of thermal conductivity

Conduction law: $\frac{Q}{t} = \frac{kA\,\Delta T}{L}$ , where L is the thickness of the slab.
Rearrange: $k = \frac{(Q/t)\,L}{A\,\Delta T}$ .
Heat is energy, so the heat flow rate is a power: $[Q/t] = \frac{[ML^{2}T^{-2}]}{[T]} = [ML^{2}T^{-3}]$ .
Thickness is [L], area is [L²] and temperature difference is [K].
Combine: $[k] = \frac{[L][ML^{2}T^{-3}]}{[K][L^{2}]} = [MLT^{-3}K^{-1}]$ .

The length powers go 2 + 1 − 2 = 1. The single most common wrong answer, [ML²T⁻³K⁻¹], comes from forgetting the thickness on top or the area below.

From the bracket to W m⁻¹ K⁻¹

Write the bracket in base units: kg m s⁻³ K⁻¹. Recognize kg m² s⁻³ as a watt and divide by one metre, and you get W m⁻¹ K⁻¹, the unit printed in data tables. Because a temperature difference of 1 K equals a difference of 1 °C, W m⁻¹ °C⁻¹ is the same unit.

Example: an old table lists a conductivity of 1 cal s⁻¹ cm⁻¹ °C⁻¹. Convert it to SI.

Use 1 cal = 4.186 J and 1 cm = 0.01 m.

$1\ \frac{\text{cal}}{\text{s cm}\ ^\circ\text{C}} = \frac{4.186\ \text{J}}{(1\ \text{s})(0.01\ \text{m})(1\ \text{K})} = 418.6\ \text{W m}^{-1}\text{K}^{-1}$ .

So the old unit is about 419 times larger than the SI unit.

A dimensional check with thermal resistance

Example: thermal resistance is defined as $R_{th} = \frac{L}{kA}$ . Show that $\frac{Q}{t} = \frac{\Delta T}{R_{th}}$ is consistent.

$[R_{th}] = \frac{[L]}{[MLT^{-3}K^{-1}][L^{2}]}$ , which simplifies to [M⁻¹L⁻²T³K].

$\frac{[K]}{[M^{-1}L^{-2}T^{3}K]} = [ML^{2}T^{-3}]$ , which is power, the bracket of Q/t. The equation passes.

Example: heat loss through a glass pane, 1.5 m² in area and 4.0 mm thick, with k = 0.80 W m⁻¹ K⁻¹ and 12 K across it.

$\frac{Q}{t} = \frac{(0.80)(1.5)(12)}{0.0040} = 3.6 \times 10^{3}\ \text{W}$ .

Writing 4.0 mm as 0.0040 m is the step most students miss.

Related heat quantities and their brackets

Thermal brackets that look alike
Quantity	Bracket	SI unit
Thermal conductivity k	[M L T⁻³ K⁻¹]	watt per metre kelvin
Heat flow rate Q/t	[M L² T⁻³]	W
Temperature gradient ΔT/L	[L⁻¹ K]	K/m
Specific heat capacity c	[L² T⁻² K⁻¹]	J kg⁻¹ K⁻¹
Boltzmann constant	[M L² T⁻² K⁻¹]	J/K

No common quantity outside heat transfer shares the bracket of k, which makes it a favorite for "find the dimensions" questions. The conduction law can be read as k = (heat flow per area) ÷ (temperature gradient).

Thermal conductivity bracket slips

Forgetting K. Temperature is a base quantity and must appear with power minus 1.
Using T for temperature. T is already time. Use K or Θ to keep the two apart.
Writing T⁻² for time. Heat flow is energy per second, which pushes the power of time to minus 3.
Mixing up conductivity and specific heat. One is about heat flow through a material, the other about heat stored per kilogram.

Thermal and mechanical brackets are listed together on the dimensional formulas hub, which helps when a JEE question mixes the two.

Frequently asked questions

What is the dimensional formula of thermal conductivity?

It is [M¹ L¹ T⁻³ K⁻¹]. Divide power times thickness by area times temperature difference, using the law of heat conduction.

What is the dimension formula of thermal conductivity with Θ?

Some books use Θ for temperature, so the same answer is printed as [M L T⁻³ Θ⁻¹]. Both letters stand for one base dimension.

Which SI unit is printed for k in data tables?

Tables give W m⁻¹ K⁻¹, read as watts per metre kelvin. Written out in base units it becomes kg m s⁻³ K⁻¹.

Is thermal conductivity a constant for a material?

k belongs to the substance and shifts a little as the temperature changes. School and entrance exam problems treat it as fixed for each material.

How is thermal conductivity different from thermal resistance?

Conductivity is a material property with [MLT⁻³K⁻¹]. Thermal resistance L/(kA) belongs to one particular slab and has [M⁻¹L⁻²T³K], measured in K/W.