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Thermal conductivity

Definition of Temperature

T = Temperature [K], [° C]

Temperature is a physical value from which the following statements about substance characteristics can be derived:

  • Thermodynamic equilibrium – if two bodies have the same temperature there is no exchange of heat between them. If they have different temperatures, heat flows from the warmer to the colder body until an equilibrium is achieved.
  • Measurement of the middle kinetic energy of gases.
  • Features of substances that depend on temperature. Examples: thermic expansion of substances, density, electric resistance.

The base unit (Si-unit) of temperature is given in Kelvin [K]. In Europe, it is also common to use °C. 1 °C = 273,15 K.

The astronomer Anders Celsius defined benchmarks for the Celsius scale. He said that the boiling point of pure water is at 0°C and the melting point at 100° C. The boiling and melting point have only been reversed after his death, though he built the foundation for today’s temperature measurements.

0th Law – temperature


“0th-Law-Temperature” Image created by Lecturio

Quantities that only depend on what a system looks like now are state variables. They only depend on the current state:

  • Examples are pressure, temperature, and volume
  • Counter-example is “the amount of added heat”

Temperature is equivalent for all materials, so if two objects are in thermal equilibrium with a third, they are in equilibrium with each other.

The absolute zero

According to its definition, absolute zero is the lowest temperature that can still be measured. It is at zero Kelvin (= – 273,15° C). The theory implies that gases get colder with decreasing speed of the particles they consists of. At the absolute zero, the speed of all particles equals zero. This means that no negative temperatures can be measured on the absolute temperature scale (Kelvin-scale).

Thermic state equations for gases

For ideal gases, proportionality between pressure and temperature exists. The following laws describe the correlations of two values while other involved values remain constant:

Law of Boyle-Mariotte

If the pressure of an ideal gas is increased and temperature and amount of substance remain the same, the pressure, is reciprocally proportional to the volume:

ρ ∼ 1 / V

Law of Gayle-Lussac

The volume of ideal gases is proportional to temperature as long as the amount of substance and pressure remain constant.

V ∼ T

Law of amontons

If ideal gases are warmed up, pressure is increased, and if cooled down—the pressure decreases.

Coefficient of thermal expansion

Different substances expand to different extents under the same increase of temperature. The responsible effect is called thermal expansion. This process does not always take place steadily and is classified depending on the extent of expansion into:

Linear coefficient of thermal expansion

This coefficient states the change in length of a substance in relation to its total length.

α Coefficient of linear expansion [10-6 / K ] at 20° C
L Length [m]
ΔL Change of length [m]
ΔT Change of temperature [K]

α = 1 / L * ΔL / ΔT

Volume-specific expansion coefficient / cubic expansion coefficient

States the change of volume of a body in relation to its total volume at a change of temperature of one Kelvin.

γ Cubic expansion coefficient  [10-3 / K] at 20° C
V0 Volume prior to heating [m³]
δV Change of volume [m³]
δT Change of temperature [K]

γ = 1/ V0 * (δV / δT)

Temperature measurements

Since the different characteristics of substances depend on temperature, these dependent values can be used to measure temperature:

Liquid thermometer

A liquid – usually mercury or dyed alcohol – is put into a thin tube with an attached scale. The temperature can be measured due to the increase and decrease in volume during heating or cooling down the liquid. The liquid in thermometer must have the following properties;

  • It should expand quickly and uniformly in a wide range of temperatures
  • Should not stick to the sides of the tube

Water is not suited due to the anomaly of water i.e. if water is heated say from -15°c, it expands but up to only 0° this point it starts to melt and it contracts up to 4°c.on further heating, the water expands uniformly up to its boiling point

Gas thermometer

A drop of mercury is put into a thin tube, similar to the liquid thermometer. The tube closes up a room containing a gas. This gas expands if warmed up or decreases its volume when being cooled down.

Bimetallic thermometer

The expansion during heating is different depending on the substance. A coiled bimetallic strip consists of two different types of metal and is attached to an indicator. If the bimetallic strip is heated up, the two metals expand with different intensity, which leads to a deflection of the indicator.

Electronic thermometer/resistance thermometer

Electric resistance highly depends on temperature, especially in the case of semiconductors. It holds: The resistance of a thermistor decreases with increasing temperature. Reducing the resistance leads to an increased current flow, which leads to a measurement of temperature changes.

Thermo colors

So-called thermo colors change their color or emit light at specific temperature changes.


Q Heat [J]
c specific heat capacity  [J/(kg*K)]

An increase in temperature results in an increase of kinetic energy of the smallest parts. Heating implies energy input. Cooling implies energy extraction.

Heat is a special type of energy. According to the principle of conservation of energy, inner energy can only be obtained by conversion of other types of energy such as mechanic, electric, chemical or nuclear energy.

Remember: Energy cannot be lost but can only be converted from one form of energy into another. (see also article Mechanics I)

Heat and inner energy are to be seen as equal with other forms of energy and at least as partly convertible into each other.

The absorbed heat of a body is proportional to its mass and to the change of temperature of the body. The specific heat capacity generally indicates what amount of heat is necessary to heat one kilogram of a specific substance by one Kelvin. A calorimeter can measure the specific heat capacity, which means the change of heat within a body and thus the change of inner kinetic energy (heat flow calorimeter, heat balance calorimeter, adiabatic calorimeter).

The calculation of heat is defined as follows:

  ΔQ = c * m * ΔT

The absorbed or emitted heat of a body equals the product of specific heat capacity, the mass of the body and the change of temperature.


“1st Law – Energy” Image created by Lecturio

1st Law – Energy

Energy in a closed system is conserved. Energy can be added or subtracted to the system but is not lost or gained within a system. Internal energy of a system: U.

Thermal output

P Thermal output [W]

If a heat source provides a specific amount of heat within a specific interval of time, the thermal output is the quotient of heat and time.

P = ΔQ / Δt

Thermal convection

Heat can be transferred in three different ways: through heat conduction, heat convection and heat radiation. In this process, heat is transferred from one body to a colder one.

If a body is heated up in one place, the density of the liquid is reduced at this location due to the increase in temperature. Buoyancy makes the warmed up liquid rise up and the colder liquid sink down to the bottom. This leads to a transport of water mass that carries heat energy.
Humans even continuously exchange heat with their surroundings through the following four mechanisms.

Conduction (transfer of heat through direct contact)

Heat wanders from places of higher temperature to neighboring (bordering) places of lower temperature. The transfer occurs through forwarding kinetic energy from molecule to molecule. The better the conductor, the more rapidly heat it will transfer.

Convection (exchange of heat through a medium (air, water))

The movement of particles in liquids and gases is also defined as convection, which is an important factor for giving off produced body heat within the process of human thermoregulation. Convection enables the transport of blood gas throughout the body with the blood flow.

Radiation (heat radiation through electromagnetic waves)

If no intermediate medium is needed for the transport of thermal energy from the warmer body to a colder one, this is referred to as heat radiation. Heat rays are not only emitted by warm bodies that emit light but also by non-light emitting bodies as soon as their own temperature is higher than the temperature of their surroundings. If heat rays hit a body with lower temperature they heat this body up.

Evaporation (loss of heat through vaporization)

Vaporization takes place via the skin in the form of sweat.

The principle of thermodynamics

W work performed [J]
H enthalpy [J]
U inner energy, has no unit
p pressure
V volume

The state of a gas is identified through the three state variables pressure, volume and temperature. Change of two or all state variables is referred to as a change of state. Each state of a system has a specific, clearly defined value of inner energy. This is where the first principle of thermodynamics applies. It holds that:

The input of heat and kinetic energy increases the inner energy of a closed system.

If Q is the amount of heat energy input, W the conducted mechanical work and ΔU the change of inner energy, it holds that:

ΔU = Q + W

In the case of ideal gases, mechanical work leads to a change in volume. The input of heat energy results in an increase of inner energy and an increase in volume. The inner energy refers to the total energy within a system. It is a state function that only depends on the states of pressure, volume, and temperature. The change of inner energy is determined only by initial state and final state.

The sum of inner energy and the product of pressure and volume is referred to as enthalpy. The product of pressure and volume equals the amount of displacement.

H = U + (ρ * V)

Thermal conduction

Since the conduction of heat is a particularly important topic in medicine, this chapter will address the issue in further details.

The easiest way of heat transfer is heat conduction. As mentioned, this is done by neighboring substances, the warmer substance transferring its energy to the colder one.

Heat conduction only takes place inside matter and requires a temperature gradient. Different materials have different heat conductivity. This heat conductivity is expressed as the coefficient of thermal conductivity. The coefficient of thermal conductivity is the amount of heat that can be transferred per time unit across two opposite sides of a cube with the edge length of 1 m and a temperature differential of one Kelvin between them. The other sides of the cube have to completely heat impermeable.

Thermal conductivityIt is also possible to present the thermal conductivity of substances in reference to a specific substance. This gives the relative conductivity.

Metals generally are good heat conductors. Bad heat conductors are gases, wool, paper and many others, which are used as thermal insulation materials.

Medicine applies different heat therapies to ensure, for example, an improvement and preservation of the function of the locomotor system, to strengthen and relax musculature, improve the muscle tone, or relieve pain.

Popular Exam Questions in Thermodynamics

The answer key can be found below the references.

1. Assume that steam is in (dynamic) equilibrium with its liquid. The isotherms of this steam are horizontal straight lines in a P-V-diagram because the pressure of a liquid’s steam only depends on temperature.

  1. Statement 1 is true, Statement 2 is true, linkage is true
  2. Statement 1 is true, Statement 2 is true, linkage is false
  3. Statement 1 is true, Statement 2 is false, no linkage possible
  4. Statement 1 is false, Statement 2 is true, no linkage possible
  5. Statement 1 is false, Statement 2 is false, no linkage possible

2. Approx. which value is the ratio T2/T1 for the respective absolute temperatures of t1 = 127 ° C and t2 = 47° C?

  1. 1/3
  2. 4/5
  3. 5/4
  4. 5
  5. None of the above mentioned is true

3. How much energy is necessary to heat 2 liters of water (cwater= 4,2 kJ * K – 1 * kg – 1) from T = 20 ° C to T =  40° C?

  1. 42 kJ
  2. 84 kJ
  3. 168 kJ
  4. 336 kJ
  5. 420 kJ
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