Law of Mixtures

• If two bodies, liquids or gases with different temperatures and heat capacitances come in touch with each other, the following law of mixtures has to be applied

$$T_\mathrm{M} = \frac{m_1c_1T_1 + m_2c_2T_2}{m_1c_1 + m_2c_2}$$ If two bodies with different temperatures are brought together, then heat migrates from the body with the higher temperature to the body with the lower temperature until, after a certain time, a mixed temperature $T_\mathrm{M}$ is reached. The energy emitted by body 1 must therefore correspond exactly to the energy absorbed by body 2: $$-\Delta Q_1 = \Delta Q_2$$ The minus sign indicates the direction of heat flow. Using the definition of the heat capacity we get $$-m_1c_1(T_1 - T_\mathrm{M}) = m_2c_2(T_2 - T_\mathrm{M})$$ By multiplying out both brackets and extracting $T_\mathrm{M}$ and then rearranging, you get the following equation for calculating the mixing temperature $$\boxed{T_\mathrm{M} = \frac{m_1c_1T_1 + m_2c_2T_2}{m_1c_1 + m_2c_2}}$$ This relationship is also called Richmann's law of mixtures, named after the physicist Georg Wilhelm Richmann. It is important to note that all temperatures are converted into the Kelvin unit if they are in degrees Celcius or another unit.

Richmann's law of mixtures is, of course, not only valid for solid bodies, but can also be applied for gases or liquids that are mixed with each other or other combinations of different phases.

• Ancient Water Boiler (++)