The Original Formula

\[t_f = \frac{(c_1 \times m_1 \times t_1) + (c_2 \times m_2 \times t_2)}{(c_1 \times m_1) + (c_2 \times m_2)}\]

Where:

• \(t_f\) is the final temperature of the mixture (\(\degree C\))

• \(c\) is the specific heat capacity of the substance (\(J/kg K\))

• \(m\) is the mass of the substance (\(kg\))

• \(t\) is the temperature of the substance (\(\degree C\))

You can use this term with as many substances as you like (simply add each term in the pattern shown).

Simplifying for water

Because the \(c\) is common to both substances (water has a specific heat capacity of \(4184 \; J/kg K\), we can simplify the above equation:

\[\begin{aligned} t_f &= \frac{c \times m_1 \times t_1 + c \times m_2 \times t_2}{c \times m_1 + c \times m_2} \\ &= \frac{c(m_1 \times t_1 + m_2 \times t_2)}{c(m_1 + m_2)} \\ &= \frac{m_1 \times t_1 + m_2 \times t_2}{m_1 + m_2} \\ \end{aligned}\]

In relation to the above statement "\(c\) is common to both substances", this is not strictly true, but it is a good enough approximation for us engineers!