A mass m₁ of a substance of specific heat capacity c₁ at temperature t₁ is mixed with a mass m₂ of other substance of specific heat capacity c₂ at a lower temperature t₂. Deduce the expression for the temperature t of the mixture. State the assumption made, if any.

Let a substance A of mass m₁, specific heat capacity c₁ at temperature t₁ is mixed with another substance B of mass m₂, specific heat capacity c₂ at a lower temperature t₂.

If the final temperature of the mixture becomes t, then

Fall in temperature of substance A = t₁ – t

Rise in temperature of substance B = t – t₂

Heat energy lost by A = m₁ × c₁ × fall in temperature

= m₁c₁ (t₁ – t)

Heat energy gained by B = m₂ × c₂ × rise in temperature

= m₂c₂ (t – t₂)

If no heat energy is lost in the surrounding, then by the principle of mixtures,

Heat energy lost by A = Heat energy gained by B

m₁c₁ (t₁ – t) = m₂c₂ (t – t₂)

m₁ c₁ t₁ - m₁c₁ t = m₂ c₂ t - m₂ c₂ t₂

m₁c₁ t₁ + m₂c₂ t₂ = m₁c₁ t + m₂c₂ t

m₁c₁ t₁ + m₂c₂ t₂ = t (m₁c₁ + m₂c₂)

Therefore,

$\text{t} = \dfrac{\text{m}$1 \text{c}1 \text{t}1 + \text{m}2 \text{c}2 \text{t}2}{\text{m}1\text{c}1 + \text{m}2\text{c}2}

The assumption made here is that there is no loss of heat energy.