Question

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.

Answer 1

A mass m₁ of a substance A of specific heat capacity c₁ at temperature T₁ is mixed with a mass m₂ of other substance B of specific heat capacity c₂ at a lower temperature T₂ and final temperature of the mixture becomes T.

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 energy 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)

or m₁ C₁ T₁ - m₁ C₁ T = m₂ C₂ T - m₂ C₂ T₂

∴ [m₂ C₂ T₂ + m₁ C₁ T₁] = m₂ C₂ T + m₁ C₁ T

[m₁ C₁ + m₂ C₂] T = [m₁ C₁ T₁ + m₂ C₂ T₂]

T = `(m_1 C_1 T_1 + m_2  C_2 T_2)/(m_1 C_1 + m_2 C_2)`

Here, we have assumed that there is no loss of heat energy.