Question

A body ‘A’ of mass m₁ of a substance of a specific heat capacity c₁ at a temperature θ₁ is mixed with a body ‘B’ of mass m₂ of other substance of specific heat capacity c₂ at a lower temperature θ₂. Deduce the expression for the temperature of the mixture. State the assumption made, if any.

Answer 1

Let the temperature of mixture be θ.

Fall in temperature of body, A = θ₁ - θ.

∴ Heat lost by the body, A = m₁ × c₁ (θ₁ - θ)

Rise in temperature of body B = θ - θ₂

∴ Heat gained by the body, B = m₂ × c₂ × (θ - θ₂) 

Assuming that there is no heat loss,

Heat lost by hot body A = Heat gained by cold body B

i.e., m₁ c₁ (θ₁ - θ) = m₂c₂ (θ - θ₂) 

or m₁c₁θ₁ + m₂c₂θ₂ = θ (m₂c₂ + m₂c₁) 

or θ = `("m"_1"c"_1theta_1 + "m"_2"c"_2theta_2)/("m"_1"c"_1 + "m"_2"c"_2)`

Assumption: It is assumed that there is no heat loss in this process.