Shaskolskaya & Eltsin Problem 198
Physics
Problem
Let t be the final total temperature; let m₁, V₁, t₁ be the mass, volume and temperature respectively of the colder water and let V₁' be the volume of this water at temperature t; also let m₂, V₂, t₂ be the mass, volume and temperature respectively of the warmer water and let V₂' be the volume of this water at temperature t. The equation of thermal balance gives cm₂(t₂-t) = cm₁(t-t₁) (1) (c, the thermal capacity, can be cancelled). On the other hand, the change of the volumes with the change of temperature will be expressed thus: V₁ = m₁/ρ₁ = m₁(1 + αt₁)/ρ₀ where ρ₁ is the density of the water at temperature t₁, ρ₀ is its density at a temperature of 0°C, and α is the coefficient of cubic expansion (taking it as constant). Similarly, V₁' = m₁(1 + αt)/ρ₀; V₂ = m₂(1 + αt₂)/ρ₀; V₂' = m₂(1 + αt)/ρ₀ Hence we shall find the alterations in volume: V₂ - V₂' = m₂α(t₂ - t)/ρ₀ (2) V₁' - V₁ = m₁α(t - t₁)/ρ₀ (2') Substituting (1) in (2) and (2'), we find: V₂ - V₂' = V₁' - V₁ or V₂ + V₁ = V₂' + V₁' i.e. the total volume of liquid will not be altered.Intuitive explanation
When two water samples at different temperatures mix, heat flows from hot to cold until they reach equilibrium—this thermal balance is straightforward. The key insight is that water's density changes with temperature through the expansion coefficient , so each sample's volume changes as it approaches the final temperature . The elegant result emerges when you substitute the thermal balance equation into the volume-change expressions: the warmer water contracts by exactly the amount the cooler water expands, making the total volume conserved despite the temperature redistribution.
Intuitive explanation
When two water samples at different temperatures mix, heat flows from hot to cold until they reach equilibrium—this thermal balance is straightforward. The key insight is that water's density changes with temperature through the expansion coefficient , so each sample's volume changes as it approaches the final temperature . The elegant result emerges when you substitute the thermal balance equation into the volume-change expressions: the warmer water contracts by exactly the amount the cooler water expands, making the total volume conserved despite the temperature redistribution.