Fisika Study Center

This problem is taken from the second IphO Budapest, Hungary, Theoretical problems, in 1968

Problem 2
There are 300 cm³ toluene of 0°C temperature in a glass and 110 cm³ toluene of 100°C temperature in another glass. (The sum of the volumes is 410 cm°C .) Find the final volume after the two liquids are mixed. The coefficient of volume expansion of toluene β = 0.001(°C)⁻¹ . Neglect the loss of heat.

Solution
If the volume at temperature t₁ is V₁, then the volume at temperature 0°C is V₁₀ = V₁/(1+βt₁). In the same way if the volume at t₂ temperature is V₂, at 0°C we have V₂₀ = V₂/(1+βt₂). Furthermore if the density of the liquid at 0°C is d, then the masses are m₁ = V₁₀ d and m₂ = V₂₀ d, respectively. After mixing the liquids the temperature is



The volumes at this temperature are V₁₀ (1+βt₂) and V₂₀ (1+βt₂). The sum of the volumes after mixing:



The sum of the volumes is constant. In our case it is 410 cm³. The result is valid for any number of quantities of toluene, as the mixing can be done successively adding always one more glass of liquid to the mixture.

Sources/Literatures :
-Waldemar Gorzkowski, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
-Péter Vankó, Institute of Physics, Budapest University of Technical Engineering, Budapest, Hungary, Problems of the 2nd and 9th International Physics Olympiads (Budapest, Hungary, 1968 and 1976)
-The Olympic home page www.jyu.fi/ipho