IPhO 1968, Теория и експеримент — Задача 2. Problem 2

• 410 cm³

If the volume at temperature $t_1$ is $V_1$, then the volume at temperature 0°C is $V_{10} = V_1/(1 + \beta t_1)$. In the same way if the volume at $t_2$ temperature is $V_2$, at 0°C we have $V_{20} = V_2/(1 + \beta t_2)$. Furthermore if the density of the liquid at 0°C is $d$, then the masses are $m_1 = V_{10}d$ and $m_2 = V_{20}d$, respectively. After mixing the liquids the temperature is

The volumes at this temperature are $V_{10}(1 + \beta t)$ and $V_{20}(1 + \beta t)$.
The sum of the volumes after mixing:

The sum of the volumes is constant. In our case it is $410\ \mathrm{cm^3}$. 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.