You've got to mix 'em
There are 300 cm^3 toluene of 0°C temperature in a glass and 110 cm^3 toluene of 100°C temperature in another glass. (The sum of the volumes is 410 cm^3.) Find the final volume after the two liquids are mixed. The coefficient of volume expansion of toluene is 0.001(°C)^(-1)=β. Neglect the loss of heat.
The answer is 410.
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1 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 t= m 1 + m 2 m 1 t 1 + m 2 t 2 The volumes at this temperature are V10 (1+βt2) and V20 (1+βt2). The sum of the volumes after mixing:v10(1+Bt)+v20(1+Bt) =(v10+v20)+B(v10+v20)t=(v10+v20)+B. d m 1 + m 2 m 2 + t 2 m 1 t 1 + m 2 t 2 =v10+v20+B(m1t1/d+m2t2/d)=v10(1+Bt1)+v20(1+Bt2). The sum of the volumes is constant. In our case it is 410 cm3.