Question

A certain aluminum alloy has a coefficient of linear expansion of 2.4×10^5. A container made of this alloy has an internal volume of 2.000L and a depth of 17.5cm. It is filled to the top with gasoline,which has an average coefficient of volume expansion of 9.60×10^-4 °C^-1. Initially the temperature is 21.0°C,and the entire system is slowly warmed until both the cylinder and gasoline are at 89.0°C.

(a).How much gasoline flows(enter your answer in cm^3).

(b).What is the volume (in cm^3)of gasoline remaining in the cylinder at 89.0°C?.(Give your answer to at least four significant figures).

(c).If the combination with this amount of gasoline is then cooled back to 21.0°C,how far below the cylinder's rim (in cm) does the gasoline's surface recede?

Answer 1

Solution;

Given;

$T_i=21°c$

$T_f=89°c$

$\alpha_{Al}=2.4×10^{-5}°c^{-1}$

$\beta_g=9.6×10^{-4}°c^{-1}$

$V_i=2000cm^3$

$h=17.5cm$

Hence;

$\beta_{Al}=3\alpha_{Al}=7.2×10^{-5}°c^{-1}$

(a)

Change in volume is given by;

$\Delta V=V_i\beta\Delta T$

Change in volume of container;

$\Delta V_c=2000×7.2×10^{-5}×68$

$\Delta V_c=9.792cm^3$

Change in volume of gasoline;

$\Delta V_g=2000×9.6×10^{-4}×68$

$\Delta V_g=130.56cm^3$

Volume of gasoline that flows;

$V_f=\Delta V_g-\Delta V_c$

$V_f=120.768cm^3$

(b)

Amount of gasoline at 89°c;

$V_{g at 89°c}=V_i- V_f$

$V_g=2000-120.768$

$V_g=1879.232cm^3$

(c)

Container cools back to initial volume.

New change in volume of gasoline;

$\Delta V_g=1879.232×9.6×10^{-4}×68$

$\Delta V_g=122.68cm^3$

Cross-sectional area of the container;

$A=\frac{V}{h}=\frac{2000}{17.5}$

Height of gasoline below the rim is;

$h_g=122.68×\frac{17.5}{2000}$

$h_g=1.073cm$