To find out how much gasoline spills out of the aluminum cylinder as both the aluminum and gasoline are heated, we need to account for the thermal expansion of both the aluminum cylinder and the gasoline.
Step 1: Understand the Problem
- Volume of the cylinder \( V_{cylinder} \) = 100 liters
- Initial Temperature \( T_i \) = 5.00 °C
- Final Temperature \( T_f \) = 60.00 °C
Step 2: Coefficients of Volume Expansion
- Coefficient of volume expansion for Aluminum \( \beta_{Al} \) ≈ \( 69 \times 10^{-6} , \text{°C}^{-1} \)
- Coefficient of volume expansion for Gasoline \( \beta_{gasoline} \) ≈ \( 950 \times 10^{-6} , \text{°C}^{-1} \)
Step 3: Calculate the Change in Temperature
\[ \Delta T = T_f - T_i = 60.00 , \text{°C} - 5.00 , \text{°C} = 55 , \text{°C} \]
Step 4: Calculate the Change in Volume of the Aluminum Cylinder
Using the formula for volume expansion: \[ \Delta V_{cylinder} = V_{cylinder} \cdot \beta_{Al} \cdot \Delta T \] \[ \Delta V_{cylinder} = 100 , \text{L} \cdot (69 \times 10^{-6} , \text{°C}^{-1}) \cdot 55 , \text{°C} \] \[ \Delta V_{cylinder} \approx 100 \cdot 69 \cdot 55 \times 10^{-6} \approx 0.3795 , \text{L} \]
Step 5: Calculate the Change in Volume of the Gasoline
Similarly, for gasoline: \[ \Delta V_{gasoline} = V_{gasoline} \cdot \beta_{gasoline} \cdot \Delta T \] Assuming the entire volume of the cylinder is filled with gasoline initially: \[ \Delta V_{gasoline} = 100 , \text{L} \cdot (950 \times 10^{-6} , \text{°C}^{-1}) \cdot 55 , \text{°C} \] \[ \Delta V_{gasoline} \approx 100 \cdot 950 \cdot 55 \times 10^{-6} \approx 5.225 , \text{L} \]
Step 6: Calculate the Volume of Gasoline that Spills Out
Now, we will find the volume of gasoline after heating:
- New volume of gasoline after heating \( V_{gasoline, final} \) = Initial volume + Change in volume \[ V_{gasoline, final} = 100 , \text{L} + 5.225 , \text{L} = 105.225 , \text{L} \]
The total volume of the cylinder after transformation is: \[ V_{cylinder, final} = 100 , \text{L} + 0.3795 , \text{L} \approx 100.3795 , \text{L} \]
Step 7: Find the Amount of Spilled Gasoline
To find out how much gasoline spills out: \[ V_{spilled} = V_{gasoline, final} - V_{cylinder, final} \] \[ V_{spilled} = 105.225 , \text{L} - 100.3795 , \text{L} \approx 4.8455 , \text{L} \]
Thus, approximately 4.85 liters of gasoline will spill out due to the heating process.