Question

On a hot summer day, a cubical swimming pool is filled to within \(1.00 \mathrm{~cm}\) of the top with water at \(21.0^{\circ} \mathrm{C}\). When the water warms to \(37.0^{\circ} \mathrm{C}\) the pool is completely full. What is the depth of the pool?

Short answer

Answer: The depth of the swimming pool is approximately \(27.98\mathrm{~cm}\).

Step-by-step solution

Identify the variables

In this problem, we have: - Initial height of the water in the pool (\(h_i\)): Unknown - Final height of the water in the pool (\(h_f\)): \(h_i + 1.00 \mathrm{~cm}\) - Initial temperature of the water (\(T_i\)): \(21.0^{\circ} \mathrm{C}\) - Final temperature of the water (\(T_f\)): \(37.0^{\circ} \mathrm{C}\) - Coefficient of volume expansion of water (\(\beta\)): \(214 \times 10^{-6} \mathrm{K}^{-1}\) - Temperature difference (\(\Delta T = T_f - T_i\)) - Volume expansion (\(\Delta V\)) Note: The coefficient of volume expansion of water (\(\beta\)) can be found in the literature.

Calculate the temperature difference

Find the temperature difference, which is needed for the formula of the volume expansion. $$ \Delta T = T_f - T_i = 37.0^{\circ}\mathrm{C} - 21.0^{\circ}\mathrm{C} = 16.0^{\circ}\mathrm{C} $$

Calculate the volume expansion

We know that the volume expansion of the water is equal to the volume of the cubical swimming pool multiplied by the temperature difference and the coefficient of volume expansion. $$ \Delta V = V\beta\Delta T = (h_i)^3\beta\Delta T $$ As mentioned earlier, the volume expansion is equal to the difference in the height of the water in the pool. $$ \Delta V = (h_f - h_i) = (h_i + 1.00 \mathrm{~cm} - h_i) = 1.00 \mathrm{~cm} $$ Now we can substitute the value of \(\Delta V\) into the equation: $$ 1.00\mathrm{~cm} = (h_i)^3\beta\Delta T $$

Solve for the initial height (depth) of the pool

To find the depth of the pool or initial height (\(h_i\)), we need to solve the equation for \(h_i\): $$ h_i = \sqrt[3]{\frac{1.00\mathrm{~cm}}{\beta\Delta T}} $$ Plug in the values of \(\beta\) and \(\Delta T\): $$ h_i = \sqrt[3]{\frac{1.00\mathrm{~cm}}{(214 \times 10^{-6} \mathrm{K}^{-1})(16.0^{\circ}\mathrm{C})}} \approx 27.98\mathrm{~cm} $$ So, the depth of the pool is approximately \(27.98\mathrm{~cm}\).

Key concepts

Coefficient of Volume Expansion

The coefficient of volume expansion, denoted as \textbf{\(\beta\)}, is a material-specific value that describes how the volume of a substance changes with temperature. Different materials will expand at different rates when subjected to the same temperature change, which is why \textbf{\(\beta\)} is so important in calculations involving thermal expansion.

For water, \textbf{\(\beta\)} is typically around \textbf{\(214 \times 10^{-6} \mathrm{K}^{-1}\)} at room temperature. This value indicates that for every degree Celsius the temperature increases, the volume of water expands by \textbf{\(214 \times 10^{-6}\)} of its original volume per Kelvin. In practical applications, like filling a swimming pool, knowing this coefficient helps us predict how much the water's volume will increase when the temperature changes.

• This coefficient is crucial for engineers and architects when designing structures that are exposed to varying temperatures to prevent damage due to expansion or contraction.
• In daily life, it helps explain phenomena such as why hot water can overflow a full container when it's heated, even if no water is added.

Temperature Difference

When solving problems related to thermal expansion, the temperature difference \textbf{\(\Delta T\)} is a key variable. It is simply the final temperature subtracted from the initial temperature. A positive \textbf{\(\Delta T\)} suggests a rise in temperature, whereas a negative value indicates a drop. Knowing the temperature difference allows us to calculate how much an object will expand or contract.

In the context of the swimming pool in the exercise, a temperature rise from \textbf{\(21.0^\circ C\)} to \textbf{\(37.0^\circ C\)} gives a \textbf{\(\Delta T\)} of \textbf{\(16.0^\circ C\)}. Temperature difference is pivotal because thermal expansion is directly proportional to it—the greater the temperature change, the more significant the expansion or contraction is likely to be.

Practical Implications

Temperature difference is a crucial consideration in applications ranging from bridge design, which must accommodate temperature-induced changes in length, to the creation of precise instruments in which tiny temperature variations can affect functionality.

Cubical Thermal Expansion

Cubical thermal expansion refers to the change in volume of a cube-shaped object when it undergoes a temperature change. Conceptually, it's an extension of the idea that objects expand when heated and contract when cooled, taking into consideration all three dimensions of a cube.

In mathematical terms, the change in volume \textbf{\(\Delta V\)} can be expressed as: \textbf{\(\Delta V = V_0\beta\Delta T\)}, where \textbf{\(V_0\)} is the original volume, \textbf{\(\beta\)} is the coefficient of volume expansion, and \textbf{\(\Delta T\)} is the temperature difference.

• It is cubical thermal expansion that causes the water level in the pool to rise when the water temperature increases.
• This concept is crucial in ensuring the proper function of devices like thermometers and in industrial contexts where material volumes might change with environmental temperatures.

Note that this expansion must be calculated comprehensively since any omission could alter the final result—a factor of grave importance in precision engineering and safety-sensitive constructions.