Question

An aluminum vessel with a volume capacity of \(500 . \mathrm{cm}^{3}\) is filled with water to the brim at \(20 .{ }^{\circ} \mathrm{C} .\) The vessel and contents are heated to \(50 .{ }^{\circ} \mathrm{C} .\) During the heating process, will the water spill over the top, will there be more room for water to be added, or will the water level remain the same? Calculate the volume of water that will spill over or that could be added if either is the case.

Short answer

If the water spills over the top, calculate the volume of water that will spill over. Answer: When the aluminum vessel filled with water is heated from 20°C to 50°C, the water will spill over the top. The volume of water that will spill over is 2.175 cm³.

Step-by-step solution

Identify the given values and the volume expansion coefficients

We are given the following information: - Initial volume of the aluminum vessel \(V_{vessel}\) = 500 cm³ - Initial volume of water \(V_{water}\) = 500 cm³ (since the vessel is filled to the brim) - Initial temperature \(T_{initial}\) = 20°C - Final temperature \(T_{final}\) = 50°C - Coefficient of volume expansion for aluminum, \(β_{aluminum}\) = \(69 × 10^{-6} °C^{-1}\) - Coefficient of volume expansion for water, \(β_{water}\) = \(214 × 10^{-6} °C^{-1}\)

Calculate the change in temperature

Now, we need to calculate the change in temperature, which is simply the difference between the final temperature and the initial temperature. \(∆T = T_{final} - T_{initial} = 50°C - 20°C = 30°C\)

Calculate the change in volume for the vessel and the water

Now we will calculate the change in the volume of the aluminum vessel and the water using the formula: \(∆V = V_0β∆T\) Change in volume of the aluminum vessel, \(∆V_{vessel} = V_{vessel} β_{aluminum} ∆T\) \(∆V_{vessel} = 500\,cm^3 × 69 × 10^{-6} °C^{-1} × 30°C = 1.035\,cm^3\) Change in volume of the water, \(∆V_{water} = V_{water} β_{water} ∆T\) \(∆V_{water} = 500\,cm^3 × 214 × 10^{-6} °C^{-1} × 30°C = 3.21\,cm^3\)

Compare the change in volume for the vessel and the water

Now we will compare the change in volume for the vessel, \(∆V_{vessel}\), and the change in volume for the water, \(∆V_{water}\). Since \(∆V_{water} > ∆V_{vessel}\), (3.21 cm³ > 1.035 cm³), the water will spill over the top when heated to 50°C.

Calculate the volume of water that will spill over

To find the volume of water that will spill over the top, we need to subtract the change in volume of the vessel from the change in volume of the water. \(V_{spillover} = ∆V_{water} - ∆V_{vessel} = 3.21\,cm^3 - 1.035\,cm^3 = 2.175\,cm^3\) Thus, when the aluminum vessel filled with water is heated from 20°C to 50°C, 2.175 cm³ of water will spill over the top.

Key concepts

Volume Expansion Coefficient

The volume expansion coefficient is a critical factor that measures how much a material's volume changes in response to a change in temperature. It varies for different substances and is particularly important when dealing with materials that expand upon heating such as metals and liquids like water. The formula to calculate the change in volume due to temperature change is expressed as \(\Delta V = V_0\beta\Delta T\), where \(\Delta V\) is the change in volume, \(V_0\) is the initial volume, \(\beta\) is the volume expansion coefficient of the material, and \(\Delta T\) is the temperature change. The unit of the volume expansion coefficient is typically \(^\circ C^{-1}\) or \(K^{-1}\). It's important to note that in order to accurately predict the behavior of materials under thermal stress, one must know their respective expansion coefficients. This knowledge allows for precise calculations and can prevent structural failures or inefficiencies in various engineering applications.

For example, in our exercise, the expansion coefficients for aluminum and water are given, and we use them to determine how much the vessel and the water will expand when heated from 20°C to 50°C. The larger the expansion coefficient, the greater the rate of volume expansion for a given temperature change.

Temperature Change

Temperature change is a fundamental concept in thermal physics, describing the difference in temperature when an object or substance is exposed to a heat source or a cooler environment. The change is usually denoted as \(\Delta T = T_{final} - T_{initial}\), where \(T_{final}\) is the final temperature and \(T_{initial}\) is the initial temperature. Matters such as water and different metals will exhibit thermal expansion when there is a temperature change. This can lead to either expansion or contraction, depending on whether the temperature increased or decreased.

In our textbook example, the 30°C temperature increase from an initial 20°C to a final 50°C produces a measurable change in volume for both the aluminum vessel and the water it contains, with different degrees of expansion due to their different volume expansion coefficients.

Material Properties in Physics

Material properties in physics are essential for understanding how various substances will behave under different conditions, such as changes in temperature, pressure, or force. Some fundamental properties include elasticity, conductivity, density, and, as relevant to our thermal expansion problem, the volume expansion coefficient. These characteristics determine the applications and limitations of materials in scientific, industrial, and everyday settings.

Materials like aluminum used in the exercise are selected for tasks based on these properties. Aluminum is known for its relatively high thermal conductivity and low density, making it an excellent material for heat transfer applications. However, it also has a lower volume expansion coefficient compared to water, which is why in our example, the water overflowed as both the metal container and the water expanded, but at different rates.

Calculations in Thermal Physics

Calculations in thermal physics often involve determining the effects of temperature changes on physical properties such as volume, pressure, and thermal energy. To successfully calculate thermal expansions and contractions, it is crucial to understand the relationship between heat, work, temperature, and material properties. The principles governing these calculations stem from thermodynamics and heat transfer.

In our example, the calculation process involved determining the volume expansion for both the aluminum vessel and the water separately by applying the given volume expansion coefficients and the identified temperature change. Such calculations are vital in various practical applications ranging from construction, where materials must be chosen and spaced to allow for thermal expansion, to manufacturing processes where temperature control is necessary for producing precision parts.