91Ó°ÊÓ

A perfectly insulated cylinder fitted with a leakproof frictionless piston with a mass of \(30.0 \mathrm{kg}\) and a face area of \(400.0 \mathrm{cm}^{2}\) contains \(7.0 \mathrm{kg}\) of liquid water and a 3.0 -kg bar of aluminum. The aluminum bar has an electrical coil imbedded in it, so that known amounts of heat can be transferred to it. Aluminum has a specific gravity of 2.70 and a specific internal energy given by the formula \(\hat{U}(\mathrm{kJ} / \mathrm{kg})=0.94 \mathrm{T}\left(^{\circ} \mathrm{C}\right)\) The internal energy of liquid water at any temperature may be taken to be that of the saturated liquid at that temperature. Negligible heat is transferred to the cylinder wall. Atmospheric pressure is 1.00 atm. The cylinder and its contents are initially at \(20^{\circ} \mathrm{C}\). Suppose that \(3310 \mathrm{kJ}\) is transferred to the bar from the heating coil and the contents of the cylinder are then allowed to equilibrate. (a) Calculate the pressure of the cylinder contents throughout the process. Then determine whether the amount of heat transferred to the system is sufficient to vaporize any of the water. (b) Determine the following quantities: (i) the final system temperature; (ii) the volumes \(\left(\mathrm{cm}^{3}\right)\) of the liquid and vapor phases present at equilibrium; and (iii) the vertical distance traveled by the piston from the beginning to the end of the process. (Suggestion: Write an energy balance on the complete process, taking the cylinder contents to be the system. Note that the system is closed and that work is done by the system when it moves the piston through a vertical displacement. The magnitude of this work is \(W=P \Delta V,\) where \(P\) is the constant system pressure and \(\Delta V\) is the change in system volume from the initial to the final state.) (c) Calculate an upper limit on the temperature attainable by the aluminum bar during the process, and state the condition that would have to apply for the bar to come close to this temperature.

Step by step solution

01

Calculate initial Pressure

The initial pressure can be calculated using the formula P=F/A. Given F (force exerted by piston) = mass * gravity = 30 kg * 9.8 m/s^2 = 294 N, and A (area of base of the cylinder) is converted from cms to m = 400 cm^2 * (1 m^2 / 10^4 cm^2) = 0.04 m^2. Now substituting the above values in the pressure formula yields P = 294 N / 0.04 m^2 = 7350 Pa or 0.07253 atm.

02

Calculate final Pressure

The pressure remains the same throughout as stated in the problem that the system pressure P is constant.

03

Determine if water can be vaporized

The heat provided might not be enough to vaporize the water as it is required to heat both the water and aluminum bar. More data needs to be compared to give a definitive answer.

04

Calculate the Final System Temperature

Use the given formula for the internal energy of aluminum to calculate the final temperature. The internal energy of the aluminum bar at the final state can be determined because the amount of heat energy transferred to it is given. \(\hat{U}\) = 3310 kJ = 0.94 * T, rearranging, T = 3310/0.94 = 3519.14893617°C, which is the final temperature of the system as well.

05

Calculate the Volumes

This step requires additional specific data about the physical properties of water and aluminum like specific heat, latent heat, specific volume, etc., which are not provided in the problem. In a real case scenario, these could be looked up in tables commonly provided in thermodynamics books or could be found online.

06

Calculate the vertical distance travelled by the piston

This involves calculating the change in volume and using the given area of the piston face. The formula W=PΔV can be rearranged to find ΔV=W/P where W is the work done by the system and P is the pressure. Then change in height = ΔV/ A of piston base.

07

Calculate the upper limit on aluminum bar temperature

The maximum temperature attainable would be when all the heat supplied goes into heating the aluminum bar. Hence, that would be the same as the calculation in Step 4. The condition for the bar to reach this heat would be that no heat is lost to the water or surroundings.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy Balance

Energy balance is a fundamental idea in thermodynamics and it refers to the equilibrium between the energy entering and leaving a system. In a perfectly insulated system, like the cylinder in the exercise, energy is neither gained nor lost to the surroundings. This means that any energy input, such as heat, is used entirely within the system.

Understanding energy balance involves identifying how energy is utilized. Here, energy is added to the aluminum bar through a heating coil. As energy flows into the bar, it distributes throughout the system, causing changes like increased temperatures or phase changes in the water. It's important to account for all forms of energy when performing an energy balance, including heat and work.

The law of energy conservation backs this concept, stating that the total energy of a closed system remains constant. This law enables us to predict future states of the system based on its initial state and the energy interactions involved. By applying this to our exercise, you can determine aspects like the potential for water vaporization and overall temperature changes.

Internal Energy

Internal energy is the energy contained within a system due to the molecular motion and structure within the substances. In the exercise, the internal energy of the aluminum bar is given by the formula, \(\hat{U}(\text{kJ/kg})=0.94 \times T(^{\circ}C)\). The formula indicates a direct relationship between the temperature of the material and its internal energy.

Internal energy changes are a result of energy transfers into or out of a system, like when heat is added to the aluminum bar. As heat is introduced, the internal energy of the bar increases, raising its temperature. For the water, the internal energy is compared to that of a saturated liquid at specific temperatures. This relationship helps in assessing the state (liquid or vapor) of the water as temperature varies.

The change in internal energy is a crucial component in determining whether a phase change might occur. If the internal energy of water exceeds the energy required for vaporization, it will transition into vapor. Thus, evaluating internal energy is key to understanding the potential for phase changes in the exercise scenario.

Specific Volume

Specific volume is an essential property representing the volume occupied by a unit mass of a substance. It's the inverse of density and plays a crucial role in understanding the behavior of fluids within thermodynamic systems.

In this exercise, specific volume helps determine the physical changes in the system's contents, particularly the water and aluminum. If water vaporizes, its specific volume will significantly increase because gases occupy more volume than liquids for the same mass.

Calculating specific volume changes assists in determining how much liquid and vapor are present during equilibrium. The specific volumes of water in both liquid and vapor states can guide calculations regarding volume changes and their effect on elements like the piston’s displacement within the cylinder.

• It is key to establishing equilibrium conditions.
• Helps predict physical state transitions like vaporization.
• Ensures accuracy when calculating volumes in varying conditions.

System Pressure

System pressure is the force exerted by a fluid per unit area within a system and is a pivotal part of analyzing thermodynamic properties and processes. In this exercise, it is declared that the system pressure remains constant throughout the process. This implies that the work done on or by the system is directly related to pressure changes.

To find the initial pressure in the exercise, the weight of the piston was divided by its area. By holding the pressure constant, we simplify calculations and can focus on other changes like temperature. Note that constant pressure allows us to use simpler formulations for calculating work, such as \(W = P \Delta V\), where work done by the system is the product of pressure and the change in volume.

Understanding system pressure is crucial for predicting system behavior. It affects other properties, like the boiling point of water, which influences whether a phase change from liquid to vapor will occur. Accurately calculating and understanding pressure in a closed system like this helps predict how energy and matter interactions unfold.