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Horatio Meshuggeneh has his own ideas of how to do things. For instance, when given the task of determining an oven temperature, most people would use a thermometer. Being allergic to doing anything most people would do, however, Meshuggeneh instead performs the following experiment. He puts acopper bar with a mass of 5.0 kg in the oven and puts an identical bar in a well- insulated 20.0-liter vessel containing 5.00L of liquid water and the remainder saturated steam at \(760 \mathrm{mm}\) Hg absolute. He waits long enough for both bars to reachthermal equilibrium with their surroundings, then quickly takes the first bar out of the oven, removes the second bar from the vessel, drops the first bar in its place, covers the vessel tightly, waits for the contents to come to equilibrium, and notes the reading on a pressure gauge built into the vessel. The value he reads is 50.1 mm Hg. He then uses the facts that copper has a specific gravity of 8.92 and a specific internal energy given by the expression \(\hat{U}(\mathrm{kJ} / \mathrm{kg})=0.36 T\left(^{\circ} \mathrm{C}\right)\) to calculate the oven temperature. (a) The Meshuggeneh assumption is that the bar can be transferred from the oven to the vessel without any heat being lost. If he makes this assumption, what oven temperature does Meshuggeneh calculate? How many grams of water evaporate in the process? (Neglect the heat transferred to the vessel wall- -i.e., assume that the heat lost by the bar is transferred entirely to the water in the vessel. Also, remember that you are dealing with a closed system once the hot bar goes into the vessel.) (b) In fact, the bar lost 8.3 kJ of heat between the oven and the vessel. What is the true oven temperature? (c) The experiment just described was actually Meshuggeneh's second attempt. The first time he tried it, the final gauge pressure in the vessel was negative. What had he forgotten to do?

Step by step solution

01

Calculate the Oven Temperature under the Meshuggeneh Assumption

Under the assumption that there is no heat loss during the bar transfer, the oven temperature is given by the equation: \[ T = \frac{{760-50.1}}{{0.36}} \] Answer: \( T \approx 1974.17 °C \]

02

Calculate the Amount of Water Evaporating

Each gram of water that changes to vapor absorbs a latent heat of vaporization (about 2260 joules/g at normal pressure and temperature). The mass of water that evaporates is then given by the heat lost by the copper bar divided by the latent heat of vaporization: \[ m_{water} = \frac{{\Delta U_{copper}}}{{L_{vap}}} \] where \(\Delta U_{copper}\) is the change in internal energy of the copper bar and \(L_{vap}\) is the latent heat of vaporization.

03

Calculate the True Oven Temperature

If the bar lost heat during the transfer, the actual oven temperature is higher. The heat lost by the bar between the oven and the vessel (\(q = 8.3 kJ\)) must be subtracted from the previous temperature calculation: \[ T_{actual} = T + \frac{{q}}{{0.36 \times m_{copper}}} \] where \(m_{copper}\) is the mass of the copper bar.

04

Determine what went wrong in the First Attempt

If the final gauge pressure was negative in the first attempt, it means the pressure inside the vessel was less than the atmospheric pressure. This could only occur if too little heat was provided to the system, meaning that the copper bar was not hot enough when placed in the vessel. Meshuggeneh might have forgotten to heat the copper bar to the necessary temperature before placing it into the vessel.

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Key Concepts

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

Thermal Equilibrium

Thermal equilibrium is a fundamental concept in thermodynamics referring to a state in which two objects in physical contact with each other cease to exchange energy by heat. At this point, both objects reach the same temperature and no further changes take place without external influence.

In the context of the given exercise, Horatio Meshuggeneh waits for the copper bars to reach thermal equilibrium with their respective environments. This means that the first bar in the oven and the second bar in the well-insulated vessel reach the same temperature as their surroundings. When Meshuggeneh transfers the first bar to the vessel quickly, he minimizes heat exchange with the external environment, assuming no heat loss in the process.

This assumption of no heat loss is quite abstract, as in real-life applications, some heat transfer to the surroundings is inevitable. Understanding thermal equilibrium helps students realize the importance of isolating systems to accurately measure temperature changes, critical for experiments and engineering applications.

Latent Heat of Vaporization

The latent heat of vaporization is the amount of heat required to convert a unit mass of a substance from the liquid phase to the vapor phase without changing its temperature. It is a characteristic property of every substance. For water, it's approximately 2260 joules/gram. This concept explains the significant amount of energy involved in phase transitions.

In the exercise, the latent heat of vaporization comes into play when the hot copper bar is placed in the vessel with water and steam, causing some of the water to evaporate. The amount of heat provided by the copper bar directly corresponds to the mass of water that can be converted into steam, which can be calculated using the heat lost by the copper and the latent heat of vaporization.

Students should note that the latent heat is also indicative of the strong molecular forces present in the liquid phase; considerable energy is needed to overcome these forces during vaporization. Moreover, it's crucial for students to understand that latent heat is a form of energy transfer that occurs at constant temperature during a phase change.

Specific Internal Energy

Specific internal energy is an intensive property of a substance, representing the internal energy per unit mass. It includes the kinetic and potential energies of the molecules of a substance. For example, the specific internal energy for copper is given by the expression \( \hat{U}(\text{kJ} / \text{kg})=0.36 T(\text{°C}) \) in the provided exercise.

This expression allows us to calculate the amount of internal energy within the copper bar based on its temperature. Specific internal energy is a crucial concept in thermal physics and chemical engineering education because it helps students understand energy changes within a system without considering the mass of the substance.

The calculation of oven temperature and the amount of water vaporized in the exercise hinge on this concept. Understanding specific internal energy enables students to track energy flows, which is essential for solving thermal system problems and designing processes involving heat transfer.