Problem 61 - University Physics with Modern Physics

Question

A vessel whose walls are thermally insulated contains 2.40 \(\mathrm{kg}\) of water and 0.450 \(\mathrm{kg}\) of ice, all at a temperature of \(0.0^{\circ} \mathrm{C}\) . The outlet of a tube leading from a boiler in which water is boiling at atmospheric pressure is inserted into the water. How many grams of steam must condense inside the vessel (also at atmospheric pressure) to raise the temperature of the system to \(28.0^{\circ} \mathrm{C}\) ? You can ignore the heat transferred to the container.

Step-by-Step Solution

Answer

214 grams of steam must condense.

Step 1: Determine the heat needed to melt the ice

To melt the 0.450 kg of ice at 0 °C, we use the latent heat of fusion for ice, which is about 334,000 J/kg.The heat needed to melt all the ice is calculated as follows:\[ Q_1 = m_\text{ice} \times L_f = 0.450 \times 334,000 = 150,300 \text{ J} \]

Step 2: Determine the heat needed to raise the water temperature

After the ice melts, we need to calculate the heat needed to raise the temperature of the resultant water (initial water plus melted ice) to 28 °C.The total mass of water is: 2.40 kg + 0.450 kg = 2.85 kg.The specific heat capacity of water is roughly 4,186 J/kg°C.\[ Q_2 = m_{\text{total}} \times c \times \Delta T = 2.85 \times 4,186 \times 28 = 333,492 \text{ J} \]

Step 3: Calculate the total heat required

The total heat required to both melt the ice and raise the temperature of the water to 28°C is the sum of the heats calculated in steps 1 and 2.\[ Q_{\text{total}} = Q_1 + Q_2 = 150,300 + 333,492 = 483,792 \text{ J} \]

Step 4: Determine the mass of steam required to supply this heat

The latent heat of vaporization of steam is approximately 2,260,000 J/kg. We will calculate the mass of steam required to supply the total heat found in Step 3.We use the equation:\[ m_{\text{steam}} = \frac{Q_{\text{total}}}{L_v} = \frac{483,792}{2,260,000} \approx 0.214 \text{ kg} \]

Step 5: Convert mass of steam from kilograms to grams

To find the mass of steam in grams, we multiply by 1,000:\[ m_{\text{steam (grams)}} = 0.214 \times 1,000 = 214 \text{ grams} \]

Key Concepts

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. This process occurs due to the temperature difference between them. When energy is transferred, the object with a higher temperature will lose heat, while the cooler object gains heat. There are three main methods of heat transfer:

• Conduction: Direct contact transfers heat between substances, often through solids.
• Convection: Heat circulates through fluids (liquids or gases).
• Radiation: Transfer of heat through electromagnetic waves without needing a medium.

In our exercise, heat transfer happens when steam condenses, transferring its thermal energy to melt ice and raise the water temperature. Understanding this process is crucial for solving thermodynamics problems and managing energy efficiently.

Latent Heat

Latent heat refers to the amount of energy required to change the phase of a substance without changing its temperature. For instance, the energy needed to convert ice to water is called the latent heat of fusion, while the energy for turning water to steam is the latent heat of vaporization. The term 'latent' signifies hidden, as this heat does not cause a temperature shift during the phase change.

In this problem, the latent heat of fusion is used for melting ice, defined as approximately 334,000 J/kg. Similarly, when steam condenses into water, it releases the latent heat of vaporization, which is about 2,260,000 J/kg. Recognizing and calculating latent heat is key to solving problems that involve phase changes, as observed in steam's condensation process to warm the mixture.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of 1 kg of a substance by 1°C. It is a unique property that varies among different materials. Water, for instance, has a specific heat capacity of roughly 4,186 J/kg°C, making it very effective in storing and transferring heat energy.

In the exercise, after the ice melts, we calculate the heat necessary to raise the entire water mass from 0°C to 28°C using specific heat capacity. Understanding specific heat ensures that we can predict how much energy is needed to change the temperature of a substance accurately. It plays a significant role in our daily life applications and industrial processes.

Phase Change

A phase change is a transition between different states of matter: solid, liquid, and gas. Common phase changes include melting, freezing, vaporization, condensation, sublimation, and deposition. These transformations occur at specific temperatures and pressures for each substance.

In the given exercise, the phase change is essential as the system involves melting ice and condensing steam. During these transformations, the temperature remains constant while the substance absorbs or releases latent heat. Understanding phase changes is critical to comprehending and controlling thermodynamic systems, whether in nature or engineering applications.

Steam Condensation

Steam condensation is the process where water vapor (steam) changes into liquid water. This phase change releases significant amounts of energy, known as the latent heat of vaporization. In this exercise, steam condensation happens inside the vessel, releasing heat to melt ice and increase the water temperature.

At atmospheric pressure, water condenses at 100°C. Thus, the release of this latent heat is efficient in warming other substances. Grasping steam condensation is vital in various applications, such as heat exchangers and distillation processes. These principles help optimize energy use in many technological and industrial tasks.