Problem 101 - University Physics with Modern Physics

Question

A copper calorimeter can with mass 0.446 \(\mathrm{kg}\) contains 0.0950 \(\mathrm{kg}\) of ice. The system is initially at \(0.0^{\circ} \mathrm{C} .\) (a) If 0.0350 \(\mathrm{kg}\) of steam at \(100.0^{\circ} \mathrm{C}\) and 1.00 atm pressure is added to the can, what is the final temperature of the calorimeter can and its contents? (b) At the final temperature, how many kilograms are there of ice, how many of liquid water, and how many of steam?

Step-by-Step Solution

Answer

The final temperature is found using energy balance, then determining the state of ice, water, and steam.

Step 1: Understand the Energy Transfer

In a calorimetry problem, we consider the energy exchanged between the substances until thermal equilibrium is reached. Assume no heat is lost to the surroundings, so the heat lost by the steam will equal the heat gained by the ice, water, and calorimeter.

Step 2: Calculate Heat Released by Condensing Steam

The steam must first condense to water at 100°C, releasing energy:\[Q_{ ext{condense}} = m_{ ext{steam}} \times L_v\]where \(m_{ ext{steam}} = 0.0350 \text{ kg}\) and \(L_v = 2260 \text{ kJ/kg}\). Substitute to find \(Q_{ ext{condense}}\).

Step 3: Calculate Heat Released by Cooling Water from 100°C to 0°C

Next, the condensed water cools from 100°C to 0°C:\[Q_{ ext{cool}} = m_{ ext{steam}} \times c_w \times \Delta T\]\(c_w = 4.18 \text{ kJ/kg°C}\) and \(\Delta T = 100°C\). Solve for \(Q_{ ext{cool}}\).

Step 4: Calculate Heat Required to Melt the Ice

The energy required to melt the ice is given by:\[Q_{ ext{melt}} = m_{ ext{ice}} \times L_f\]where \(m_{ ext{ice}} = 0.0950 \text{ kg}\) and \(L_f = 334 \text{ kJ/kg}\). Substitute to find \(Q_{ ext{melt}}\).

Step 5: Calculate Heat Required to Raise Temperature of Resulting Water and Calorimeter from 0°C to Final Temperature

Calculate for water and calorimeter:\[Q_{ ext{warm}} = (m_{ ext{water}} \times c_w + m_{ ext{calorimeter}} \times c_c) \times (T_f - 0)\]where \(m_{ ext{water}} = m_{ ext{ice}} + m_{ ext{melted ice}}\), \(m_{ ext{calorimeter}} = 0.446 \text{ kg}\), and \(c_c = 0.385 \text{ kJ/kg°C}\).

Step 6: Formulate the Energy Balance Equation

Using energy balance:\[Q_{ ext{condense}} + Q_{ ext{cool}} = Q_{ ext{melt}} + Q_{ ext{warm}}\]Solve for the final temperature \(T_f\).

Step 7: Analyze Phases at Final Temperature

Assess the energy changes and compare to phase transition energies to determine if all ice has melted. If \(T_f > 0°C\), all ice has melted and the energy distribution changes only with temperature.

Key Concepts

Energy Transfer

Energy transfer plays a key role in understanding calorimetry problems. Calorimetry involves examining how energy moves between different substances. The key is to remember that energy cannot be created or destroyed, only transferred. In the problem, energy is shifting from the steam to the ice, water, and calorimeter. We assume that no energy escapes to the surroundings.

This means that all the energy lost by the steam is gained by the other elements. The steam condenses, giving off energy, which is absorbed by the ice, water, and the copper can.

By applying these principles, one can analyze how much energy is transferred through various processes such as condensation, cooling, and warming up until thermal equilibrium is reached.

Thermal Equilibrium

Thermal equilibrium is achieved when all parts of a system have the same temperature, and there is no net flow of thermal energy between them. In our calorimetry problem, this means the steam, water, ice, and calorimeter ultimately arrive at a uniform temperature.

To solve the exercise, it's important to calculate the energy exchanges that happen during the energy transfer processes mentioned. These calculations help determine the temperature at which thermal equilibrium is reached.

The process begins with steam at a high temperature losing energy, while the ice and calorimeter gain energy. The final step confirms equilibrium when the total energy lost and gained balances out, showing that the entire system has arrived at the same temperature.

Phase Transition

Phase transitions occur when substances change states, such as from solid to liquid or liquid to gas. These transitions require specific amounts of energy to occur. In this problem, the major phase transition is the melting of ice and condensation of steam.

When steam condenses into water, it releases a significant amount of latent heat, which is then used to melt the ice and warm the rest of the system. Calculating the energy involved in these phase changes is crucial for understanding how the temperature of the entire system changes.

Each phase change consumes or releases heat, affecting how energy is shared amongst the substances. By understanding the energies linked to these phase transitions, you can predict changes in states, like whether all the ice has become water by the end of the process.