What mass of steam at 100°C must be added to 1.00 kg of ice at 0°C to yield liquid water at 30°C?

검증된 단계별 안내

Step 1: Identify the energy changes involved. The problem involves three energy exchanges: (1) the ice at 0°C melting into water at 0°C, (2) the resulting water heating from 0°C to 30°C, and (3) the steam at 100°C condensing into water and cooling to 30°C.

Step 2: Write the energy equations for each process. For the ice melting, the energy required is given by \( Q_\text{ice} = m_\text{ice} L_f \), where \( L_f \) is the latent heat of fusion of ice. For the water heating, the energy required is \( Q_\text{water} = m_\text{water} c_\text{water} \Delta T \), where \( c_\text{water} \) is the specific heat capacity of water and \( \Delta T \) is the temperature change. For the steam condensing and cooling, the energy released is \( Q_\text{steam} = m_\text{steam} L_v + m_\text{steam} c_\text{water} \Delta T \), where \( L_v \) is the latent heat of vaporization of steam.

Step 3: Set up the energy balance equation. The energy gained by the ice and water must equal the energy lost by the steam: \( Q_\text{ice} + Q_\text{water} = Q_\text{steam} \). Substitute the expressions for \( Q_\text{ice} \), \( Q_\text{water} \), and \( Q_\text{steam} \) into this equation.

Step 4: Solve for the mass of steam. Rearrange the energy balance equation to isolate \( m_\text{steam} \). This will involve substituting the known values for \( L_f \), \( L_v \), \( c_\text{water} \), and the temperature changes, as well as the given mass of ice (1.00 kg).

Step 5: Perform the calculations. Plug in the numerical values for the constants and solve for \( m_\text{steam} \). Ensure that the units are consistent throughout the calculation (e.g., use joules for energy, kilograms for mass, and degrees Celsius for temperature).

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주요 개념

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Phase Changes

Phase changes refer to the transitions between solid, liquid, and gas states of matter. In this problem, steam (gas) condenses to water (liquid), and ice (solid) melts to water. Understanding the energy involved in these phase changes, particularly the latent heat of fusion and vaporization, is crucial for calculating the mass of steam needed.

Heat Transfer

Heat transfer is the process of thermal energy moving from one object to another due to a temperature difference. In this scenario, heat is transferred from the steam to the ice, causing the ice to melt and the resulting water to warm up. The principle of conservation of energy dictates that the heat lost by the steam must equal the heat gained by the ice and water.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius. This concept is essential for calculating the temperature change of the water produced from the melted ice and the steam. Each substance involved (ice, water, and steam) has a different specific heat capacity, which must be considered in the energy balance of the system.