Chapter 17: Problem 96
Styrofoam bucket of negligible mass contains 1.75 kg of water and 0.450 kg of ice. More ice, from a refrigerator at -15.0\(^\circ\)C, is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is 0.884 kg. Assuming no heat exchange with the surroundings, what mass of ice was added?
Short Answer
Step by step solution
Understand the Problem
Identify Heat Transfer Components
Define Heat Transfer Equations
Calculate Heat Required to Convert Added Ice to Water
Set Up Heat Balance Equation
Calculate Heat Balance
Solve for the Mass of Added Ice
Verify Calculation
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Thermal Equilibrium
During this equilibrium process, the heat gained by the ice from the water is equal to the heat lost by the water. This balance of heat, without any leak to the surroundings, means that the energy transferred is solely used to reach a uniform temperature across the system.
Understanding thermal equilibrium helps us predict how a system will behave over time if left undisturbed. In our exercise, it's the equilibrium state that reveals how much ice was needed initially. Employing this principle allows us to solve for unknown quantities based on the conservation of energy.
- No net heat loss to surroundings
- Temperatures of water and ice adjust until equal
- Energy exchanged internally
Specific Heat Capacity
In our exercise, the specific heat capacity determines how much heat (Q_1) is needed to bring the temperature of the added ice to 0°C. It's calculated as follows: \( Q_1 = mc_i(T_f - T_i) \), where \(m\) is the mass of the ice, \(c_i\) is the specific heat capacity, and \(T_f\) and \(T_i\) are the final and initial temperatures respectively.
This concept highlights a material's resistance to temperature change as a result of heat energy input. By using specific heat capacity, we can accurately calculate the amount of energy required to change a substance’s temperature. In our case, it helps determine part of the job done by the ice in raising its temperature.
- Ice’s specific heat: 2.09 J/g°C
- Heat required to change temperature without phase change
- Direct relation between heat added and temperature increase
Latent Heat of Fusion
When calculating the energy used to melt some of the ice, we use \( Q_2 = mL_f \), where \( m \) is the mass of the ice that melts, and \( L_f \) is the latent heat of fusion. This energy calculation is vital for the added ice to metamorphose from its solid state into liquid as part of its energy transformation.
The latent heat of fusion is essential because the transformation happens at a constant temperature (0°C for ice). Thus, the entire process involves significant energetic exchanges without temperature shifts, a crucial insight for understanding phase changes in physics.
- Heat needed for phase change: solid to liquid
- Temperature remains constant during phase change
- Energy calculated by mass and latent heat of fusion