Question

Ice at \(0^{\circ} \mathrm{C}\) is placed in a Styrofoam cup containing \(361 \mathrm{~g}\) of a soft drink at \(23^{\circ} \mathrm{C}\). The specific heat of the drink is about the same as that of water. Some ice remains after the ice and soft drink reach an equilibrium temperature of \(0^{\circ} \mathrm{C}\). Determine the mass of ice that has melted. Ignore the heat capacity of the cup.

Short answer

Approximately 104 g of ice has melted.

Step-by-step solution

Understand the Problem

We have a Styrofoam cup containing 361 g of a soft drink at \( 23^{\circ} \text{C} \). Ice at \( 0^{\circ} \text{C} \) is added and the final equilibrium temperature is \( 0^{\circ} \text{C} \). We need to find the mass of ice that has melted.

Identify Known Values

The known values are: \( c = 4.18 \text{ J/g} \cdot \text{°C} \) (specific heat capacity), \( m_{drink} = 361 \text{ g} \), initial temperature of the drink \( T_i = 23^{\circ} \text{C} \), final temperature \( T_f = 0^{\circ} \text{C} \), and latent heat of fusion for ice \( L_f = 334 \text{ J/g} \).

Calculate Heat Lost by the Drink

Use the formula \( Q = mc\Delta T \) to calculate the heat lost by the drink.\( \Delta T = T_i - T_f = 23^{\circ} \text{C} - 0^{\circ} \text{C} = 23^{\circ} \text{C} \).\[ Q_{lost} = 361 \text{ g} \times 4.18 \text{ J/g°C} \times 23^{\circ} \text{C} \].

Substitute Values and Calculate

Substitute the values into the equation:\[ Q_{lost} = 361 \times 4.18 \times 23 \]Solving gives \( Q_{lost} = 34686.34 \text{ J} \).

Calculate Mass of Ice Melted

Since the heat lost by the drink melts the ice, use the formula \( Q = mL_f \) to calculate the mass of ice melted. \[ 34686.34 \text{ J} = m \times 334 \text{ J/g} \].

Solve for Mass of Ice

Rearrange the equation to solve for \( m \):\[ m = \frac{34686.34}{334} \]Solving gives \( m \approx 103.89 \text{ g} \). Thus, the mass of ice that has melted is approximately 104 g.

Key concepts

Heat Capacity

Heat capacity is a fundamental concept in thermodynamics. It describes the amount of heat required to change the temperature of a substance by a given amount. In general, the heat capacity of a material is given as its specific heat capacity, which is the heat energy required to raise the temperature of one gram of the substance by one degree Celsius. This formula is expressed as:

• Specific Heat Capacity: \[c = \frac{Q}{m \Delta T}\]where:
  • \( c \) = specific heat capacity
  • \( Q \) = heat energy (in joules)
  • \( m \) = mass (in grams)
  • \( \Delta T \) = change in temperature (in Celsius)

For the soft drink in our exercise, the specific heat capacity is the same as water's, which is approximately 4.18 J/g°C. This allows us to calculate the heat lost or gained by the substance during a temperature change.

Phase Change

Phase change refers to the transition of a substance from one state of matter to another, such as solid, liquid, or gas. Common phase changes include melting, freezing, vaporization, and condensation. During a phase change, a substance absorbs or releases energy without a change in temperature. This is because the energy is used to alter the molecular bonds rather than increase kinetic energy. In the given problem, the phase change of interest is the melting of ice. The heat absorbed by ice to transition from solid to liquid at the melting point is crucial to determining how much ice melts. The temperature does not change during this process, despite heat transfer, highlighting the unique nature of phase changes.

Latent Heat

Latent heat is the amount of energy absorbed or released by a substance during a phase change without a change in temperature. For melting, this specific form of latent heat is called the latent heat of fusion. It indicates the heat needed to convert a substance from a solid to a liquid at its melting point.The latent heat of fusion for ice is known to be \( 334 \) J/g, meaning each gram of ice requires 334 joules of energy to melt. In this exercise, the heat previously calculated as lost by the soft drink is used for melting the ice. Utilizing the latent heat formula:

• Latent Heat of Fusion:\[Q = m \times L_f\]where:
  • \( Q \) = heat energy (in joules)
  • \( m \) = mass of the substance
  • \( L_f \) = latent heat of fusion (334 J/g for ice)

This allows us to determine how much ice melted during the process.

Equilibrium Temperature

Equilibrium temperature occurs when two substances in thermal contact reach a uniform temperature. At this point, the rate of heat transfer between the substances becomes equal, and there is no net change in the energy exchanged. For the problem at hand, the equilibrium temperature is reached at \(0^{\circ}C\). This suggests that the system's heat loss and gain have balanced out. The soft drink loses heat, which is absorbed by the ice, leading to melting. The equilibrium temperature gives us insight into the heat interactions and phase transitions occurring within the system. Understanding equilibrium temperature is vital for problems involving thermal energy transfer, as it indicates a stable final state of the system after energy exchanges.