Problem 70 - Chemistry and Chemical Reactivity

Question

Three 45 -g ice cubes at \(0^{\circ} \mathrm{C}\) are dropped into \(5.00 \times 10^{2} \mathrm{mL}\) of tea to make iced tea. The tea was initially at \(20.0^{\circ} \mathrm{C} ;\) when thermal equilibrium was reached, the final temperature was \(0^{\circ} \mathrm{C} .\) How much of the ice melted, and how much remained floating in the beverage? Assume the specific heat capacity of tea is the same as that of pure water.

Step-by-Step Solution

Answer

125 g melted, 10 g remained.

Step 1: Understand the Problem

To solve this problem, we need to determine how much of the ice cubes will melt in the tea until thermal equilibrium is reached at 0°C. Knowing the mass of the tea (500 g for 500 mL), its initial temperature (20°C), and the final temperature (0°C), and assuming the specific heat capacity of tea is the same as water, helps us calculate the energy needed to cool the tea to 0°C. This energy comes from the ice melting.

Step 2: Calculate the Energy to Cool the Tea

We need to calculate the energy required to cool the tea from 20°C to 0°C. This can be done using the formula for heat transfer: \[ q = m imes c imes \Delta T \]where \( m \) is the mass of the tea (500 g), \( c \) is the specific heat capacity of water (4.18 J/g°C), and \( \Delta T \) is the change in temperature (20°C). Substituting the values, we get \[ q = 500 imes 4.18 imes 20 \] \[ q = 41800 \text{ J} \]

Step 3: Calculate Ice Melt Potential Using Latent Heat

The latent heat of fusion for ice is 334 J/g. We need to know how much ice can be melted using the 41800 J of energy calculated: \[ q = m \times L \]where \( m \) is the mass of ice that melts, and \( L \) is the latent heat of fusion. Solving for \( m \): \[ 41800 = m \times 334 \] \[ m \approx 125 \text{g} \]

Step 4: Compare Mass of Melted Ice to Initial Mass

The initial mass of the three ice cubes is 45 g each, totaling 135 g. Since 125 g of ice melts, we need to determine how much ice remains: - Mass of ice melted: 125 g - Total initial mass of ice: 135 g - Mass of ice remaining: 135 g - 125 g = 10 g.

Key Concepts

Heat Transfer

Heat transfer is the process through which thermal energy moves from a hotter substance to a cooler one. This is a fundamental principle in thermodynamics that helps explain things like why your iced tea gets cold when you add ice cubes.
In our scenario, the hot tea loses heat to the ice cubes, causing them to melt. This happens because the tea initially has a temperature of 20°C, but needs to reach thermal equilibrium at 0°C when it comes into contact with the ice.
The transfer follows this rule: energy moves from a higher temperature to a lower temperature until both reach the same temperature, which is known as thermal equilibrium. Understanding this concept is crucial in solving problems where temperature changes are involved.

Specific Heat Capacity

Specific heat capacity is a measure of how much heat energy is required to change the temperature of a substance's mass by a small amount, typically 1 degree Celsius.
In this problem, we're dealing with water (and tea, which behaves similarly), which has a specific heat capacity of 4.18 J/g°C. This means that each gram of the tea needs 4.18 Joules of energy to raise its temperature by 1°C.
Knowing the specific heat capacity allows us to calculate how much energy the tea must lose to drop from its initial temperature down to 0°C. This calculation helps in determining how much ice will melt as the tea transfers its heat energy to the ice cubes.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy required to change a substance from solid to liquid without changing its temperature. For ice, this value is 334 J/g.
While ice absorbs heat from the tea, it uses this energy to change from solid to liquid at a constant temperature, which in our exercise is 0°C.
Knowing the latent heat of fusion allows us to calculate how much ice can melt with the given amount of energy absorbed from the tea. In this case, the ice absorbs 41800 J of energy from the tea, which is enough to melt 125 grams of it. This allows for translating energy values into physical changes that are observable, like ice cubes melting.

Phase Change

Phase change refers to the transition of a substance from one state of matter to another. In thermodynamics, this involves energy transfer but not a change in temperature during the actual phase change process.
For our situation, the phase change occurs as the ice transitions from solid to liquid at 0°C. This is crucial because while the ice is melting, it maintains the same temperature but uses the energy supplied to transition into a new phase.
This principle helps explain why, even as the ice absorbs heat, the temperature does not increase until the entire ice cube has melted. Understanding the concept of phase changes can make it easier to solve problems involving substances transitioning through different states of matter.