Question

A \(20.0-\mathrm{g}\) sample of ice at \(-10.0^{\circ} \mathrm{C}\) is mixed with \(100.0 \mathrm{~g}\) water at \(80.0^{\circ} \mathrm{C}\). Calculate the final temperature of the mixture assuming no heat loss to the surroundings. The heat capacities of \(\mathrm{H}_{2} \mathrm{O}(s)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\) are \(2.03\) and \(4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\), respectively, and the enthalpy of fusion for ice is \(6.02 \mathrm{~kJ} / \mathrm{mol}\).

Short answer

The final temperature of the mixture can be found by following these steps: 1. Calculate the energy required to heat the ice to 0°C: \(q_{ice} = m_{i} C_{i} (T_{i_{final}} - T_{i_{initial}})\). 2. Calculate the energy required to melt the ice: \(q_{fusion} = n \Delta H_{fusion} \cdot 1000\). 3. Calculate the energy exchanged between the melted ice and the water until they reach the final temperature: \((m_{i} + n \Delta H_{fusion}) \cdot C_{w} \cdot (T_{final} - 0^{\circ}C) + m_{w} C_{w} (T_{final} - T_{w_{initial}})=0\). By solving the last equation, we can determine the final temperature of the system, \(T_{final}\).

Step-by-step solution

Calculate the energy required to raise the ice temperature to 0°C

To raise the ice's temperature from -10.0°C to 0°C, we need to find the energy by using the formula \(q = mC\Delta T\), where \(q\) is the heat energy, \(m\) is the mass, \(C\) is the heat capacity, and \(\Delta T\) is the temperature difference. Given: - Ice mass: \(m_{i} = 20.0 g\) - Ice initial temperature: \(T_{i_{initial}} = -10.0 ^{\circ}C\) - Ice final temperature: \(T_{i_{final}} = 0 ^{\circ}C\) - Heat capacity of ice: \(C_{i} = 2.03 \frac{J}{g \cdot ^{\circ}C}\) Using the formula, we get \(q_{ice} = m_{i} C_{i} (T_{i_{final}} - T_{i_{initial}})\).

Calculate the energy required to melt the ice

To find the energy required to melt the ice, we have to use the formula \(q = n \Delta H\), where \(q\) is the heat energy, \(n\) is the number of moles, and \(\Delta H\) is the enthalpy change (in this case, the enthalpy of fusion). Given: - Enthalpy of fusion for ice: \(\Delta H_{fusion} = 6.02 \frac{kJ}{mol}\) - Molar mass of water: \(M_{water} = 18.02 \frac{g}{mol}\) The number of moles of ice is given by \( n = \frac{m_{i}}{M_{water}}\), and the energy required is \(q_{fusion} = n \Delta H_{fusion} \cdot 1000\) (we multiply by 1000 to convert kJ to J).

Calculate the energy exchanged until the system reaches the final temperature

Let the final temperature of the system be \(T_{final}\). Given: - Water mass: \(m_{w} = 100.0 g\) - Water initial temperature: \(T_{w_{initial}} = 80.0 ^{\circ}C\) - Heat capacity of liquid water: \(C_{w} = 4.18 \frac{J}{g \cdot ^{\circ}C}\) The energy exchanged between the melted ice and the water, assuming no heat loss, can be calculated as: \(q_{ice\_liquid} + q_{water} = 0\) Using the formula, we get the following relation as: \((m_{i} + n \Delta H_{fusion}) \cdot C_{w} \cdot (T_{final} - 0^{\circ}C) + m_{w} C_{w} (T_{final} - T_{w_{initial}})=0\) Now, we can solve the equation to find the final temperature of the system, \(T_{final}\). Note: Ensure all values are in their proper units when performing calculations.

Key concepts

Understanding Specific Heat Capacity

Specific heat capacity is a property that determines how much energy is needed to raise the temperature of a substance by a unit degree. This concept is crucial in thermochemistry as it helps us understand how different substances react to heat.

Using the formula \(q = mC\Delta T\), where \(q\) is the heat energy in joules, \(m\) is the mass of the substance in grams, \(C\) is the specific heat capacity in joules per gram per degree Celsius (\(\frac{J}{g \cdot ^{\circ}C}\)), and \(\Delta T\) is the change in temperature in degrees Celsius, we can calculate the amount of energy needed for a temperature change.

For instance, ice, with a specific heat capacity of \(2.03 \frac{J}{g \cdot ^{\circ}C}\), will require less energy to raise its temperature by one degree as compared to water, which has a higher specific heat capacity of \(4.18 \frac{J}{g \cdot ^{\circ}C}\). This information is vital when mixing substances at different temperatures, as it tells us how the temperature of each component will change.

Enthalpy of Fusion Explained

Enthalpy of fusion is the amount of energy needed to change a substance from solid to liquid at its melting point. It is an important aspect of phase changes in thermochemistry.

To calculate the total energy required to melt a solid, we use the formula \(q = n\Delta H\), where \(q\) is the heat absorbed or released in joules, \(n\) is the number of moles, and \(\Delta H\) is the enthalpy change—in this case, the enthalpy of fusion.

Typically, the enthalpy of fusion is expressed in kilojoules per mole (\(\frac{kJ}{mol}\)), so it's crucial to convert it into joules when calculating the total heat energy, especially when other values are also in joules or joules per gram. For example, the enthalpy of fusion for ice is \(6.02 \frac{kJ}{mol}\), which tells us how much energy is required to melt one mole of ice without changing its temperature.

Heat Transfer Calculations in Thermochemistry

Heat transfer calculations are employed to determine the flow of heat energy during chemical or physical processes. Understanding this concept is key to solving problems involving temperature changes and phase transitions in thermochemistry.

The fundamental principle of heat transfer in these scenarios is the conservation of energy, which states that energy cannot be created or destroyed, only transferred. Using the formula \(q_{ice\_liquid} + q_{water} = 0\), we set up a balance equation where the heat gained by one part of the system is equal to the heat lost by the other.

To calculate the final temperature of a mixture, we need to consider the specific heat capacities, the mass of the substances involved, and the enthalpy changes for any phase transitions that occur. By combining these with the conservation of energy, we arrive at a relation that allows us to solve for the unknown temperature.