Question

The enthalpy of fusion of water is \(6.01 \mathrm{~kJ} / \mathrm{mol}\). Sunlight striking Earth's surface supplies \(168 \mathrm{~W}\) per square meter \((1 \mathrm{~W}=\) 1 watt \(=1 \mathrm{~J} / \mathrm{s}\) ). (a) Assuming that melting of ice is only due to energy input from the Sun, calculate how many grams of ice could be melted from a 1.00 square meter patch of ice over a 12 hour day. (b) The specific heat capacity of ice is \(2.032 \mathrm{~J} / \mathrm{g}^{\circ} \mathrm{C}\). If the initial temperature of a 1.00 square meter patch of ice is \(-5.0^{\circ} \mathrm{C},\) what is its final temperature after being in sunlight for 12 hours, assuming no phase changes and assuming that sunlight penetrates uniformly to a depth of \(1.00 \mathrm{~cm}\) ?

Short answer

The mass of ice melted from a 1.00 square meter patch over a 12-hour day is calculated by first finding the total energy input from sunlight (168 W per square meter) in Joules and dividing it by the enthalpy of fusion of water ($6.01 \cdot 10^3 \frac{\mathrm{J}}{\mathrm{mol}}$), then multiplying the result by the molar mass of water (18 g/mol). The final temperature of a 1.00 square meter patch of ice with an initial temperature of -5°C, after being in sunlight for 12 h, can be calculated by first finding the heat gained by the ice and then using the heat gained equation: ΔT = $\frac{Heat\:gained}{m \cdot c}$. The final temperature can then be determined by adding the initial temperature to ΔT.

Step-by-step solution

Calculate the total energy input from the Sun in 12 hours

Since the energy from the sun is given in watts, we need to convert it to Joules first. We are given that sunlight intensity is 168 W per square meter, which is equal to 168 J/s per square meter. To find the energy in a 12-hour day, we need to multiply this value by the total time in seconds: 12 hours = 12*60*60 seconds Total energy input = Sunlight intensity x total time= \(168 \frac{\mathrm{J}}{\mathrm{s}} \cdot 12 \cdot 60 \cdot 60 \) s

Calculate the number of moles of ice that can be melted

To find the number of moles of ice that can be melted, we need to use the enthalpy of fusion of water, which is given as \(6.01 \frac{\mathrm{kJ}}{\mathrm{mol}}\). Converting this value into Joules, we get \(6.01 \frac{\mathrm{kJ}}{\mathrm{mol}} \cdot 10^{3}\frac{\mathrm{J}}{\mathrm{kJ}}\), and dividing the total energy input by the enthalpy of fusion of water, we obtain the number of moles melted: Number of moles = \(\frac{Total\:energy\:input}{Enthalpy\:of\:fusion}\)

Calculate the mass of ice melted

To find the mass of ice melted, we multiply the number of moles calculated in Step 2 by the molar mass of water, which is approximately 18 g/mol: Mass of ice melted = Number of moles x molar mass of water Part (b)

Calculate the heat gained by the ice

The equation for the heat gained is given by: Heat gained = m x c x ΔT where m is the mass of the ice, c is the specific heat capacity, and ΔT is the change in temperature. To find the heat gained by the ice, we use the total energy input from sunlight calculated in part (a): Heat gained = Total energy input

Calculate the final temperature of the ice

Since we know that the initial temperature of the ice is -5°C, we need to find the final temperature. Using the heat gained equation, ΔT = \(\frac{Heat\:gained}{m \cdot c}\) Final temperature = Initial temperature + ΔT

Key concepts

Enthalpy of Fusion

When we talk about the enthalpy of fusion, we're referring to the amount of energy required to change a substance from a solid to a liquid at its melting point without changing its temperature. For water, this value is quite significant—6.01 kJ/mol. It's a measure of the energy needed to overcome the forces holding the solid lattice together, allowing the molecules to move more freely in the liquid state.

Understanding the enthalpy of fusion is crucial when calculating how much ice can melt when exposed to a heat source, such as sunlight in our exercise. By knowing this value and the energy input, we can determine the mass of ice that will transition from solid to liquid over a given period.

Specific Heat Capacity

The specific heat capacity is a property that tells us how much heat energy is needed to raise the temperature of a unit mass of a substance by one degree Celsius. The specific heat capacity of ice is 2.032 J/g°C, indicating that ice requires 2.032 joules of energy to raise the temperature of one gram by one degree Celsius.

This concept is used in the second part of our exercise, where we calculate the final temperature of ice after it absorbs sunlight. Here, the specific heat capacity helps us understand the relationship between the energy absorbed and the resultant rise in temperature, assuming no phase change occurs and the energy input is uniformly distributed.

Phase Change

A phase change is the transformation of a substance from one state of matter to another, for example, from solid to liquid (melting), or liquid to gas (evaporation). Each phase change occurs at a specific temperature and often requires or releases energy. In the context of our exercise, melting is the phase change of interest, and it occurs at 0°C for ice. However, before a phase change can happen, the substance must be heated to the phase change temperature, which is where specific heat capacity plays a role.

To completely understand the problem at hand, it is vital to consider both the warming of the ice to its melting point and the melting process itself, each of which consumes energy from the sunlight.

Heat Transfer Calculations

Heat transfer calculations allow us to quantify the energy movement from one body to another as a result of temperature differences. These calculations are essential for determining the effect of sunlight on the ice in our exercise. By knowing the intensity of sunlight and the duration of exposure, we can work out the total energy delivered to the ice. This total energy can then be divided into the portion that will be used to raise the temperature of the ice (based on its specific heat capacity) and the portion that will be used to melt it (based on the enthalpy of fusion).

Ultimately, the heat transfer calculations enable us to link the energy input from the sun to the observable physical changes in the ice, which are the increase in temperature and the change from solid to liquid state.