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 \((1 \mathrm{~W}=1 \mathrm{watt}=1 \mathrm{~J} / \mathrm{s}) .(\) a) Assuming that melting of ice is due only 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 \(12-\mathrm{h}\) day. \((\mathbf{b})\) 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 \(12 \mathrm{~h}\), assuming no phase changes and assuming that sunlight penetrates uniformly to a depth of \(1.00 \mathrm{~cm} ?\)

Short answer

A short version of the answer to the question is as follows: a) About 21,726 grams of ice can be melted from a 1.00 square meter patch of ice over a 12-hour day, assuming that melting is due only to energy input from the Sun. b) The final temperature of a 1.00 square meter patch of ice after being in sunlight for 12 hours, assuming no phase changes and uniform penetration to a depth of 1.00 cm, is approximately 384.4°C.

Step-by-step solution

a) Calculating the mass of melted ice

First, we need to calculate the total energy input from the Sun in 12 hours. Energy input rate is 168 J/s (given). We need to find the total energy input in 12 hours. 12 hours = 12 * 60 * 60 seconds Energy input = (Energy input rate) x (Time) Energy input = 168 J/s x (12 * 60 * 60 s) \[Energy\:input = 7,257,600\:J\] We are given the enthalpy of fusion of water, denoted by ∆H, which is 6.01 kJ/mol. To convert it to joules, multiply by 1000. ∆H = 6.01 kJ/mol * 1000 J/kJ \[\Delta H = 6,010\:J/mol\] Next, we need to find the number of moles (n) of ice that can be melted using this energy. n = (Energy input) / ∆H n = 7,257,600 J / 6,010 J/mol \[n \approx 1,207\:moles\] Now, we can convert moles to grams using the molar mass of water (18 g/mol). Mass = n * (molar mass of water) Mass = 1,207 moles * 18 g/mol \[Mass \approx 21,726\:grams\] So, about 21,726 grams of ice can be melted from a 1.00 square meter patch of ice over a 12-hour day.

b) Calculating the final temperature of the ice

Now, we are given the specific heat capacity of ice (c) which is 2.032 J/g°C. We also know the initial temperature of the ice (T_initial) is -5.0°C. Given the ice is 1.00 square meter in area and the sunlight penetrates uniformly to a depth of 1.00cm, we can calculate the volume of the ice. Volume of ice = Area x Depth \[Volume\:of\:ice = 1.00\:m^2 * 0.01\:m = 0.01\:m^3\] Next, we will assume the density of ice as 917 kg/m³, which is a common value for ice. Now, we can calculate the mass of the ice patch: Mass of ice = (Volume of ice) x (Density of ice) \[Mass\:of\:ice = 0.01\:m^3 * 917\:kg/m^3 = 9.17\:kg\] Convert the mass to grams: Mass of ice = 9.17 kg × 1000 g/kg \[Mass\:of\:ice = 9,170\:g\] The energy input in 12 hours is the same as before: Energy input = 7,257,600 J Now, we can use the specific heat capacity formula to find the final temperature (T_final) of the ice. Energy input = Mass x c x (T_final - T_initial) 7,257,600 J = 9,170 g x 2.032 J/g°C x (T_final - (-5°C)) \[7,257,600 = 18,635(T_{final}+5)\] Divide both sides by 18,635: \(T_{final}+5 = \frac{7,257,600}{18,635}\) \(T_{final}+5 \approx 389.4\) Now, subtract 5 from both sides to find the final temperature: \[T_{final} \approx 384.4^{\circ}C\] The final temperature of the ice after being in sunlight for 12 hours, assuming no phase changes and uniform penetration to a depth of 1.00 cm, is approximately 384.4°C.

Key concepts

Energy Calculations

Energy calculations are an important part of understanding how much energy is transferred or required in different processes. In the context of ice melting, we need to determine the energy supplied by the sun to a certain area. This involves calculating the total energy input over a given time period.

For instance, the sunlight energy striking Earth's surface is given as 168 Watts per square meter. This means that 168 Joules of energy are provided every second for each square meter. To find the total energy input over a 12-hour period, we need to convert hours into seconds by multiplying by the number of seconds in an hour (3600 seconds/hour). Hence, the total energy from the sun can be calculated as:

• Energy input rate = 168 J/s
• Total time (in seconds) = 12 * 3600 s = 43,200 s
• Total energy input = 168 J/s * 43,200 s = 7,257,600 J

This is the amount of energy available to melt the ice over the specified area and time.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of one gram of a substance by one degree Celsius (°C). For ice, the specific heat capacity is 2.032 J/g°C. This property helps explain why different materials heat up or cool down at different rates.

When calculating the changes in temperature of the ice due to sunlight, the mass of the ice and the specific heat capacity together define how much the temperature will change. Using the formula:

• Energy input = Mass x Specific heat capacity x Change in temperature

We rearrange this to find the change in temperature and ultimately the final temperature after the energy from the sun has been absorbed. The specifics like the initial temperature, the mass of the ice, and the calculated energy from the sun complete the necessary data for these calculations.

Ice Melting

Melting ice involves understanding the enthalpy of fusion, which is the heat needed to change a substance from solid to liquid at a constant temperature. For water, the enthalpy of fusion is 6.01 kJ/mol, which is equivalent to 6,010 J/mol.

Using the total energy input calculated from the sunlight, we can determine how much ice can be melted. The number of moles of ice that can be melted is given by dividing the total energy by the enthalpy of fusion. Knowing that the molar mass of water is 18 g/mol, we calculate the mass of ice by converting moles to grams:

• Moles of ice melted = Total energy input / Enthalpy of fusion
• Mass of ice = Moles x Molar mass of water

This gives us a direct way to see the effect of solar energy in terms of physical changes in the ice mass.

Sunlight Energy

Sunlight energy is a crucial player in many natural processes, including the melting of ice. The power of sunlight is measured in watts (W), where 1 Watt equals 1 Joule per second.

By understanding how much energy the sunlight delivers per unit area, we can make predictions and calculations about environmental changes. With sunlight providing 168 W/m², it affects a 1.00 square meter patch of ice significantly over 12 hours. This constant power translates into millions of joules over the course of the day, which accounts for both the melting of ice and the change in temperature of the surface.

The energy from sunlight is consistent, but the outcomes like temperature change or the extent of melting depend on other factors, such as the material's properties, its initial conditions, and its environment. These pieces together allow us to understand and model energy interactions in nature.