The heat capacity of liquid water is 75.6J/mol K, while the enthalpy of fusion of ice is 6 kj/mol.what is thesmallest number of ice cubes at 0^0c,each containing 9g of water,needed to cool 500g of liquid water from 20^0c to 0^0c?

To find out the smallest number of ice cubes required to cool 500 grams of liquid water from 20°C to 0°C, we need to consider the heat transfer involved during this process. This involves both the cooling of the liquid water and the melting of the ice cubes. Let’s break it down step by step.

Understanding the Problem

We have two main components here: the liquid water that we want to cool and the ice cubes that will absorb heat as they melt. The liquid water will lose heat, while the ice cubes will gain heat to melt and eventually reach thermal equilibrium.

Key Values

• Heat capacity of liquid water: 75.6 J/mol·K
• Enthalpy of fusion of ice: 6 kJ/mol
• Mass of water to be cooled: 500 g
• Initial temperature of water: 20°C
• Final temperature of water: 0°C
• Mass of one ice cube: 9 g

Calculating Heat Loss by Water

First, let’s calculate the amount of heat lost by the 500 g of liquid water when it cools from 20°C to 0°C. The formula for heat transfer is:

q = m × C × ΔT

Where:

• q = heat transferred (in Joules)
• m = mass of the water (in grams)
• C = specific heat capacity (in J/g·K)
• ΔT = change in temperature (in °C)

For water, the specific heat capacity is approximately 4.18 J/g·K. Thus, we calculate:

q = 500 g × 4.18 J/g·K × (20°C - 0°C)

q = 500 g × 4.18 J/g·K × 20 K

q = 41800 J (or 41.8 kJ)

Calculating Heat Absorbed by Ice Cubes

Now let's calculate how much heat each ice cube can absorb as it melts. First, we need to know how many moles of water are in one ice cube:

Molar mass of water = 18 g/mol

Number of moles in one ice cube = 9 g / 18 g/mol = 0.5 mol

Next, we can determine the heat absorbed by one ice cube when it melts:

Heat absorbed = moles × enthalpy of fusion

Heat absorbed by one ice cube = 0.5 mol × 6 kJ/mol = 3 kJ

Finding the Number of Ice Cubes

Now we know the total heat lost by the water (41.8 kJ) and the heat absorbed per ice cube (3 kJ). To find the number of ice cubes needed, we divide the total heat loss by the heat absorbed per ice cube:

Number of ice cubes = total heat loss / heat absorbed per ice cube

Number of ice cubes = 41.8 kJ / 3 kJ = 13.93

Since you can't have a fraction of an ice cube, you will need to round up to the nearest whole number, which means you will need at least 14 ice cubes to cool the water down from 20°C to 0°C.

Final Thoughts

To summarize, to cool 500 grams of water from 20°C to 0°C using 9-gram ice cubes at 0°C, you need a minimum of 14 ice cubes. This calculation illustrates the principles of heat transfer, showing how energy is conserved in a closed system involving phase changes.