A layer of ice 10cm is formed on a pond. If temperature of air is -10℃ calculate how long will it take for the thickness of ice to increase by 1mm. Density of ice=0.9g/cm3. Thermal conductivity of ice=.005cal/s M K

To determine how long it will take for the thickness of ice on a pond to increase by 1 mm when the air temperature is -10℃, we can use the principles of heat transfer and the properties of ice. Let's break this down step by step.

Understanding the Problem

We know the following:

• Initial thickness of ice: 10 cm
• Desired increase in thickness: 1 mm (which is 0.1 cm)
• Density of ice: 0.9 g/cm³
• Thermal conductivity of ice: 0.005 cal/s·m·K
• Air temperature: -10℃

Converting Units

First, we need to ensure our units are consistent. The thickness increase we want is 0.1 cm. The thermal conductivity is given in cal/s·m·K, so we should convert the thickness of ice from centimeters to meters for our calculations:

• 0.1 cm = 0.001 m

Calculating the Heat Transfer Rate

The rate of heat transfer through the ice can be calculated using Fourier's law of heat conduction, which states:

Q = k * A * (ΔT / d)

Where:

• Q = heat transfer rate (cal/s)
• k = thermal conductivity (0.005 cal/s·m·K)
• A = area of the ice surface (we can assume 1 m² for simplicity)
• ΔT = temperature difference (temperature of ice surface - air temperature)
• d = thickness of the ice (initially 10 cm or 0.1 m)

Assuming the temperature of the ice surface is 0℃ (the melting point), the temperature difference is:

ΔT = 0℃ - (-10℃) = 10 K

Substituting Values

Now we can substitute the values into the equation:

Q = 0.005 cal/s·m·K * 1 m² * (10 K / 0.1 m)

Calculating this gives:

Q = 0.005 * 1 * 100 = 0.5 cal/s

Calculating the Volume of Ice Formed

Next, we need to find out how much ice is formed when the thickness increases by 1 mm (0.1 cm). The volume of ice formed can be calculated as:

Volume = Area * Thickness

For 1 m² of surface area:

Volume = 1 m² * 0.001 m = 0.001 m³

Since the density of ice is 0.9 g/cm³, we convert this to kg/m³ for easier calculations:

Density = 0.9 g/cm³ = 900 kg/m³

The mass of the ice formed is:

Mass = Density * Volume = 900 kg/m³ * 0.001 m³ = 0.9 kg

Calculating the Heat Required to Freeze the Water

To freeze this mass of water, we need to know the latent heat of fusion for ice, which is approximately 80 cal/g. Therefore, for 0.9 kg (or 900 g):

Heat required = Mass * Latent heat = 900 g * 80 cal/g = 72000 cal

Time Calculation

Finally, to find the time it takes to freeze this amount of water, we can use the formula:

Time = Heat required / Heat transfer rate

Substituting the values we have:

Time = 72000 cal / 0.5 cal/s = 144000 seconds

To convert this into hours:

Time = 144000 seconds / 3600 seconds/hour = 40 hours

Final Thoughts

Therefore, it will take approximately 40 hours for the thickness of the ice on the pond to increase by 1 mm at an air temperature of -10℃. This calculation highlights the interplay between thermal conductivity, temperature differences, and the physical properties of ice.