Question

At time t = 0, a vessel contains a mixture of 18 kg of water and an unknown mass of ice in equilibrium at 0°C. The temperature of the mixture is measured over a period of an hour, with the following results: During the first 45 min, the mixture remains at 0°C; from 45 min to 60 min, the temperature increases steadily from 0°C to 2.0°C. Neglecting the heat capacity of the vessel, determine the mass of ice that was initially placed in the vessel. Assume a constant power input to the container.

Answer 1

To determine the mass of ice initially placed in the vessel, use the principle of conservation of energy. Calculate the energy gained by the ice melting and the energy lost by the ice warming up. Set up an equation to solve for the unknown mass of ice.

Step-by-step explanation:

First, we calculate the energy gained by the ice melting:
Energy gained = Mass of ice (unknown) × Latent heat of fusion of ice

Next, we calculate the energy lost by the ice warming up:
The energy lost = Mass of ice (unknown) × Specific heat capacity of ice × Change in temperature

Since the energy gained and lost are equal, we can set up the equation:
Mass of ice × Latent heat of fusion = Mass of ice × Specific heat capacity of ice × Change in temperature

Simplifying the equation, we can cancel out the masses of ice and solve for the unknown mass of ice:

Latent heat of fusion = Specific heat capacity of ice × Change in temperature

Finally, we plug in the known values:
Latent heat of fusion = 334,000 J/kg
Specific heat capacity of ice = 2,090 J/(kg·°C)
Change in temperature = 2.0°C - 0°C = 2.0°C

Answer 2

To determine the mass of ice that was initially placed in the vessel, we need to analyze the changes in temperature over time. The ice melts at 0°C, so we can calculate the mass of ice by equating the heat transfer required to melt the ice with the heat transferred during the time period. Plugging in the values, the mass of ice is approximately 0.012 kg.

Step-by-step explanation:

To determine the mass of ice that was initially placed in the vessel, we need to analyze the changes in temperature over time. From 0 to 45 minutes, the mixture remains at 0°C, indicating that the ice is melting but no temperature change occurs. From 45 to 60 minutes, the temperature increases steadily from 0°C to 2.0°C, implying that the ice has completely melted and only water remains.

Since the ice melts at 0°C, the heat transfer required to melt the ice can be calculated using the heat equation: Q = mL, where Q is the heat transfer, m is the mass of the ice, and L is the latent heat of fusion for water (334,000 J/kg). Assuming a constant power input, the heat transferred during the 15 minutes is Q = Pt, where P is the power and t is the time in seconds.

By equating the two expressions for Q, we can solve for the mass of the ice: m = Pt / L. Plugging in the values (P = 60 W, t = 15 min = 900 s, L = 334,000 J/kg), we can calculate the mass of ice that was initially placed in the vessel to be approximately 0.012 kg.