In a physics experiment, a 0.100 kg aluminum calorimeter cup holding 0.200 kg of ice is removed from the freezer, where both ice and cup have been cooled to -5.00 degrees Celsius. Next, 0.0500 kg of steam at 100 degrees Celsius is added to the ice in the cup. What will be the equilibrium temperature of the system after the ice has melted? (c_aluminum=920 J/kgC, ice melts at 0 degrees Celsius
To find the equilibrium temperature of the system after the ice has melted, we can use the principle of conservation of energy.
First, let's calculate the heat transfer required to heat the ice from -5.00 degrees Celsius to 0 degrees Celsius and then melt it. We will assume that the ice absorbs all the heat from the aluminum calorimeter cup.
The heat transfer required to raise the temperature of the ice from -5.00 degrees Celsius to 0 degrees Celsius can be calculated using the formula:
Q1 = m_ice * c_ice * ΔT_ice
where:
m_ice = mass of the ice = 0.200 kg
c_ice = specific heat capacity of ice = 2100 J/kg°C (approximately)
ΔT_ice = change in temperature = (0 - (-5.00)) °C = 5.00 °C
Q1 = 0.200 kg * 2100 J/kg°C * 5.00°C
Q1 = 2100 J/°C * kg * 5.00 kg
Q1 = 52,500 J
Next, let's calculate the heat transfer required to melt the ice. The heat required to melt ice can be calculated using the formula:
Q2 = m_ice * L_fusion
where:
L_fusion = latent heat of fusion of ice = 333,000 J/kg (approximately)
Q2 = 0.200 kg * 333,000 J/kg
Q2 = 66,600 J
Now, let's calculate the heat transfer when 0.0500 kg of steam at 100 degrees Celsius condenses and cools down to 0 degrees Celsius. The formula to calculate heat transfer is:
Q3 = m_steam * c_steam * ΔT_steam
where:
m_steam = mass of the steam = 0.0500 kg
c_steam = specific heat capacity of steam = 2020 J/kg°C (approximately)
ΔT_steam = change in temperature = (0 - 100) °C = -100 °C
Q3 = 0.0500 kg * 2020 J/kg°C * (-100)°C
Q3 = -10,100 J
According to the principle of conservation of energy, the total heat transfer in the system should be zero when the system reaches equilibrium.
Therefore, the total heat transfer can be calculated as:
Q_total = Q1 + Q2 + Q3
0 = 52,500 J + 66,600 J - 10,100 J
Simplifying this equation, we find:
0 = 109,000 J
Since the total heat transfer is not equal to zero, the system has not reached equilibrium yet. This means that some heat still needs to be transferred to reach the equilibrium temperature.
We can assume that all the heat transfer comes from the steam as it cools down to the equilibrium temperature.
Using the formula for heat transfer, we have:
Q3 = m_steam * c_steam * ΔT_eq
where:
ΔT_eq = change in temperature from 0 degrees Celsius to the equilibrium temperature
Substituting the known values, we get:
-10,100 J = 0.0500 kg * 2020 J/kg°C * ΔT_eq
Simplifying this equation, we find:
ΔT_eq = -10,100 J / (0.0500 kg * 2020 J/kg°C)
ΔT_eq ≈ -10.00°C
Since the change in temperature is negative, this means that the equilibrium temperature is 10.00 degrees Celsius below 0 degrees Celsius.
Therefore, the equilibrium temperature of the system after the ice has melted will be approximately -10.00 degrees Celsius.
To find the equilibrium temperature of the system after the ice has melted, we need to calculate the heat gained by the ice and the heat lost by the steam when they come into contact and reach thermal equilibrium.
First, let's calculate the heat gained by the ice. We'll use the specific heat capacity of ice (c_ice) and the mass of the ice (m_ice).
The equation for calculating heat is: Q = m * c * ΔT
Where:
Q is the heat gained or lost
m is the mass of the substance
c is the specific heat capacity
ΔT is the change in temperature
Given:
Specific heat capacity of aluminum (c_aluminum) = 920 J/kg°C
Specific heat capacity of ice (c_ice) = ?
Mass of ice (m_ice) = 0.200 kg
Initial temperature of ice (T_initial) = -5.00°C
Final temperature of ice after melting (T_final) = 0°C
We know that the ice will absorb heat until it reaches its melting point (0°C), so the change in temperature (ΔT) is T_final - T_initial.
Therefore: ΔT = 0°C - (-5.00°C) = 5.00°C
Now we plug in the values to calculate the heat gained:
Q_ice = m_ice * c_ice * ΔT
Since ice melts at 0°C, we know that during this process, the ice will absorb heat without any temperature change, so ΔT would be 0. So the equation becomes:
Q_ice = m_ice * c_ice * ΔT = 0
Since the heat lost by the ice is zero, the heat gained by the steam must also be zero in order to preserve energy. This means that no energy will be transferred from the steam to the ice after they reach thermal equilibrium.
Therefore, the equilibrium temperature after the ice has melted will remain at 0°C.