A cube of ice is taken from the freezer at -8.5°C and placed in a 15g aluminium calorimeter filled with 310g of water at room temperature of 20°C. The final temperature of water is observed to be 17°C What is the mass of the ice cube? 18] [Given cice 21001/kg C, CA 9001/kg°C, L-3.35 x 10 J/kgl
To solve this problem, we can use the principle of conservation of energy.
First, we need to calculate the heat lost by the ice cube to cool down from -8.5°C to its melting point:
Q1 = m_ice * c_ice * (0°C - (-8.5°C))
Where:
Q1 = heat lost by the ice cube (in Joules)
m_ice = mass of the ice cube (unknown)
c_ice = specific heat capacity of ice (given as 2100 J/(kg°C))
Next, we need to calculate the heat lost by the ice cube to melt:
Q2 = m_ice * L
Where:
Q2 = heat lost by the ice cube to melt (in Joules)
L = latent heat of fusion of ice (given as 3.35 x 10^5 J/kg)
The total heat lost by the ice cube is the sum of Q1 and Q2:
Q_total = Q1 + Q2
Now, we need to calculate the heat gained by the water to increase its temperature from 20°C to 17°C:
Q_water = m_water * c_water * (17°C - 20°C)
Where:
Q_water = heat gained by the water (in Joules)
m_water = mass of the water (given as 310 g, or 0.31 kg)
c_water = specific heat capacity of water (given as 900 J/(kg°C))
According to the principle of conservation of energy, the heat lost by the ice cube is equal to the heat gained by the water:
Q_total = Q_water
Substituting the expressions for Q_total, Q1, Q2, and Q_water, we can solve for the unknown mass of the ice cube:
m_ice * c_ice * (0°C - (-8.5°C)) + m_ice * L = m_water * c_water * (17°C - 20°C)
m_ice * (c_ice * (0°C - (-8.5°C)) + L) = m_water * c_water * (17°C - 20°C)
m_ice = (m_water * c_water * (17°C - 20°C)) / (c_ice * (0°C - (-8.5°C)) + L)
Plugging in the given values:
m_ice = (0.31 kg * 900 J/(kg°C) * (17°C - 20°C)) / (2100 J/(kg°C) * (0°C - (-8.5°C)) + 3.35 x 10^5 J/kg)
Calculating this expression will give you the mass of the ice cube.
To find the mass of the ice cube, we can use the principle of conservation of energy. The heat lost by the ice cube will be equal to the heat gained by the water and the calorimeter.
1. Calculate the heat lost by the ice cube:
- Mass of the ice cube (mice) = ?
- Specific heat capacity of ice (cice) = 2100 J/kg°C (given)
- Initial temperature of the ice (Tice_initial) = -8.5°C (given)
- Final temperature of the ice (Tice_final) = 0°C (since it melts at 0°C)
The heat lost by the ice cube can be calculated using the formula:
Qlost = mice * cice * (Tice_final - Tice_initial)
2. Calculate the heat gained by the water and the calorimeter:
- Mass of water (mwater) = 310 g
- Specific heat capacity of water (cwater) = 900 J/kg°C (given)
- Initial temperature of the water (Twater_initial) = 20°C
- Final temperature of the water (Twater_final) = 17°C (observed)
The heat gained by the water and the calorimeter can be calculated using the formula:
Qgain = (mwater + mass of calorimeter) * cwater * (Twater_final - Twater_initial)
3. Equate the heat lost by the ice cube to the heat gained by the water and the calorimeter:
Qlost = Qgain
mice * cice * (Tice_final - Tice_initial) = (mwater + mass of calorimeter) * cwater * (Twater_final - Twater_initial)
4. Rearrange the equation to solve for the mass of the ice cube (mice):
mice = ((mwater + mass of calorimeter) * cwater * (Twater_final - Twater_initial)) / (cice * (Tice_final - Tice_initial))
Now, substitute the given values into the equation and calculate the mass of the ice cube.