You drop 454 g of solid ice at −35°C into a bucket of water (1,200 g) at 45°C. Eventually, the ice all melts, and the entire system comes into thermal equilibrium. Find the final temperature of the system.
To find the final temperature of the system, we can use the principle of conservation of energy.
The heat gained by the water can be calculated using the formula 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 of the substance, and
- ΔT is the change in temperature.
Let's calculate the heat gained by the water:
Q_water = m_water * c_water * ΔT_water
= 1200 g * 4.18 J/g°C * (T_final - 45°C)
Next, we need to calculate the heat lost by the ice. As the ice melts, it absorbs energy without a change in temperature. The heat lost by the ice can be calculated using the formula Q = m * ΔH_f, where:
- Q is the heat lost or gained,
- m is the mass of the substance, and
- ΔH_f is the heat of fusion.
Since all the ice melts, the heat lost by the ice is equal to the heat gained by the water:
Q_water = Q_ice
By setting these two equations equal to each other and isolating T_final, we can solve for the final temperature:
1200 g * 4.18 J/g°C * (T_final - 45°C) = 454 g * ΔH_f
Let's assume the heat of fusion for ice is 334 J/g:
1200 g * 4.18 J/g°C * (T_final - 45°C) = 454 g * 334 J/g
Now, we can solve for T_final:
4.18 * (T_final - 45°C) = 334 * (454/1200)
T_final - 45°C = 8.26
T_final = 8.26 + 45 = 53.26°C
Therefore, the final temperature of the system is approximately 53.26°C
To find the final temperature of the system after the ice has melted and the thermal equilibrium is established, we need to apply the principle of conservation of energy.
The change in thermal energy of a substance can be calculated using the formula:
Q = mcΔT
Where:
Q is the thermal energy transferred (in joules)
m is the mass of the substance (in grams)
c is the specific heat capacity of the substance (in J/g°C)
ΔT is the change in temperature (in °C)
In this case, we have ice and water, so we need to calculate the thermal energy each substance gains or loses during the process.
For ice:
Q_ice = m_ice * c_ice * ΔT_ice
For water:
Q_water = m_water * c_water * ΔT_water
Since the ice melts at 0°C and comes into thermal equilibrium with the water, the change in temperature for both substances will be the same, denoted as ΔT.
During the process, the heat lost by the water is equal to the heat gained by the ice, so we have:
Q_water = -Q_ice
We can substitute the formulas and solve for the final temperature of the system:
m_water * c_water * ΔT = -m_ice * c_ice * ΔT
Since ΔT is common on both sides of the equation, we can cancel it out:
m_water * c_water = -m_ice * c_ice
Now, let's substitute the given values into the equation:
m_water = 1200 g (mass of water)
c_water = 4.18 J/g°C (specific heat capacity of water)
m_ice = 454 g (mass of ice)
c_ice = 2.09 J/g°C (specific heat capacity of ice)
1200 * 4.18 = -454 * 2.09
Simplifying the equation gives us:
5016 = -949.86
Since the equation cannot be balanced, it means there is an error in the given information or calculations. Please double-check the data provided, and we can calculate the final temperature accordingly.