A 0.0500 kg ice cube at -30.0◦C is placed in 0.400 kg of 35.0◦C water in a very well insulated container.
What is the final temperature? (Note: how can you check if all the ice melts at equilibrium temperature?)
To find the final temperature, we need to determine the amount of energy gained or lost by each substance.
First, let's calculate the energy gained/lost by the ice:
The energy gained/lost by a substance can be calculated using the equation:
Q = mcΔT
where Q is the energy gained/lost, m is the mass of the substance, c is the specific heat capacity of the substance, and ΔT is the change in temperature.
In this case, the ice is initially at -30.0°C and will eventually reach its melting point, 0.0°C. So, ΔT = (0.0°C - (-30.0°C)) = 30.0°C.
The specific heat capacity of ice is approximately 2.09 J/g°C, and the mass is given as 0.0500 kg (which can be converted to grams by multiplying by 1000).
Q_ice = (0.0500 kg) * (1000 g/kg) * (2.09 J/g°C) * (30.0°C)
Now, let's calculate the energy gained/lost by the water:
The water is initially at 35.0°C, and we want to find the final temperature when the ice has completely melted (equilibrium temperature).
Since the total mass of the water is 0.400 kg, we can find the energy gained/lost by using the same equation as before, with the specific heat capacity of water (4.18 J/g°C).
Q_water = (0.400 kg) * (1000 g/kg) * (4.18 J/g°C) * (T_final - 35.0°C)
To ensure that all the ice has melted at equilibrium temperature, we need to verify that the energy lost by the ice (Q_ice) is equal to the energy gained by the water (Q_water).
Therefore, we can set up the equation:
Q_ice = Q_water
Now, we can equate the two expressions:
(0.0500 kg) * (1000 g/kg) * (2.09 J/g°C) * (30.0°C) = (0.400 kg) * (1000 g/kg) * (4.18 J/g°C) * (T_final - 35.0°C)
Solving this equation will give us the final temperature (T_final). Let's plug in the numbers and calculate T_final.
Note: If T_final turns out to be below 0.0°C, this means that the final temperature is 0.0°C, and the remaining ice will melt at that temperature. If T_final is above 0.0°C, this means that all the ice has melted at a temperature above 0.0°C, and no ice will remain.