What is the final temperature after a 21.8 g piece of ice is placed into a styrofoam cup containing 114 g of hot water at 68.2 degrees C.
Please explain because I don't even understand how to start.
heat to melt ice + heat to raise T of melted ice from zero to final T + heat lost from hot water = 0
heat to melt ice is mass x heat fusion.
heat to raise T of melted ice is mass ice x specific heat water x (Tfinal-Tinitial).
heat lost from hot water is mass x specific heat water x (Tfinal-Tinitial)
Add all of those up and set to zero, solve for Tfinal.
I think something like 45 C is the final T but that's just approximate.
To solve this problem, we can use the principles of heat transfer and the concept of specific heat capacity. The heat gained by the ice must equal the heat lost by the hot water in order to reach thermal equilibrium. We can use the equation:
(heat gained by ice) = (heat lost by hot water)
First, let's find the heat gained by the ice:
(heat gained by ice) = (mass of ice) × (specific heat capacity of ice) × (change in temperature)
Given:
- Mass of ice = 21.8 g
- Specific heat capacity of ice = 2.09 J/g°C (this is how much energy is required to raise the temperature of 1 gram of ice by 1 degree Celsius)
- Change in temperature = final temperature of the ice - initial temperature of the ice
Since the initial temperature of the ice is not given, we assume it is at its melting point, 0 degrees Celsius. Therefore, the change in temperature is the final temperature of the ice.
Now, let's find the heat lost by the hot water:
(heat lost by hot water) = (mass of hot water) × (specific heat capacity of water) × (change in temperature)
Given:
- Mass of hot water = 114 g
- Specific heat capacity of water = 4.18 J/g°C (this is how much energy is required to raise the temperature of 1 gram of water by 1 degree Celsius)
- Initial temperature of the hot water = 68.2°C
- Change in temperature = initial temperature of the hot water - final temperature of the hot water
Now, we can set up the equation:
(mass of ice) × (specific heat capacity of ice) × (final temperature of the ice - 0°C) = (mass of hot water) × (specific heat capacity of water) × (68.2°C - final temperature of the hot water)
Now, solve for the final temperature of the ice by rearranging the equation and substituting the known values:
(21.8 g) × (2.09 J/g°C) × (final temperature of the ice) = (114 g) × (4.18 J/g°C) × (68.2°C - final temperature of the hot water)
This equation can be solved algebraically to find the final temperature of the ice.
To find the final temperature after placing the ice into the cup of hot water, we can use the concept of heat transfer and the principle of conservation of energy.
First, let's break down the problem into two parts. Part one involves determining the amount of heat gained by the ice to reach its melting point, and part two involves determining the amount of heat lost by the hot water to reach the final temperature.
Part 1: Heat gained by the ice
To start, we need to calculate the heat gained by the ice to warm up from its initial temperature, which is 0 degrees Celsius, to its melting point at 0 degrees Celsius. We can use the specific heat capacity formula:
Q = m * c * ΔT,
where Q is the heat gained, m is the mass of the ice, c is the specific heat capacity of ice (2.09 J/g°C), and ΔT is the change in temperature.
In this case, the mass of the ice is given as 21.8 grams, so we have:
Q1 = 21.8 g * (2.09 J/g°C) * (0 - 0°C) = 0 J.
Since the temperature of the ice does not change during this phase, no heat is gained.
Next, we need to calculate the heat gained by the ice during its phase change from solid to liquid (melting). We can use the formula:
Q = m * ΔHf,
where Q is the heat gained, m is the mass of the ice, and ΔHf is the enthalpy of fusion for ice (334 J/g).
In this case, we have:
Q2 = 21.8 g * 334 J/g = 7299.2 J.
Part 2: Heat lost by the hot water
We also need to determine how much heat the hot water loses to reach the final temperature.
Using the same formula as before, Q = m * c * ΔT, we can calculate the heat lost by the hot water.
In this case, the mass of the hot water is given as 114 grams, and the initial temperature is 68.2 degrees Celsius. We are trying to find the final temperature, so let's denote it as T. The change in temperature is ΔT = 68.2°C - T.
Q3 = −114 g * (4.18 J/g°C) * (T - 68.2°C).
The negative sign indicates heat lost by the hot water.
Applying the principle of energy conservation, where the total heat gained by the ice is equal to the total heat lost by the hot water:
Q1 + Q2 = -Q3,
0 J + 7299.2 J = -114 g * (4.18 J/g°C) * (T - 68.2°C).
Simplifying the equation, we have:
7299.2 J = -478.92 J/g°C * (T - 68.2°C).
Now, let's solve for T.
First, divide both sides of the equation by -478.92 J/g°C:
(7299.2 J) / (-478.92 J/g°C) = T - 68.2°C.
Next, add 68.2°C to both sides of the equation:
T = [(7299.2 J) / (-478.92 J/g°C)] + 68.2°C.
Finally, calculate T using the given values and substituting them into the equation.
After obtaining the value of T, you will have the final temperature reached after placing the ice into the styrofoam cup containing hot water.