a cubic piece of uranium metal (specific eat capacity= .117 J/g degree celcius) at 200.0 degrees celcius is droped into 1.00 L of deuterium oxide ("heavy water", specific heat capacity=4.211 J/g degree celcius) at 25.5 degrees celcius. the final temperature of the uranium and deuterium oxide mixture is 28.5 degrees celcius. Given the densities of uranium (19.05 g/cm cubed) and deuterium oxide (1.11 g/mL), what is the edge length of the cube of uranium?
[mass U x specific heat U x (Tfinal-Tinitial)] + [mass water x specific heat water x (Tfinal-Tintial)] = 0
Use density D2O to determine mass heavy water.
Solve for mass U, then use density to calculate volume. Take the cube root of the volume to determine the edge length.Note the correct spelling of celsius.
THX!!!!
To find the edge length of the cube of uranium, we will use the principle of conservation of heat.
1. Calculate the heat absorbed by the deuterium oxide (D2O):
Q1 = mass1 × specific heat capacity1 × change in temperature1
Since the density of D2O is given in grams/mL, we need to convert 1.00 L to grams:
Mass of D2O = volume × density = 1.00 L × 1.11 g/mL = 111 g
Q1 = 111 g × 4.211 J/g°C × (28.5°C - 25.5°C)
Q1 = 111 g × 4.211 J/g°C × 3.0°C
Q1 = 1405.2 J
2. Calculate the heat released by the uranium:
Q2 = mass2 × specific heat capacity2 × change in temperature2
The mass of the uranium can be determined using its density:
Density = mass2 / volume
Mass of uranium = Density × volume = 19.05 g/cm³ × (edge length)^3
The change in temperature for the uranium is (28.5°C - 200°C).
Q2 = (19.05 g/cm³ × (edge length)^3) × 0.117 J/g°C × (28.5°C - 200°C)
Q2 = (19.05 g/cm³ × (edge length)^3) × 0.117 J/g°C × -171.5°C
Q2 = -3658.745 J × (edge length)^3
3. Apply the conservation of heat principle:
According to the conservation of heat, the heat absorbed by the D2O equals the heat released by the uranium:
Q1 = Q2
1405.2 J = -3658.745 J × (edge length)^3
Divide both sides by -3658.745 J:
-1405.2 J / -3658.745 J = (edge length)^3
0.384 = (edge length)^3
Take the cube root of both sides:
edge length = ∛0.384
Therefore, the edge length of the cube of uranium is approximately 0.72 cm.
To find the edge length of the cube of uranium, we need to calculate the heat gained by the uranium and the heat lost by the deuterium oxide when they reach the final temperature.
We can use the formula for heat transfer:
Q = mcΔT
Where:
Q is the heat transferred
m is the mass of the substance
c is the specific heat capacity
ΔT is the change in temperature
First, let's find the mass of the uranium cube. We know the density of uranium is 19.05 g/cm³. Since it is a cube, its volume will be (edge length)³.
Density = mass / volume
Mass = Density * Volume
Since the volume is (edge length)³, we can rewrite the equation as:
Mass = Density * (edge length)³
Next, let's find the heat gained by the uranium. The initial temperature of the uranium is 200.0°C, and the final temperature is 28.5°C. We know the specific heat capacity of uranium is 0.117 J/g°C.
Heat gained by uranium = mass * specific heat capacity * temperature change
Now, let's find the mass of the deuterium oxide. We know the density of deuterium oxide is 1.11 g/mL, and we have 1.00 L, which is equal to 1000 mL.
Mass = density * volume
Finally, let's find the heat lost by the deuterium oxide. The initial temperature of the deuterium oxide is 25.5°C, and the final temperature is 28.5°C. We know the specific heat capacity of deuterium oxide is 4.211 J/g°C.
Heat lost by deuterium oxide = mass * specific heat capacity * temperature change
To find the edge length of the cube, we need to equate the heat gained by uranium to the heat lost by deuterium oxide, since they eventually reach the same temperature:
Heat gained by uranium = Heat lost by deuterium oxide
Plug in the values and solve the equation to find the edge length of the cube of uranium.