A cubic piece of uranium metal (specific heat capacity = .0.117 J /^∘ C ·g) at 200.0^∘ C is drop

step 1 So we have two species there, really, that we're considering. We're going to have some uranium,

A cubic piece of uranium metal (specific heat capacity =...

A cubic piece of uranium metal (specific heat capacity $=$ $\left.0.117 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\right)$ at $200.0^{\circ} \mathrm{C}$ is dropped into $1.00 \mathrm{L}$ deuterium oxide ("heavy water," specific heat capacity $\left.=4.211 \mathrm{J} /^{\circ} \mathrm{C} \cdot \mathrm{g}\right)$ at $25.5^{\circ} \mathrm{C}$. The final temperature of the uranium and deuterium oxide mixture is $28.5^{\circ} \mathrm{C}$. Given the densities of uranium (19.05 $\mathrm{g} / \mathrm{cm}^{3}$ ) and deuterium oxide (1.11 g/mL), what is the edge length of the cube of uranium?

Key Concepts

Specific Heat Capacity

Specific heat capacity is a property of a substance that indicates the amount of heat required to raise the temperature of a unit mass of that substance by one degree Celsius. This concept is central to understanding how different materials absorb and release heat during temperature changes, which is essential in heat transfer and calorimetry problems.

Conservation of Energy and Calorimetry

The conservation of energy principle states that energy cannot be created or destroyed, only transferred. In calorimetry, this means that the heat lost by a hot substance must be equal to the heat gained by a cooler one when they reach thermal equilibrium. This concept allows us to set up equations that relate the temperature changes of the substances involved to their masses and specific heat capacities.

Density

Density is defined as mass per unit volume and is a fundamental property used to relate volume and mass. In many problems, including those involving geometric shapes, knowing the density allows one to convert between mass and volume, which is essential when the problem involves calculating one from the other.

Cube Geometry

Cube geometry involves understanding the relationships between volume and edge length in a cube. Specifically, the volume of a cube is the cube of its edge length, and therefore the edge length can be determined by taking the cube root of the volume. This relationship is key when the problem involves finding an unknown dimension of a cube based on its mass or volume.

Transcript

00:01 So we have two species there, really, that we're considering.

00:04 We're going to have some uranium, which i'm just going to rise in the big u, and we're going to have our d2o.

00:10 And so our d2o starts at 25 .5 degrees celsius.

00:18 And our uranium starts, t initial, t1, is at 200 degrees celsius.

00:25 And they both go and become 28 .5 degrees celsius.

00:33 Now, since what's happening here is it's really about heat being transferred from the uranium to the d2o.

00:43 And so the amount of heat that is lost by the uranium that causes this change in temperature has to be the same as the amount of heat that is gained by the d2o to cause this change in temperature.

00:58 And so what we can do is we can set the, remember the q equals mc delta t, equal to, equation for each of these two reactants equal to each other.

01:10 So we could have a qmc delta t where, you know, we have different changes in temperature, different cs and different ms, but overall they have to equal each other.

01:21 And so what we're going to get is that the mass of the uranium times the specific heat of the uranium times the change in temperature of the uranium is equal to the mass of the water times the specific heat of the water, times a change in temperature of the water.

01:45 And so now let's plug in everything that we have, and we'll see that we're the only thing that we don't have is the mass of uranium so then we can get that, and then we'll solve the rest of the problem from there.

01:54 So, like i said, mass of uranium, we can't figure that out.

01:59 But we're given the specific heat, which is 0 .117.

02:05 I'm going to ignore units for now, just because everything here is in joules or grams or celsius, and so you have to make sure you're careful making sure the units are the same, but here they are.

02:16 The change in temperature is going to be 200 minus 28 .5.

02:25 That's the difference.

02:26 It's really technically it should be negative.

02:28 You should be final minus initial, but ultimately we're just sitting with an absolute value...

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