A combination of 0.250 kg of water at 20.0^∘ C, 0.400 kg of aluminum at 26.0^∘ C, and 0.100 kg o

A combination of 0.250 kg of water at 20.0^∘ C, 0.400 kg of...

A combination of 0.250 $\mathrm{kg}$ of water at $20.0^{\circ} \mathrm{C}, 0.400 \mathrm{kg}$ of aluminum at $26.0^{\circ} \mathrm{C},$ and 0.100 $\mathrm{kg}$ of copper at $100^{\circ} \mathrm{C}$ is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final temperature of the mixture.

A combination of 0.250 $\mathrm{kg}$ of water at $20.0^{\circ} \mathrm{C}, 0.400 \mathrm{kg}$ of
aluminum at $26.0^{\circ} \mathrm{C},$ and 0.100 $\mathrm{kg}$ of copper at $100^{\circ} \mathrm{C}$ is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final temperature of the mixture.

Key Concepts

Thermal Equilibrium

Thermal equilibrium is the state in which all parts of a system eventually reach the same temperature, so that no net heat flow occurs between them. In the context of mixing substances with different initial temperatures, the system evolves to this uniform temperature due to the equalization of thermal energy.

Specific Heat Capacity

Specific heat capacity is a material property that quantifies the amount of heat required to raise the temperature of a unit mass of a substance by one degree Celsius (or Kelvin). It is crucial in calculating how much heat is absorbed or released when a substance's temperature changes.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed in an isolated system. When substances at different temperatures are mixed, the heat lost by the hotter substances must equal the heat gained by the cooler ones, allowing for the calculation of the final equilibrium temperature.

Heat Transfer

Heat transfer refers to the movement of thermal energy from a hotter object or region to a cooler one. This process occurs through conduction, convection, or radiation, but in an insulated system, only the internal transfers of heat between substances occur without any external losses.

Transcript

00:01 Today we're considering if we took some water, aluminum, and copper, different masses, different temperatures, and through them in an insulated container, what the final temperature would be.

00:10 The given mass of the water is 0 .250 kilograms.

00:17 The mass of the aluminum is 400 grams or 0 .440 kilograms, and the mass of the copper is also 0 .10 .0 kilograms.

00:31 The initial temperature of the water is 20 degrees celsius.

00:38 The initial temperature of the aluminum is 26 degrees celsius, and the initial temperature of the copper is 100 degrees celsius.

00:52 And what we're going to do mostly looking at here, we know we're going to be end up using this relationship that the energy that goes into the internal energy of the substance, i .e.

01:08 Heat added, is given to us by, can be described by the mass times the specific heat, times the change in temperature, where delta t is the final t, which we're solving for, minus the initial temperature.

01:23 And we can go ahead and from conservation of energy, go ahead and write ourselves a little chalometry equation that the q cold is equal to negative of the q heat.

01:35 And by that i mean just the energy that goes to increasing the temperature of any of our substances comes directly from the energy that was lost from our substances, just a conservation of energy.

01:54 Now, i don't necessarily know what's if the, i assume the water's going to heat up, so the water's going to be in the heat cold.

02:01 I assume the copper is going to cool down because it's in q heat, but i don't actually know that if the aluminum is going to heat up, cool down, it's going to all depend on their specific heats.

02:12 But i'm actually kind of safe in this because this statement allows me, you know, if t initial is actually, if i put on the cold side and t initial is actually higher than t final, then it'll determine...

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