A mountain glacier ends in a small lake that holds 185,000 m^3 of water, initially at 6.40^∘ C

A mountain glacier ends in a small lake that holds 185,000...

Key Concepts

Phase Change Processes

Phase change processes involve transitions between different states of matter—solid, liquid, and gas—often requiring or releasing energy without temperature change. In thermal problems, understanding these processes is critical because the energy used in a phase change (like melting) is expressed in terms of latent heat, and it influences the final equilibrium state of the system.

Latent Heat of Fusion

The latent heat of fusion is the amount of heat energy required to change a substance from solid to liquid at its melting point, without changing its temperature. This concept is key in problems involving phase transitions, such as melting ice, because it allows for the calculation of the energy required for the phase change separate from the energy required to change the temperature of the substance.

Specific Heat Capacity

Specific heat capacity is a property of a substance that describes the amount of heat required to change the temperature of a unit mass by one degree Celsius. In thermal calculations, knowing the specific heat capacity allows us to quantify how much energy must be absorbed or released to raise or lower the temperature of a substance, making it essential for understanding temperature changes during energy exchange.

Conservation of Energy in Calorimetry

The principle of energy conservation in calorimetry is fundamental when analyzing how heat is exchanged between substances. This concept states that the total energy of an isolated system remains constant, and in the context of thermal exchanges, it is used to equate the heat lost by one part of the system to the heat gained by another, facilitating the calculation of final temperatures and phase changes.

Thermal Equilibrium

Thermal equilibrium is the state in which two or more bodies reach a common temperature after exchanging heat energy. In problems involving mixing objects at different temperatures, the assumption of thermal equilibrium allows us to set up an energy balance where the heat lost by warmer substances equals the heat gained by cooler ones, ignoring any external energy exchanges.

Transcript

00:01 For this problem, we're told that there's a lake that holds some water that's initially at some temperature and that some ice from an iceberg ends up floating in the lake.

00:17 And we want to assume that no energy exchanges except between the lake water and the ice and to determine what's the final temperature and how much if any ice remains.

00:29 So i thought that the best way to approach this is to sort of go in steps.

00:36 So first, i want to write down all of the information and make sure it is organized.

00:42 So i'm going to specifically do this using two different colors.

00:46 So we'll make the lake red.

00:50 And so we're told the volume of the lake.

00:53 The volume of the lake is 185 ,000 cubic meters.

01:00 We're also told the initial temperature of the lake, it is 6 .4 degrees celsius.

01:11 Thinking about this problem, we're probably going to need to use the equation, q is equal to mc delta t, or q is equal to ml.

01:27 And so i immediately want to figure out what the mass of water in the lake is.

01:33 Because we'll probably need to use that.

01:37 So mass is generally given as density times volume.

01:44 We know the volume of the lake and the lake is filled with water.

01:48 So we can just multiply by the density of water, which is 1 ,000 kilograms per cubic meter.

01:58 So multiplying by 1 ,000, we get that the mass of the water is 1 .85.

02:03 Times 10 to the 8 kilograms.

02:10 For the iceberg, we'll make it blue.

02:17 The iceberg, we're told, has a mass m sub i of 17 .3 times 10 to the 6 kilograms, and an initial temperature t sub i of negative 10 degrees celsius.

02:37 Some additional pieces of information that will probably be useful, again, assuming that we'll have to use one of those two equations that i gave up there, is the specific heat of water, which is 4 .184 times 10 to the third joules per kilogram kelvin, as well as the specific heat for ice, which is 4 .184 times 10 to the third, joules per kilogram kelvin, 2 .093 times 10 to the third joules per kilogram kelvin.

03:18 And lastly, the latent heat of fusion for water, which is 334 kilojoules per kilogram.

03:31 So my first thought was there are a few things that can't happen.

03:38 We firstly need to be able to cool the water to zero degrees celsius.

03:48 So how much energy is needed to cool the water to zero degrees celsius? so q to cool the water is going to be mc delta t.

04:09 And in this case, we're considering the lake.

04:13 So this will be the mass of the lake times the specific heat of water.

04:18 Times the change in temperature.

04:20 It starts at 6 .4 degrees celsius and ends at 0 degrees celsius.

04:26 So delta t is going to be 6 .4.

04:33 And so when you plug all of that end, you get that we need 4 .954 times 10 to the 12 joules of energy to cool the water to zero degrees celsius.

04:49 Now it's a matter of figuring out if that actually happens.

04:55 So the first thing once the ice hit the water is that the ice also needs to reach zero degrees celsius.

05:03 So next i want to figure out how much energy is needed to warm the ice to zero degrees celsius.

05:13 And so we're going to use that same equation...

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