Twenty grams of ice at 0^∘ C are dropped into a beaker containing 120 grams of water initially a

Twenty grams of ice at 0^∘ C are dropped into a beaker...

Twenty grams of ice at $0^{\circ} \mathrm{C}$ are dropped into a beaker containing 120 grams of water initially at $70^{\circ} \mathrm{C}$. Find the final temperature of the mixture neglecting the heat capacity of the beaker. Heat of fusion of ice is $80 \mathrm{cal} / \mathrm{g}$.

Key Concepts

Latent Heat of Fusion

This is the amount of energy required to change a substance from a solid to a liquid state at its melting point without changing its temperature. In problems involving melting, such as ice turning into water, the latent heat of fusion indicates the energy absorbed to overcome the attractive forces holding the solid together.

Specific Heat Capacity

An intrinsic property of substances, this refers to the amount of energy needed to raise the temperature of a unit mass of a substance by one degree Celsius. In water, which has a high specific heat capacity, this concept is crucial when calculating the energy changes as the temperature of the water increases or decreases.

Heat Transfer

This involves the movement of thermal energy from a higher temperature object to a lower temperature one until thermal equilibrium is reached. It is governed by the principle of conservation of energy, ensuring that the total heat lost by one part of the system is equal to the heat gained by another.

Thermal Equilibrium

This is the state reached when two or more bodies in thermal contact cease to exchange energy due to having the same temperature. In such scenarios, the final uniform temperature is determined by balancing the heat lost by the warmer components with the heat gained by the cooler components.

Transcript

00:01 In this problem, we are told that we have solid water in the form of ice, and we have 20 grams of that ice at a temperature of negative 10 degrees celsius.

00:10 We mix this with liquid water at 80 degrees celsius, and we have 100 grams of that.

00:16 We want to determine what the final temperature is after we mix these two forms of water together.

00:23 We know that when we mix these two, that the hotter water at 80 degrees celsius will become cooler.

00:31 And when it does that it will release energy.

00:34 That energy that it releases will be used by the ice in order to heat up and melt the ice and then raise its temperature.

00:43 So we first need to determine how much energy is required from the water when it is releasing energy to decrease its temperature in order to heat up and melt that ice.

00:58 And so the energy that will correspond to will be first the energy that is required to increase the temperature of that ice from negative 10 degrees celsius to the melting point of ice, which is zero degrees celsius.

01:16 That is equal to mc delta t.

01:18 So we have 20 grams of that ice that we need to heat up.

01:25 The specific heat capacity of ice is two.

01:30 0 .03 joules per gram times degrees celsius.

01:39 And the final temperature is 0 degrees celsius.

01:44 The initial temperature is given as negative 10 degrees celsius.

01:51 And then once we bring it to the melting point of ice, we need to add the energy that is required to change the phase of ice from a solid to a liquid.

02:04 And we use the delta h of fusion in order to suffer that energy.

02:10 So we again have 20 grams of ice.

02:16 We convert that to moles of water.

02:19 From the periodic table, we know that one mole of water corresponds to 18 .016 grams.

02:28 And the delta h of fusion is 6 .02 times 10 to the 3rd, joules.

02:39 Per mole.

02:45 And so that will come out to the total energy that is required to first warm that ice from negative 10 degrees celsius to zero degrees celsius and then to convert that ice, all of that ice into the liquid form of water.

03:00 And when we take those amounts of energy separately and we add them together, then we get a total of about 7 ,088...

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