Highschool Question on Thermal Equilibrium

[chris1]

Hi guys,
I'm having a bit of trouble with following thermal equilibrium question...

Homework Statement

Create an equation representing energy transfers of thermal energy equilibrium, that will enable one to determine the latent heat of fusion for ice.

One conducts an experiment mixing ice and water in a jug, recording the changes in mass and temperature:

The mass of the ice used ( m$_{i}$ ) = 0.10kg

The initial temperature of the ice ( T$_{i}$ ) = -15°C

The initial mass of the water ( m$_{w}$ ) = 0.45kg

The initial temperature of the water ( T$_{w}$ ) = 23°C

Final Temperature of mixture ( T$_{f}$ ) = 16°C

Specific heat capacity of water ( c$_{w}$ ) = 4.2 kJ kg$^{-1}$°C$^{-1}$

Specific heat capacity of ice ( c$_{i}$ ) = 2.1 kJ kg$^{-1}$°C$^{-1}$So one must set up an equation of thermal equilibrium, that will allow for the value for the latent heat of fusion of ice (l) can be determined.

Homework Equations

c = $\frac{Q}{mΔT}$

For changes of state:
Q = ml

The Attempt at a Solution

Here is my attempt of working it out:

(m$_{i}$ * c$_{i}$ * 15°C) + (m$_{i}$ * l) + (m$_{i+w}$ * c$_{w}$ * 16°C) = (m$_{i+w}$ * c$_{w}$ * 7°C)

I know my solution isn't correct due to cancelling of terms but I don't know what to do to fix it. The answer isn't in the textbook either.

Thanks for the feedback guys

 

[mukundpa]

heat absorbed by ice and the water due to fusion of ice will be

(mi * ci * 15°C) + (mi * l) + (mi * cw * 16°C)

and heat given up by initially hot water will be

(mw * cw * 7°C)

 

[chris1]

heat absorbed by ice and the water due to fusion of ice will be

(mi * ci * 15°C) + (mi * l) + (mi * cw * 16°C)

and heat given up by initially hot water will be

(mw * cw * 7°C)

So would that mean:

(mi * ci * 15°C) + (mi * l) + (mi * cw * 16°C) = (mw * cw * 7°C)

But wouldn't the mass on the right side of the equation be constantly changing as the ice melts, or is that irrelevant ?

 

[Chestermiller]

So would that mean:

(mi * ci * 15°C) + (mi * l) + (mi * cw * 16°C) = (mw * cw * 7°C)

But wouldn't the mass on the right side of the equation be constantly changing as the ice melts, or is that irrelevant ?

Imagine that the material comprising the original mass of the ice, and the mass comprising the original mass of the water are kept separate from one another by a thin invisible membrane. Both masses are allowed to thermally equilibrate with one another. Afterwards the membrane is removed.

 

[Nickie]

Hello,

To solve this problem, we need to use the principle of thermal equilibrium, which states that in a closed system, the total energy must remain constant. This means that the energy lost by one component (in this case, the ice) must equal the energy gained by the other component (the water).

First, let's calculate the energy lost by the ice as it goes from -15°C to 0°C (where it starts to melt):

Q_{ice} = m_{ice} * c_{ice} * (0°C - (-15°C)) = 0.10kg * 2.1 kJ kg^{-1}°C^{-1} * 15°C = 31.5 kJ

Next, we need to calculate the energy gained by the water as it goes from 23°C to 0°C (where it starts to freeze):

Q_{water} = m_{water} * c_{water} * (0°C - 23°C) = 0.45kg * 4.2 kJ kg^{-1}°C^{-1} * (-23°C) = -42.525 kJ

Note that we use a negative sign here because the water is losing energy (cooling down).

Now, in order to reach thermal equilibrium, the energy lost by the ice must equal the energy gained by the water. So we can set up the following equation:

Q_{ice} = Q_{water}

31.5 kJ = -42.525 kJ

Next, we need to consider the energy needed for the ice to melt. This is given by the equation Q = ml, where m is the mass of the substance and l is the latent heat of fusion. We can rearrange this equation to solve for l:

l = Q/m

Substituting in the values we have calculated, we get:

l = 31.5 kJ / 0.10kg = 315 kJ kg^{-1}

So the latent heat of fusion for ice is 315 kJ kg^{-1}. I hope this helps!