**1. The problem statement, all variables and given/known data**

You are asked to mix equal masses of of ice and steam at 0.00°C and 100.00°C respectively, What will the final temperature of the mixture be? The mixture is in a perfect insulator.

m_{s}=mass of steam

m_{i}=mass of ice

L_{HV,s}=Heat of vaporization of steam=2257J/g

L_{HF,i}=Heat of fusion of ice=334J/g

T_{i,s}=100°C=373K

T_{i,i}=0.00°C=273K

c_{w}=4.19J/(g*K)

**2. Relevant equations**

Q=mL

Q=mcΔT

**3. The attempt at a solution**

I am fairly confident that my thought process is correct. I just keep making algebra mistakes(I assume) but I can’t find what my error is. Please let me know where my process breaks down please.

I did this in 3 parts

Part 1)

So first I consider the situation.

Since the mixture is in a perfect insulator, that means no heat is lost to the surroundings.

**Q=0**

That means that the heat lost by the steam is gained by the ice. Q for the steam will be negative since it is losing heat.

**Q=0=Q _{gained}+Q_{lost}=Q_{gained}+(-Q_{lost})**

**Q _{lost}=Q_{gained}**

From here, the Q_{lost} is given by the sum of energy used to turn steam into water, then to bring the steam-turned-water to thermal equilibrium

**Q _{lost}**=m

_{s}c

_{w}(T

_{i,s}-T

_{eq})+m

_{s}L

_{HV}

and **Q _{gained}** is given by the sum of energy used to melt the ice, then to bring the ice-turned-water to thermal equilibrium.

**Q _{gained}**=m

_{i}c

_{w}(T

_{eq}-T

_{i,i})+m

_{i}L

_{HF}

Again, since there is no heat lost to the surroundings, and the mixture is allowed to reach thermal equilibrium then…

**Q _{lost}=Q_{gained}**

Part 2)

Now we plug in to be able to solve for T_{eq}

m_{s}c_{w}(T_{i,s}-T_{eq})+m_{s}L_{HV}=m_{i}c_{w}(T_{eq}-T_{i,i})+m_{i}L_{HF}

m_{s}c_{w}T_{i,s}-m_{s}c_{w}T_{eq}+m_{s}L_{HV}=m_{i}c_{w}T_{eq}-m_{i}c_{w}T_{i,i}+m_{i}L_{HF}

Moving all terms containing T_{eq} to one side

m_{s}c_{w}T_{i,s}+m_{i}c_{w}T_{i,i}+m_{s}L_{HV}-m_{i}L_{HF}=m_{i}c_{w}**T _{eq}**+m

_{s}c

_{w}

**T**

_{eq}Factor out **T _{eq}**

m_{s}c_{w}T_{i,s}+m_{i}c_{w}T_{i,i}+m_{s}L_{HV}-m_{i}L_{HF}=(m_{i}c_{w}+m_{s}c_{w})**T _{eq}**

and finally solving for **T _{eq}** results in

**T _{eq}**=m

_{s}c

_{w}T

_{i,s}+m

_{i}c

_{w}T

_{i,i}+m

_{s}L

_{HV}-m

_{i}L

_{HF}/(m

_{i}c

_{w}+m

_{s}c

_{w})

We can do one more thing, since the masses are equal, we can factor out the mass.

**T _{eq}**=

**m**c

_{w}T

_{i,s}+

**m**c

_{w}T

_{i,i}+

**m**L

_{HV}–

**m**L

_{HF}/(

**m**c

_{w}+

**m**c

_{w})

**T _{eq}**=

**m**(c

_{w}T

_{i,s}+c

_{w}T

_{i,i}+L

_{HV}-L

_{HF})/

**m**(c

_{w}+c

_{w})

**T _{eq}**=(c

_{w}T

_{i,s}+c

_{w}T

_{i,i}+L

_{HV}-L

_{HF})/2c

_{w}

Part 3)

Plug in known values

**T _{eq}**=[(4.19J/(g*K)(373K)+(4.19J/(g*K))(273K)+(2257J/g)-(334J/g)]/[(2*4.19J/(g*K))]

**T _{eq}**=552K Obviously incorrect

Please let me know where I am making my mistake. I seperated my work into 3 parts to make it easier to point out where i made an error. Thank you I have a test on Friday I NEEEEEED to understand how to do this properly. Thank you

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