- #1

castrodisastro

- 82

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## Homework Statement

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)

## Homework Equations

Q=mL

Q=mcΔT

## 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**=m

_{lost}_{s}c

_{w}(T

_{i,s}-T

_{eq})+m

_{s}L

_{HV}

and

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

_{gained}**Q**=m

_{gained}_{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 into 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**+m

_{eq}_{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**results in

_{eq}**T**=m

_{eq}_{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**=(c

_{eq}_{w}T

_{i,s}+c

_{w}T

_{i,i}+L

_{HV}-L

_{HF})/2c

_{w}

Part 3)

Plug in known values

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

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

_{eq}Please let me know where I am making my mistake. I separated 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