Heat Capacity of NaCl: 500K-1074K & 1074K-1500K

[krootox217]

Homework Statement

The heat capacity of the solid NaCl from 500 K to 1074 K is given by [52.996 J*K^-1*mol^-1 – (7.86*10^-3J*K^-2*mol^-1)*T + (1.97*10^-5J*K^-3*mol^-1)*T2 ] and that of liquid NaCl from 1074 K to 1500 K is given by [125.637 J*K^-1mol^-1 – (8.187*10^-2 J*K^-2*mol^-1)*T + (2.85*10^-5 J*K^-3*mol^-1)*T2 ]. If Δfus H^Θ_{T, m} = 28.158 kJ*mol^-1 at 1074K, Please determine H^Θ1500K - H^Θ500K for NaCl.

Homework Equations

The Attempt at a Solution

I tried to solve this task, but I don't understand what they exactly want me to do.

Do I have to calculate the ΔH from 500K to 1074K with the given heat capacities? And what is Δ_fusH^Θ?

 

[MexChemE]

You need to calculate two integrals. The integral of the solid heat capacity function with respect to temperature, from 500 K to 1074 K, and the integral of the liquid heat capacity function from 1074 K to 1500 K. Δ_fusH° is the heat of fusion of NaCl. The first integral is the sensible heat required by solid NaCl to reach its melting point, then some amount of latent heat is required to melt NaCl at 1074 K, and the second integral is the sensible heat required by liquid NaCl to reach 1500 K. Add these three heats (don't neglect minus signs) and you'll have solved the problem.

 

[Chestermiller]

This is exactly the same kind of problem we did with the enthalpy chance of H2O in going from (373,l) to (393,v). The differences are that we are dealing with melting instead of vaporization, and the heat capacities depend on temperature. Since the heat capacities do depend on temperature, you have to integrate the heat capacity over the relevant temperature change to get the change in enthalpy.

Chet

 

[krootox217]

I don't exactly understand what you mean with "integrate the heat capacity over the relevant temperature change", but maybe that's because I'm not a native English speaker.

What is the change in the formula compared to the H2O problem?

Does that mean, that H(T2)=H(T1)+Cp*(T2-T1) becomes H(T2)=H(T1)+(Cp(product)-Cp(starting material))*(T2-T1)?

 

[Chestermiller]

I don't exactly understand what you mean with "integrate the heat capacity over the relevant temperature change", but maybe that's because I'm not a native English speaker.

What is the change in the formula compared to the H2O problem?

Does that mean, that H(T2)=H(T1)+Cp*(T2-T1) becomes H(T2)=H(T1)+(Cp(product)-Cp(starting material))*(T2-T1)?

No.

$$H(1074,s)=H(500,s)+\int_{500}^{1074}{C_p(T,s)dT}$$
$$H(1074,l)=H(1074,s)+\Delta H_{fus}$$
$$H(1500,l)=H(1074,l)+\int_{1074}^{1500}{C_p(T,l)dT}$$

Do you see the similarity to our water problem now?

Chet

 

[krootox217]

Yes I do, thank you!

I try to calculate it when I'm home

 

[krootox217]

I got:

34'186.93 J/mol for H(1074,s), while I set H(500,s) = 0

315'766.93 J/mol for H(1074,l)

and

549'775.31 J/mol for H(1500,l)

Are these values correct? They seem pretty big compared to those of the H20, but since it is a salt, I think it's possible?

Now the task says: "Please determine HΘ1500K - HΘ500K for NaCl."

Does this mean, that I have to add H(1074,s)+H(1074,l)+H(1500,l) to get de final ΔHΘ, which I think is asked?

 

[MexChemE]

Are these values correct? They seem pretty big compared to those of the H20, but since it is a salt, I think it's possible?

It takes a lot of energy to "loosen" (and break) ionic bonds.

Now the task says: "Please determine HΘ1500K - HΘ500K for NaCl."

Does this mean, that I have to add H(1074,s)+H(1074,l)+H(1500,l) to get de final ΔHΘ, which I think is asked?

Yes.

 

[Chestermiller]

I got:

34'186.93 J/mol for H(1074,s), while I set H(500,s) = 0

315'766.93 J/mol for H(1074,l)

If H(1074,s)=34,186.93 J/mol, shouldn't H(1074,l) = 34186.93 + 28158 J/mol?

and

549'775.31 J/mol for H(1500,l)

Are these values correct?

Apparently not.

They seem pretty big compared to those of the H20, but since it is a salt, I think it's possible?

The values for water were per kg, right?

Now the task says: "Please determine HΘ1500K - HΘ500K for NaCl."
Does this mean, that I have to add H(1074,s)+H(1074,l)+H(1500,l) to get de final ΔHΘ, which I think is asked?

No. This difference is equal to H(1500,l)-H(500,s).

Chet

 

[krootox217]

Right, I saw my mistake.

Now I got:

34'186.93 J/mol for H(1074,s), while I set H(500,s) = 0

34186.93 + 28158 J/mol = 62'344.93 J/mol for H(1074,l)

and

296353.31 J/mol for H(1500,l)

This means, that ΔHΘ = H(1500,l)-H(500,s) = 296353.31 J/mol - 0 J/mol = 296353.31 J/mol?

 

[Chestermiller]

Right, I saw my mistake.

Now I got:

34'186.93 J/mol for H(1074,s), while I set H(500,s) = 0

34186.93 + 28158 J/mol = 62'344.93 J/mol for H(1074,l)

and

296353.31 J/mol for H(1500,l)

This means, that ΔHΘ = H(1500,l)-H(500,s) = 296353.31 J/mol - 0 J/mol = 296353.31 J/mol?

Yes, that's better. I didn't check your arithmetic on the integrations, so your answer is correct if the integrations were done correctly. A way to check is to calculate the heat capacity at the average temperature for each temperature interval, and multiply each by the corresponding temperature difference. The results should be reasonable approximations to the values of the integrals.

Chet

 

[krootox217]

Ok, thanks for the help!