Melting point of lead at 100atm

[curto]

Homework Statement

Homework Equations

use clausius-clapeyron equation: [itex]\frac{dp}{dt}[/itex] = [itex]\frac{L}{T(V_{2}-V_{1})}[/itex]
which can be rearranged (i think) in the following: [itex]p = p_{0}+\frac{L}{\Delta V}ln(\frac{T}{T_{0}})[/itex]

The Attempt at a Solution

so using the above equation i solved for T in which i got: [itex]T = T_{0}e^\frac{(P-P_{0})\Delta V}{L}[/itex]

and then since density is equal to mass per unit volume i solved for V and got my two volumes by dividing the densities given by 11340. also 1 atm is roughly 10^6 pa so i used that and got a massive answer for T which is obviously wrong. then i just tried doing it in terms of atm in which i got a very small increase in temperature. isn't the melting point meant to become lower if the pressure is increased?? so how can T_{0} times e^{whatever} equal a number less than T_{0}?? does that mean my formula is wrong?

 

[ehild]

Your formula is right, but how did you calculate ΔV?

ehild

 

[curto]

Your formula is right, but how did you calculate ΔV?

ehild

ΔV i assumed would just be the difference between my two volumes, which i found to be 0.971m^3 and 0.939m^3 by dividing the densities by the density of lead which is 11340kg/m^3. and yeah so my ΔV was just 0.032m^3

 

[ehild]

ΔV i assumed would just be the difference between my two volumes, which i found to be 0.971m^3 and 0.939m^3 by dividing the densities by the density of lead which is 11340kg/m^3. and yeah so my ΔV was just 0.032m^3

Dividing density by density is not volume. What do you mean on the density of lead as 11340 kg/m³? The densities were given at 327 C° both for the solid phase (1.101˙10⁴ kg/m³) and for the liquid phase (1.065˙10⁴ kg/m³). As the latent heat refers to 1 kg material, you need the volumes of 1 kg lead both in the solid and in the liquid state.
The volume change is much less than the one you got.
More: 1 atm is not near to 10⁶ Pa.

ehild

 

[curto]

Dividing density by density is not volume. What do you mean on the density of lead as 11340 kg/m³? The densities were given at 327 C° both for the solid phase (1.101˙10⁴ kg/m³) and for the liquid phase (1.065˙10⁴ kg/m³). As the latent heat refers to 1 kg material, you need the volumes of 1 kg lead both in the solid and in the liquid state.
The volume change is much less than the one you got.
More: 1 atm is not near to 10⁶ Pa.

ehild

oh ok (btw i meant 10^5 for 1atm not 10^6 lol, sorry). so to find volume from the latent heat and density.. i need to? sorry I've never found volume from density and latent heat before. i haven't really had to use latent heat much at all so I am not very familiar with how to use it sorry. could i have another hint?

 

[ehild]

The latent heat is the energy needed to melt a given amount of material. Here the specific latent heat is given, the energy needed to melt 1 kg material. The mass is not given, so you can use 1 kg, and then you have to find the volume of 1 kg lead.

ehild

 

[curto]

The latent heat is the energy needed to melt a given amount of material. Here the specific latent heat is given, the energy needed to melt 1 kg material. The mass is not given, so you can use 1 kg, and then you have to find the volume of 1 kg lead.

ehild

i still don't understand how the latent heat can be used to find volume if we have density. the units arent even going to cancel unless i add some other factor into the conversion. i can get numbers in terms of m^3 per J.. but that's not volume.

 

[ehild]

The latent heat IS NOT used to find the volume.

ehild

 

[ehild]

You are given the specific latent heat. You get the latent heat of one kg material by multiplying L with 1 kg. So the unit of the denominator is joule=Nm.
You have pressure multiplied by volume in the numerator. What is the unit of pressure?

ehild

 

[curto]

You are given the specific latent heat. You get the latent heat of one kg material by multiplying L with 1 kg. So the unit of the denominator is joule=Nm.
You have pressure multiplied by volume in the numerator. What is the unit of pressure?

ehild

i see how those units are going to cancel, and I am going to accept that we can just change the specific latent heat into the latent heat by just multiplying it by 1(units of the denominator) .. ? and so i guess I am back to the initial problem which is how to find deltaV? thanks for the help so far.

 

[ehild]

You get the latent heat for 1 kg lead by multiplying the specific latent heat by 1 kg.

I have written it several times that you need the volume of 1 kg material. How is the density defined?

ehild

 

[curto]

You get the latent heat for 1 kg lead by multiplying the specific latent heat by 1 kg.

I have written it several times that you need the volume of 1 kg material. How is the density defined?

ehild

so the latent heat for 1kg of lead.. has the units J per nothing and the specific heat of lead has the units J per 1kg. that's why I am getting confused if you can see what i mean. density is defined to be the mass per unit of volume. so we have p = m/V hence V = m/p but we don't know the mass unless we assume the mass to be 1kg or something. i know this question is easy since its the first one from the chapter in the textbook and my lecturer even said its easy and it LOOKS easy and when I am having this much trouble with something, from experience its just one tiny little concept I've forgetting or something and once i know that everything just clicks and i get it 100%. but so far atm i still don't get how we can find the volume without knowing the mass. sorry if youre getting frustrated

 

[ehild]

The melting point does not depend on mass, do you agree? The mass is the same whether it is liquid or solid. The mass was not specified, why not take it 1 kg? You can calculate with m=3.5kg, if you prefer it, but then you have to multiply the specific latent heat with that mass and you divide 3.5 kg with the densities to get the volumes. We just choose the simplest value when we take the mass equal to 1 kg.

ehild

 

[curto]

The melting point does not depend on mass, do you agree? The mass is the same whether it is liquid or solid. The mass was not specified, why not take it 1 kg? You can calculate with m=3.5kg, if you prefer it, but then you have to multiply the specific latent heat with that mass and you divide 3.5 kg with the densities to get the volumes. We just choose the simplest value when we take the mass equal to 1 kg.

ehild

thanks heaps! it all makes sense now, cheers

 

[ehild]

You are welcome

ehild