Thermodynamics-Find final pressure of a piece of lead

[Kelsi_Jade]

Problem:
The temperature of a piece of lead is increased from 20C to 35C while its volume is held constant by increasing the hydrostatic pressure. If the initial pressure is 1atm, what is the final pressure?

Honestly, I'm not sure where to begin here. Everything we have done thus far has been in relation to gases and this isn't an ideal gas here.

I know if a volume is constant it is considered a isochoric process and the process does no pressure-volume work. Also, Q=mCvΔT.
But I am not sure where to go from here, much less how to relate this to a process involving a solid.
Any help is much appreciated!

 

[DrClaude]

Problem:
The temperature of a piece of lead is increased from 20C to 35C while its volume is held constant by increasing the hydrostatic pressure. If the initial pressure is 1atm, what is the final pressure?

Honestly, I'm not sure where to begin here. Everything we have done thus far has been in relation to gases and this isn't an ideal gas here.

Indeed, not much of a gas! You need to look at thermal expansion coefficients.

 

[Chestermiller]

Indeed, not much of a gas! You need to look at thermal exapansion coefficients.

...and bulk modulus of Pb.

 

[Kelsi_Jade]

...and bulk modulus of Pb.

So, Bulk modulus of a solid is B=ΔP/(ΔV/V). But if the process is done by keeping the volume constant, that would give B=(Pf-1atm)/0=undefined?

I'm thinking, then, that there is a way to relate volume to temperature?
From reading about expansion in solids,
Volume is related to temp by :ΔV/V=αvΔT where αv= 1/V* dV/dT. I'm not sure if this is how to properly relate the two since you would still have to have a numerical value for V?

 

[Chestermiller]

So, Bulk modulus of a solid is B=ΔP/(ΔV/V). But if the process is done by keeping the volume constant, that would give B=(Pf-1atm)/0=undefined?

I'm thinking, then, that there is a way to relate volume to temperature?
From reading about expansion in solids,
Volume is related to temp by :ΔV/V=αvΔT where αv= 1/V* dV/dT. I'm not sure if this is how to properly relate the two since you would still have to have a numerical value for V?

Not really. You can use α to get ΔV/V from the temperature rise, and then use B to figure out how much ΔP you need to exactly cancel out the ΔV/V from the temperature rise.

 

[Kelsi_Jade]

You can use α to get ΔV/V from the temperature rise

How do you go about doing that? Sorry, I'm just not seeing how this fits together yet.

 

[DrClaude]

How do you go about doing that? Sorry, I'm just not seeing how this fits together yet.

You wrote the equations yourself in a previous post:

B=ΔP/(ΔV/V)
ΔV/V=αvΔT

 

[Kelsi_Jade]

Oh, I think I understand. I was trying to still solve for a value of α but I found there is a standard volumetric thermal expansion coefficient for lead. Thanks!