How to calculate the transition temperature in this problem?

In summary: Can you please tell me your way of solving?I said ΔH = TΔS at equilibrium, so calculated ΔH at 291K and 1 atm.
  • #1
Sayantan21
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Summary: The transition Sn(s, gray) ⇌ Sn(s, white) is in equilibrium at 18°C and 1 atm pressure. If ΔS = 8.811K mol for the transition at 18°C and if the densities are 5.75 g/cm3 for gray tin and 7.28 g/cm3 for white tin, calculate the transition temperature under 100 atm pressure

20190806_170226.jpg
The transition Sn(s, gray) ⇌ Sn(s, white) is in equilibrium at 18°C and 1 atm pressure. If ΔS = 8.811K mol for the transition at 18°C and if the densities are 5.75 g/cm3 for gray tin and 7.28 g/cm3 for white tin, calculate the transition temperature under 100 atm pressure
 
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  • #2
You need to show an attempt of your own before receiving help, this is a forum rule.

There's no point repeating the same paragraph 3 times. And if you're going to, copy it right. ΔS = 8.8 J/K mol, not 8.811K mol.

What quantities determine the transition temperature? How might they vary with pressure?
 
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  • #3
dp/dT = ΔS /ΔV
dp/dT = ΔS /ΔV

(P2-P1)/(T2-T1) = ΔS/(M/d1-M/d2); where d1 and d2 are densities of gray tin and white tin, M is equal to molecular weight of tin.
P2 = 100 atm ; P1= 1atm T2=? T1= 273+ 18 =291K ;
(100-1)/(T2-291) = 8.811/ (118.71/5.75 - 118.71/ 7.28)
 
  • #4
That looks OK in principle; I did it a bit differently, but it should come to the same answer. Just a few points you should be careful about:
- ΔS = 8.8, not 8.811, as I've already said.
- ΔV is Vwhite - Vgrey, make sure you get the sign right.
- Careful with units; what are the units of P? ΔV? ΔS? Be consistent.
 
  • #5
Can you please tell me your way of solving?
 
  • #6
I said ΔH = TΔS at equilibrium, so calculated ΔH at 291K and 1 atm.
ΔH = ΔU + PΔV
Assuming ΔU and ΔV don't change with pressure, d(ΔH)/dP = ΔV
So I calculated ΔH at 100 atm, and then T from T = ΔH/ΔS.
(I could not remember a dP/dT formula off the top of my head.)
 
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  • #7
Since they start out at equilibrium, the Gibbs free energy of the grey start out equal the Gibbs free energy of the white. For small changes in pressure and temperature, we have $$dG=-SdT+VdP$$. So, for the changes in pressure and temperature here, we have for the grey $$dG_g=-S_gdT+V_gdP$$and, for the white,$$dG_w=-S_wdT+V_wdP$$But to maintain equilibrium, these changes must be equal. Therefore, $$\frac{dT}{dP}=\frac{V_w-V_g}{S_w-S_g}=\frac{\Delta V}{\Delta S}$$Of course, specific volume is the reciprocal of density. And, of course, this calculation must be done using consistent units.
 

Related to How to calculate the transition temperature in this problem?

What is the definition of transition temperature?

The transition temperature is the temperature at which a substance undergoes a change in its physical or chemical properties, such as a phase transition.

What factors affect the calculation of transition temperature?

The factors that affect the calculation of transition temperature include the chemical composition of the substance, external pressure, and the presence of impurities or defects.

What is the mathematical equation used to calculate transition temperature?

The mathematical equation used to calculate transition temperature varies depending on the specific problem, but it typically involves the use of thermodynamic principles and experimental data.

How do I determine the correct units for the transition temperature calculation?

The correct units for the transition temperature calculation depend on the specific problem and the units used in the given data. It is important to carefully check and convert units to ensure accurate calculations.

Can the transition temperature be accurately predicted?

In most cases, the transition temperature cannot be accurately predicted due to the complex nature of the problem. It is typically determined through experimentation and data analysis.

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