What is the change in entropy when gallium melts in your hand?

[Art_Vandelay]

Homework Statement

If 25.0 g of gallium melts in your hand, what is the change in entropy of the gallium?

What about the change of entropy in your hand? Is it positive or negative? Is its magnitude greater or less than that of the change in entropy of the gallium?

The melting temperature of gallium is 29.8°C, and its heat of fusion is 8.04 * 10⁴ J/kg.

Homework Equations

ΔS = Q/Temp

The Attempt at a Solution

ΔS_gal = ((.025 kg)(8.04*10⁴ J/kg))/(20+273 K) assuming air temp is 20C
ΔS_gal = 6.86 J/K

ΔS_hand = -((.025 kg)(8.04 * 10⁴ J/kg))/(29.8+273 K)
ΔS_hand = -6.63 J/K Negative since heat is leaving hand, reducing entropy

 

[Astronuc]

I'm not sure why air temperature is used.

The Ga melts in the hand, not in the air.

The Ga melts at the melting temperature, no?

Heat is transferred from hotter to cooler. What is the source of the thermal energy to melt the Ga? What might be the temperature of the hand be?

 

[Art_Vandelay]

I'm not sure why air temperature is used.

The Ga melts in the hand, not in the air.

The Ga melts at the melting temperature, no?

Heat is transferred from hotter to cooler. What is the source of the thermal energy to melt the Ga? What might be the temperature of the hand be?

The source of the thermal is the hand, which must be equal to or greater than 29.8°C. So, the changes in entropy of the hand and the gallium are equal and opposite?

ΔS_gal = ((.025 kg)(8.04*10⁴ J/kg))/(29.8+273 K)
ΔS_gal = 6.64 J/K

ΔS_hand = -((.025 kg)(8.04 * 10⁴ J/kg))/(29.8+273 K)
ΔS_hand = -6.64 J/K

 

[Chestermiller]

In the case of the gallium, the answer is pretty clearcut, because it takes place reversibly. In the case of the hand, the process is not reversible (because the temperature at the boundary (29.8) is less than the bulk temperature of the hand (37)). The initial and final temperatures of the hand are both very close to 37C, but a finite amount of heat has been transferred. You need to dream up a reversible path for this heat transfer , and calculate the entropy change from this path. One such path is where the interface is held only slightly below 37, and the same amount is transferred over a longer amount of time. For this reversible path, what is the entropy change?

 

[Art_Vandelay]

Would I then divide the Q_Hand by 310 K? I'm not sure how to apply reversibility and irreversibility.

 

[Art_Vandelay]

In the case of the gallium, the answer is pretty clearcut, because it takes place reversibly. In the case of the hand, the process is not reversible (because the temperature at the boundary (29.8) is less than the bulk temperature of the hand (37)). The initial and final temperatures of the hand are both very close to 37C, but a finite amount of heat has been transferred. You need to dream up a reversible path for this heat transfer , and calculate the entropy change from this path. One such path is where the interface is held only slightly below 37, and the same amount is transferred over a longer amount of time. For this reversible path, what is the entropy change?

ΔS_gal = ((.025 kg)(8.04*10⁴ J/kg))/(29.8+273 K)
ΔS_gal = 6.64 J/K

ΔS_hand = -((.025 kg)(8.04 * 10⁴ J/kg))/(37+273 K)
ΔS_hand = -6.48 J/K

 

[Chestermiller]

ΔS_gal = ((.025 kg)(8.04*10⁴ J/kg))/(29.8+273 K)
ΔS_gal = 6.64 J/K

ΔS_hand = -((.025 kg)(8.04 * 10⁴ J/kg))/(37+273 K)
ΔS_hand = -6.48 J/K

Yes. Please understand that getting the entropy change for the hand by this approach is not very accurate because of a number of complexities that the problem statement glosses over. The hand is attached to a body (unless it's been chopped off), and there is interchange of heat with the rest of the body. There is also heat generated by chemical reactions in the body. And, there is heat transfer from the hand to the surrounding air, which may be affected by the cooling provided by the gallium to the hand. The approach we have taken here is to simply approximate the hand as a constant temperature reservoir at 37C. Getting the actual entropy change of the hand as a result of contact with the gallium is a much more complicated analysis which, IMHO, is not worth the effort.

Chet

 

[Art_Vandelay]

Yes. Please understand that getting the entropy change for the hand by this approach is not very accurate because of a number of complexities that the problem statement glosses over. The hand is attached to a body (unless it's been chopped off), and there is interchange of heat with the rest of the body. There is also heat generated by chemical reactions in the body. And, there is heat transfer from the hand to the surrounding air, which may be affected by the cooling provided by the gallium to the hand. The approach we have taken here is to simply approximate the hand as a constant temperature reservoir at 37C. Getting the actual entropy change of the hand as a result of contact with the gallium is a much more complicated analysis which, IMHO, is not worth the effort.
Chet

Thank you very much! I assume it is supposed to be an ideal hand, possibly Thing from The Addam's Family. :)

 

[Chestermiller]

Thank you very much! I assume it is supposed to be an ideal hand, possibly Thing from The Addam's Family. :)

Ha! Great joke. Incidentally, Art, please give my regards to Jerry, Elaine, and Kramer.

Chet