Help Please!! Sign of Entropy

Question

You probably know that when two objects at different temperatures are brought into contact, heat always travels from the hot to the cold, never the reverse. Imagine then that you have two gold metal blocks, one with mass 390.0 grams at a temperature of 126.0deg C and another with mass 329.0 grams at a temperature of 37.5deg C that are brought into contact in a perfectly insulated container

What is the sign of the entropy change for the heat transfer that occurs? The specific heat of gold = 0.129 JK-1g-1.

Is it Positive OR Negative

Answer 1

Answer is that the entropy change is positive.  You have a lot of numbers in this problem that you really don't need to find the sign, so the problem can be a little confusing.

Start with the equation that give the thermodynamic definition of entropy:

dS = dq/T,

Where dS is the entropy change, dq is the heat change, and T is the Kelvin temp.  You have two blocks at different temps in a closed, insulated system, so the entropy change is:

dS = dq1/T1 + dq2/T2,

where 1 refers to the hot block and 2 to the cold block.  No heat is lost to the outside, so

dq1 = -dq2 = dq,

and we can write:

dS = dq(1/T2 - 1/T1)

Since T1>T2, the quantity (1/T2 - 1/T1) is positive, and so therefore is dS.