An object with a higher temperature can have less thermal energy than an object with a lower tempature

Question

A question from my study guide

Answer 1

I will assume this is a TRUE/FALSE question.  Thermal energy is also known as internal energy, which is an extensive property, represented by U.  It is possible to have a small "hot" system that has less internal energy than a large "cool" system.  Here is an example:

 

The internal energy of an ideal gas consisting of n moles of "hard sphere" molecules at absolute temperature T is

 

U = (3/2)nRT

 

where R is the universal gas constant.  Suppose you have object 1 consisting of 2 moles of this type of gas at 300°K and you have object 2 consisting of 20 moles of this type of gas at 290°K.  Object 1 has a higher temperature than object 2.

 

T₁ > T₂

 

Now consider

 

U₁ = (3/2)n₁RT₁

 

U₂ = (3/2)n₂RT₂

 

U₁/U₂ = [(3/2)n₁RT₁] / [(3/2)n₂RT₂] = n₁T₁/(n₂T₂) = (2)(300)/[(20)(290)] ≅ 0.103 < 1

 

U₁/U₂ < 1 ⇒ U₁ < U₂

 

So object 1 has less internal energy than object 2, even though the temperature of object 1 is higher than the temperature of object 2.  Both the temperature and the size of the object matter.  However, heat will always flow from the object of higher temperature to the object of lower temperature, regardless of their sizes.

 

This example shows the statement is TRUE in some cases.