Relation between heat capacity and internal energy

In summary, the conversation discusses deriving an expression for the heat capacity of a system at constant volume, and determining that at high temperatures it is inversely proportional to the square of temperature. The correct expression for the total internal energy of the system is also provided, and the correct steps for differentiation are discussed. There is a clarification about the incorrect v' and the use of the chain rule.
  • #1
trelek2
88
0
The task is to derive an expression for the heat capacity of a system at constant volume and show that at high temperatures it is inversely proportional to the square of temperature.

As far as I'm concerned the relation between internal energy and heat capacity is:
Cv=dE/dT
However with with i cannot get a reasonable answer.

The expression for the total internal energy of a system is given by:
[tex]E= \frac{N \Delta E}{exp( \Delta E/kT) +1 } }[/tex]

Where N is the total number of particles, T is the temperature, k is the Boltzmann constant.

When i differentiate it I get an expression which is negative and inversely proportional to the temperature, not to the square temperature...
 
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  • #2
I do not think you are differentiating correctly. I took the partial in a big T limit and got a positive inverse T relationship. What are your steps?
 
  • #3
But it is clearly supposed to be proportional to inverse square T...
I use the quotient rule E'=(u'v-uv')/v^2
and take u=NdeltaE, u'=0
v=exp(deltaE/kT)+1 v'=(deltaE/kT)exp(deltaE/kT).
This clearly gives a negative inverse T proportional.
Show me what I am supposed to do.
 
  • #4
Sorry, typo. I meant to say positive inverse square T. Your v' is incorrect. Use the chain rule.
 
  • #5
oh ****, you're right. How stupid I am.
 
  • #6
No worries :approve:
 

1. What is the definition of heat capacity?

Heat capacity is a measure of the amount of heat energy required to raise the temperature of a substance by 1 degree Celsius.

2. How is heat capacity related to internal energy?

Internal energy is the total energy of a substance, including both its kinetic and potential energies. Heat capacity is directly proportional to the internal energy of a substance, as it measures the amount of energy needed to increase its temperature.

3. How does the heat capacity of a substance affect its temperature change?

The higher the heat capacity of a substance, the more energy it can absorb before its temperature increases. This means that substances with higher heat capacities will experience smaller temperature changes for a given amount of heat energy.

4. How does the heat capacity of a substance vary with temperature?

The heat capacity of most substances increases with temperature. This is because as the temperature increases, the particles within the substance have more energy and are able to vibrate more, resulting in a greater capacity to absorb heat.

5. What factors can affect the heat capacity of a substance?

The heat capacity of a substance can be influenced by factors such as its mass, composition, and physical state (solid, liquid, or gas). In addition, the heat capacity may also vary depending on the temperature range being considered.

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