Converting partial derivative w.r.t. T to partial derivative w.r.t. 1/T

In summary, the conversation discusses the operator identity (\partial/ \partialT) = (-1/T) * (\partial/ \partial(1/T)) and how it can be shown to be true. By defining F = 1/T and using the chain rule, it is proven that the identity holds for any suitable function g. Thus, the operator identity can be written as T(\partial/ \partialT) = (-1/T)(\partial/ \partial(1/T)).
  • #1
knulp
6
0
Hi, I have a question about a certain step in the following problem/derivation, which you'll see in square brackets:

Show that T * ([tex]\partial[/tex]/ [tex]\partial[/tex]T) = (-1/T) * ([tex]\partial[/tex]/ [tex]\partial[/tex](1/T))
["[tex]\partial[/tex]/[tex]\partial[/tex]T" is the operator that takes the partial derivative of something with respect to T]

Showing that this is true is a little tricky. For example, we can define F = 1/T. Then ([tex]\partial[/tex]F/ [tex]\partial[/tex]T) = -1/T^2 and ([tex]\partial[/tex]F/ [tex]\partial[/tex]F) = 1. So we can write

([tex]\partial[/tex]F/ [tex]\partial[/tex]T) = (-1/T^2) ([tex]\partial[/tex]F/ [tex]\partial[/tex]F).

[In the next step he drops the F, so it's now an operator for an arbitrary function, but still with respect to F… Is this really okay?]

([tex]\partial[/tex]/ [tex]\partial[/tex]T) = (-1/T^2)([tex]\partial[/tex]/ [tex]\partial[/tex]F)

= (-1/T^2) ([tex]\partial[/tex]/ [tex]\partial[/tex](1/T)).

Multiplying by T, T([tex]\partial[/tex]/ [tex]\partial[/tex]T) = (-1/T)([tex]\partial[/tex]/ [tex]\partial[/tex](1/T)) and we're done.
 
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  • #2
[In the next step he drops the F, so it's now an operator for an arbitrary function, but still with respect to F… Is this really okay?]


The operator identity holds by definition if, when acting on suitable functions, both sides of the identity give the same result. So in my understanding it is not quite right to conclude the operator identiy from only having shown that when acting on F they give the same result. However you can easily cure this. It is just the chain rule. Let g(T) be some function, F(T) = 1/T,. then

[tex]
\frac{\partial g}{\partial T} = \frac{\partial F}{\partial T}\frac{\partial g}{\partial F} = -\frac{1}{T^2}\frac{\partial g}{\partial F}
[/tex]

This is true for any suitable function g so

[tex]
\frac{\partial }{\partial T} = -\frac{1}{T^2}\frac{\partial }{\partial F}
[/tex]

or

[tex]
T\frac{\partial }{\partial T} = -\frac{1}{T}\frac{\partial }{\partial (1/T)}
[/tex]
 
Last edited:
  • #3
Thanks, it makes complete sense now. So simple, doh!
 

1. How do you convert a partial derivative with respect to T to a partial derivative with respect to 1/T?

To convert a partial derivative with respect to T to a partial derivative with respect to 1/T, you can use the chain rule. First, rewrite the expression as a function of 1/T instead of T. Then, take the derivative with respect to 1/T and multiply it by the derivative of 1/T with respect to T, which is -1/T^2. This will give you the final expression for the partial derivative with respect to 1/T.

2. Why would you need to convert a partial derivative with respect to T to a partial derivative with respect to 1/T?

There are various reasons why you may need to convert a partial derivative with respect to T to a partial derivative with respect to 1/T. One common reason is when dealing with thermodynamic equations, where temperature (T) and its inverse (1/T) are both important variables. In some cases, it may be easier to work with 1/T instead of T, so converting the partial derivative allows for simpler calculations.

3. Can you give an example of converting a partial derivative with respect to T to a partial derivative with respect to 1/T?

One example is in the ideal gas law, where the partial derivative of pressure (P) with respect to temperature (T) is given by P = nRT/V. To convert this to a partial derivative with respect to 1/T, you can use the chain rule to get dP/d(1/T) = (nR/V) * (-1/T^2) = -nRT^2/V. This can be useful when working with the ideal gas law at constant volume, where 1/T is a more suitable variable than T.

4. Are there any limitations to converting a partial derivative with respect to T to a partial derivative with respect to 1/T?

Yes, there are some limitations to this conversion. It works best when the original equation is a simple function of T, and when the inverse of T is easy to calculate. In more complex equations, the conversion may not be as straightforward and may require additional steps. Additionally, the conversion may not be applicable in all situations, so it is important to carefully consider if it is appropriate for a specific problem.

5. How can converting a partial derivative with respect to T to a partial derivative with respect to 1/T be applied in real-world scenarios?

One real-world application of this conversion is in thermodynamics, particularly in the study of phase transitions. For example, in the van der Waals equation of state, the partial derivative of pressure with respect to temperature can be converted to a partial derivative with respect to 1/T to better understand the behavior of a substance near its critical point. This conversion can also be useful in chemical reactions involving temperature as a variable, where 1/T may be a more relevant measure of the reaction rate.

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