In thermodynamics, independent variable, as contrasted with a dependent variable, is a variable not dependent on any other variable.

Ideal gas law
In the case in which the state of a given body can be described by the ideal gas law, such as of the form:

 PV = RT \,

where the n, the molar particle count is set equal to one, then one can treat any one of the three variables, P,V, or T, as a function of the other two, an example being P as a function of V and T. In this case, the generic function notation would be:

 P = f(V,T) \,

or in a general sense:

 \text{dependent variable} = f(\text{independent variables}) \,

The independent variables then fix the condition of the gas in a given state. [1]

Partial differential equations
In a simple homogeneous fluid system, the state of the system can be defined by any two of the three variables V, P, and T. In this case, and given state function, such as internal energy U, will then be a function of two variables which have been chosen to represent the state. In order to avoid any misunderstanding as to which are the independent variables when it is necessary to differentiate partially, i.e. hold certain variable constant during the differentiation, the common practice is to enclose the partial derivative symbol in a parenthesis and place the variable that is to be held constant in the partial differentiation at the foot of the parenthesis. Hence the notation:

Independent variables vs dependent variables (partial derivatives)

means the derivative of U with respect to T, keeping V constant, when T and V are taken as independent variables. [2] In this case, according to the differential rule (find name), the differential of internal energy becomes:

 dU = \Bigg ( \frac{\partial U}{\partial T} \Bigg )_V + \Bigg ( \frac{\partial U}{\partial V} \Bigg )_T \,

See also
Conjugate variables
State variable
Intensive variable
Extensive variable

1. Clausius, Rudolf. (1850). “On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom”, Communicated to the Academy of Berlin, Feb.; Published in Poggendorff’s Annalen, March-April, Vol. lxxix, pgs. 368, 500, and Translated in the Philosophical Magazine, July 1851, Vol. ii. pgs. 1, 102.
2. Fermi, Enrico. (1936). Thermodynamics (pgs. 19-20). Prentice Hall.

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