# Tension

In thermodynamics, a tension, symbol X, is a partial derivative of the internal energy U with respect to an extensity, symbol x, other extensive quantities being kept constant. [1] Subsequently, with any extensity xi it is always possible to associate a tension variable Xi:

$X_i = \frac{\partial U}{\partial x_i}$

which is called the conjugate. As such, according to the first law, the change in internal energy of a system is given by:

$dU = \sum_{i=1}^k X_i dx_i$

Stated verbially, extensity is an energy “transfer variable”, defined such that when two systems A and B are brought into contact or interact, the assembly (A + B) being isolated, each system having different tensions, an exchange of energy results via a transfer of an extensity, such that the transfer ends when the tensions have equalized. [2]

First law example
The simplest example being the addition of a quantity of heat
δQ to a body, whereby the body expands, according to Boerhaave's law, pushing outward on the surrounding atmosphere, thus doing an amount of pressure volume work δW. The first law thus gives:

dU = δQ δW

or stated in terms of conjugate variables (with substitution of the integrating factor formulation of heat and work):

dU = Tds Pdv

where T and P are the tensions and S and V are the extensities. To restate, the measure of p
ressure P, for instance, is a quantity of tension whose conjugated extensity is volume V. In the case of transfer, an amount of energy in the form of pressure-volume work dW can be exchanged between two interactive systems (the body of steam, for example, and the body of surrounding atmosphere):

$dW = -P dV \,$

where the minus sign is the consequence of the "system-based" sign convention of dW > 0 if dV < 0.

References
1. Perrot, Pierre. (1998). A to Z of Thermodynamics. Oxford: Oxford University Press.
2. Richet, Pascal. (2001). The Physical Basis of Thermodynamics: with Applications to Chemistry, (pg. 11). Springer.