Exact Differential - Some Useful Equations Derived From Exact Differentials in Two Dimensions | Equations Derived Exact Differentials Dimensions

Some Useful Equations Derived From Exact Differentials in Two Dimensions

(See also Bridgman's thermodynamic equations for the use of exact differentials in the theory of thermodynamic equations)

Suppose we have five state functions, and . Suppose that the state space is two dimensional and any of the five quantities are exact differentials. Then by the chain rule

$(1)~~~~~ dz = \left(\frac{\partial z}{\partial x}\right)_y dx+ \left(\frac{\partial z}{\partial y}\right)_x dy = \left(\frac{\partial z}{\partial u}\right)_v du +\left(\frac{\partial z}{\partial v}\right)_u dv$

but also by the chain rule:

$(2)~~~~~ dx = \left(\frac{\partial x}{\partial u}\right)_v du +\left(\frac{\partial x}{\partial v}\right)_u dv$

and

$(3)~~~~~ dy= \left(\frac{\partial y}{\partial u}\right)_v du +\left(\frac{\partial y}{\partial v}\right)_u dv$

so that:

$(4)~~~~~ dz = \left[ \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial u}\right)_v + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial u}\right)_v \right]du$

$+ \left[ \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial v}\right)_u + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial v}\right)_u \right]dv$

which implies that:

$(5)~~~~~ \left(\frac{\partial z}{\partial u}\right)_v = \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial u}\right)_v + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial u}\right)_v$

Letting gives:

$(6)~~~~~ \left(\frac{\partial z}{\partial u}\right)_y = \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial u}\right)_y$

Letting gives:

$(7)~~~~~ \left(\frac{\partial z}{\partial y}\right)_v = \left(\frac{\partial z}{\partial y}\right)_x + \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial y}\right)_v$

Letting, gives:

$(8)~~~~~ \left(\frac{\partial z}{\partial y}\right)_x = - \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial y}\right)_z$

using ( $\partial a/\partial b)_c = 1/(\partial b/\partial a)_c$ gives the triple product rule:

$(9)~~~~~ \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x =-1$

Read more about this topic: Exact Differential

Famous quotes containing the words derived, exact and/or dimensions:

“Ex oriente lux may still be the motto of scholars, for the Western world has not yet derived from the East all the light which it is destined to receive thence.”
—Henry David Thoreau (1817–1862)

“The first moments of sleep are an image of death; a hazy torpor grips our thoughts and it becomes impossible for us to determine the exact instant when the “I,” under another form, continues the task of existence.”
—Gérard De Nerval (1808–1855)

“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)