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:

“Jesus wept; Voltaire smiled. From that divine tear and from that human smile is derived the grace of present civilization.”
—Victor Hugo (1802–1885)

“To conclude, The Light of humane minds is Perspicuous Words, but by exact definitions first snuffed, and purged from ambiguity; Reason is the pace; Encrease of Science, the way; and the Benefit of man-kind, the end.”
—Thomas Hobbes (1579–1688)

“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)