Total and Exact Differentials

  • We now know that \(C_V \equiv \left( \frac {\delta U}{\delta T} \right)_V\)
  • This suggests that \(U\) is very dependent on \(V\) and \(T\)
  • This suggests that to change \(U\) we can change these variables
  • This gives us a total differential \[dU = \left( \frac{\delta U}{\delta V} \right)_T dV + \left( \frac{\delta U}{\delta T} \right)_V dT\]
  • This means that \[\Delta U = \int_{V_1}^{V_2} \left( \frac{\delta U}{\delta V} \right)_T dV+ \int_{T_1}^{T_2} \left( \frac{\delta U}{\delta T} \right)_V dT\]

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Exact Differentials

  • Let’s say we have a differential \[ df(x,y) = P\,dx + Q\,dy \]
  • If these variable obey the Euler Relation \[ \left(\frac{\partial P}{\partial y}\right)_x = \left(\frac{\partial Q}{\partial x}\right)_y \] then that differential will be an exact differential
  • We can illustrate this with the ideal gas law \[ P(V,T) = \frac{RT}{V} \]
  • The differentials of all the state functions will be exact.

Compressivity and Expansivity

Isothermal Compressibility (\(\kappa_T\))

  • It is important that we understand how compressible substances are
  • To quantify how compressible substances are we can look at the fractional differential change in volume due to a change in pressure \[ \kappa_T \equiv -\frac{1}{V} \left(\frac{\partial V}{\partial P}\right)_T \]
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Isobaric Thermal Expansivity (\(\alpha\))

  • It is also important to understand how the volume of a substance responds to temperature
  • To quantify this, we define the isobaric thermal expansivity (sometimes called the expansion coefficient) \[ \alpha \equiv \frac{1}{V} \left(\frac{\partial V}{\partial T}\right)_P \]

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Partial Derivatives: The Reciprocal Rule

  • Consider a function \(F\) of three variables \(x\), \(y\), and \(z\) such that we can write \(F(x,y,z)=0\)
  • This means that we can specify the system by knowning two of the three variables, or \[ dz = \left(\frac{\partial z}{\partial x}\right)_y dx + \left(\frac{\partial z}{\partial y}\right)_x dy \text{ and } dy = \left(\frac{\partial y}{\partial x}\right)_z dx + \left(\frac{\partial y}{\partial z}\right)_x dz \]
  • After a lot of calculus we find that \[ 1=\left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial z}\right)_x \text{ or } \left(\frac{\partial z}{\partial y}\right)_x = \frac{1}{\left(\frac{\partial y}{\partial z}\right)_x}\]

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Partial Derivatives: The Cyclic Permutation Rule

  • Along similar lines to the last we, we can generate a very useful partial differential relationship

\[ \left(\frac{\partial z}{\partial x}\right)_y = -\left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial x}\right)_z \]

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Example 4.1

Derive an expression for \[ \frac{\alpha}{\kappa_T} \] in terms of derivatives of thermodynamic functions using out partial derivative relations.

The Joule Experiment