# Physics:Partial specific volume

The partial specific volume $\displaystyle{ \bar{v_i}, }$ express the variation of the extensive volume of a mixture in respect to composition of the masses. It is the partial derivative of volume with respect to the mass of the component of interest.

$\displaystyle{ V=\sum _{i=1}^n m_i \bar{v_i}, }$

where $\displaystyle{ \bar{v_i} }$ is the partial specific volume of a component $\displaystyle{ i }$ defined as:

$\displaystyle{ \bar{v_i}=\left( \frac{\partial V}{\partial m_i} \right)_{T,P,m_{j\neq i}}. }$

The PSV is usually measured in milliLiters (mL) per gram (g), proteins > 30 kDa can be assumed to have a partial specific volume of 0.708 mL/g.[1] Experimental determination is possible by measuring the natural frequency of a U-shaped tube filled successively with air, buffer and protein solution.[2]

## Properties

The sum of partial specific volumes of a mixture or solution is an inverse of density of the mixture namely the specific volume of the mixture.

$\displaystyle{ v = \sum_i w_i\cdot \bar{v_i} = \frac {1}{\rho} }$
$\displaystyle{ \sum_i \rho_i \cdot \bar{v_i} = 1 }$

## Notes

1. Buxbaum, Engelbert (2011). "25: Centrifugation". Biophysical Chemistry of Proteins. New York, Dordrecht, Heidelberg, London: Springer. pp. 237–249. doi:10.1007/978-1-4419-7251-4_25. ISBN 978-1-4419-7250-7.
2. Kratky, O.; Leopold, H.; Stabinger, H. (1973). "5: The determination of the partial specific volume of proteins by the mechanical oscillator technique". Meth. Enzymol. Vol. 27: Enzyme Structure Part D. pp. 98–110. ISBN 978-0-12-181890-6.