= Intensive quantity
= Intensive variable
{synonym}
An <intensive quantity> is unchanged under replication at the same <thermodynamic equilibrium>. <Temperature>, <pressure>, <chemical potential> and the <mass density> of a uniform substance are examples. Independence from sample size does not mean independence from state: a material's <mass density> can change with <temperature> or <pressure>.
The relation to <extensive quantities> is particularly elegant for a differentiable bulk fundamental relation. Put $\mathbf x=(S,V,n)$ and write
$$
U(\lambda\mathbf x)=\lambda U(\mathbf x).
$$
Taking a <partial derivative> with respect to $x_i$ and cancelling $\lambda$ gives
$$
\frac{\partial U}{\partial x_i}(\lambda\mathbf x)
=\frac{\partial U}{\partial x_i}(\mathbf x).
$$
The equilibrium differential $dU=T\,dS-p\,dV+\mu\,dn$ identifies these derivatives with <temperature>, minus <pressure> and molar <chemical potential>. They are degree-zero <homogeneous functions>. The same degree-one assumption yields the <Euler relation in thermodynamics>, but a <surface energy> contribution requires extra care.
An <intensive quantity> generally has no meaningful sum across samples: two equal <temperatures> do not add to a doubled <temperature>. Neither is the final <temperature> after thermal contact generally an unweighted average; <heat capacities> and the <first law of thermodynamics> determine the balance. Ratios of additive inventories lead to the useful weighted rules in <normalizing extensive quantities>.
Back to article page
