= Solution
Let the stellar radius be $R$ and assume zero external surface pressure. A spherical shell has mass $dm=4\pi r^2\rho\,dr$ and contributes $-Gm(r)\,dm/r$ to the <gravitational potential energy>. Using hydrostatic pressure support,
$$
E_{\rm grav}=-\int_0^R4\pi Gm\rho r\,dr
=4\pi\int_0^Rr^3P'\,dr
=-12\pi\int_0^Rr^2P\,dr
=\boxed{-3\langle P\rangle V}.
$$
The center boundary term vanishes and $P(R)=0$ eliminates the outer term. Nonzero external pressure would add $3P(R)V$.
The total energy is $(\alpha-3)\langle P\rangle V$, so energetic gravitational binding requires \b[$\alpha<3$]. For an isotropic gas, kinetic theory gives $P=(1/3)\int p v\,dn$. Nonrelativistic <kinetic energy> is $pv/2$, giving $U=3PV/2$ and $\alpha=3/2$. Ultrarelativistic energy is $pv$, giving $U=3PV$ and $\alpha=3$: the ultrarelativistic limit is marginal in this Newtonian virial balance.
For a white dwarf, <electron degeneracy pressure> becomes ultrarelativistic as density increases. Its equation of state then scales as $P=K\rho^{4/3}$. Pressure support and gravitational stress have the same radius scaling, leaving a limiting mass of order $(K/G)^{3/2}$ rather than allowing arbitrary mass to be supported by contraction. The <Chandrasekhar limit> is this maximum mass for a cold electron-degenerate star at specified composition, of order $(\hbar c/G)^{3/2}/(\mu_e m_p)^2$.
The virial calculation by itself does not establish a universal numerical maximum mass for every stellar composition and equation of state; its precise consequence is loss of binding/support as the pressure-bearing particles become ultrarelativistic. An accreting carbon-oxygen white dwarf approaching its limiting mass typically undergoes runaway carbon burning and thermonuclear disruption, a <Type Ia supernova>. For other compositions or evolution, electron captures can instead lead to collapse into a neutron star. The subsequent evolution needs that physical distinction.
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