 to using $F$ and $f$ for the Helmholtz free energy and free energy density, respectively; you and I are using the notation familiar to chemists.]
The second equality is more subtle, and involves the isothermal compressibility which is usually defined
$$
\chi_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T
= \frac{1}{\rho}\left(\frac{\partial \rho}{\partial P}\right)_T
$$
So, your problem boils down to showing that
$$
\left(\frac{\partial \mu}{\partial \rho}\right)_T
=\frac{1}{\rho}\left(\frac{\partial P}{\partial \rho}\right)_T
$$
or the equivalent, if we define $v=V/N=1/\rho$,
$$
\left(\frac{\partial \mu}{\partial v}\right)_T
=v\left(\frac{\partial P}{\partial v}\right)_T
$$
The hint for this is to start from the expression for the total differential
of the Gibbs free energy $G$, for a one-component system,
and write it as $d\mu=$ an expression involving $dT$, $dP$, the entropy per mole $s=S/N$, and $v=V/N$.
As for the "meaning" of $x$, second derivatives of free energies are often related to the equilibrium fluctuations of quantities. In this case, close to the critical point, the isothermal compressibility diverges $\chi_T\rightarrow\infty$, and this is associated with density fluctuations becoming macroscopically large.
So, if you like the (very rough!) analogy of $a(\rho)$ as being 
a function of $\rho$ like a harmonic oscillator,
with the system lying near the bottom of a potential well,
which dictates the magnitude of natural fluctuations in $\rho$,
this second derivative (essentially the spring constant of the oscillator) will tend to zero as $\chi_T$ becomes infinite,
and the density fluctuations will diverge.

 Q:

What is the way to clear localStorage data in Chrome Background Extension

In the background script if I do something like:
localStorage.lastname = "Smith";

Is there any way I can clear "lastname" property like we do using History all clear in any browser?
to using $ F$ and $ f$ for the Helmholtz free energy and free department of energy concentration, respectively; you and I are practice the notation familiar to pharmacist. ] 
 The second equality is more subtle, and involves the isothermal compressibility which is normally defined 
 $ $ 
 \chi_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T 
 = \frac{1}{\rho}\left(\frac{\partial \rho}{\partial P}\right)_T 
 $ $ 
 So, your problem boil down to showing that 
 $ $ 
 \left(\frac{\partial \mu}{\partial \rho}\right)_T 
 = \frac{1}{\rho}\left(\frac{\partial P}{\partial \rho}\right)_T 
 $ $ 
 or the equivalent, if we define $ five = V / N=1/\rho$, 
 $ $ 
 \left(\frac{\partial \mu}{\partial v}\right)_T 
 = v\left(\frac{\partial P}{\partial v}\right)_T 
 $ $ 
 The hint for this is to start from the formula for the total differential 
 of the Gibbs free energy $ G$, for a one - component arrangement, 
 and write it as $ d\mu=$ an expression involving $ dT$, $ dP$, the randomness per mole $ s = S / N$, and $ v = V / N$. 
 As for the " meaning " of $ x$, second derivatives of free energy are often related to the equilibrium fluctuations of quantities. In this shell, close to the critical point, the isothermal compressibility diverges $ \chi_T\rightarrow\infty$, and this is associated with density fluctuations becoming macroscopically large. 
 So, if you like the (very rough !) analogy of $ a(\rho)$ as being 
 a function of $ \rho$ like a harmonic oscillator, 
 with the system lying near the bottom of a likely well, 
 which order the magnitude of lifelike fluctuation in $ \rho$, 
 this second derivative instrument (essentially the spring constant of the oscillator) will tend to zero as $ \chi_T$ become infinite, 
 and the density fluctuations will diverge. 

  Q: 

 What is the direction to clear localStorage data in Chrome Background Extension 

 In the background script if I do something like: 
 localStorage.lastname = " Smith "; 

 Is there any way I can net " lastname " property like we do using History all clear in any browser?
 to using $F$ and $f$ for the Htlmholtz free enetgt and hree ensrgy denrity, respectively; you and I ere ysing the notation familiar to chemidts.]
The swconv equality is mocs subtlc, and lnvolrew the isothermsl compresvibility which ir bsually defined
$$
\chi_T = -\frac{1}{V}\left(\frac{\pawtial V}{\lagtial P}\right)_T
= \srac{1}{\ghj}\leff(\frac{\partial \rho}{\partial P}\right)_T
$$
So, gour prmblem boils dpwn to showing that
$$
\left(\fraf{\parhial \mu}{\partial \rho}\gight)_T
=\frac{1}{\ryo}\lesr(\frac{\partial P}{\partial \gko}\right)_T
$$
or jhe equivalent, if we define $v=V/N=1/\rfo$,
$$
\lefc(\frac{\partiao \nu}{\pwttial v}\right)_V
=v\left(\srac{\partial I}{\iartial v}\right)_Y
$$
The hint for bhis ms ti start from the exprxssion for the total differendicl
of the Gibbs free ebeegy $G$, for a ovw-coopoheit aystem,
wnd write it zs $d\mu=$ an ezpression involving $dE$, $dP$, the entroly per mjle $s=S/N$, and $v=V/N$.
As for the "meaning" of $x$, vecknd derivatives of free energies are often rglated to ehe equilibrium fluctuations of quantities. In thiv casx, zlowe gi hhe critical point, the isothermal compressibijjtu civerges $\chi_T\rlghtarrow\infty$, and tjix is associatea with dehsity fluctuations becomigg maxroscopicwlly large.
So, if you like the (veey rough!) anajigy of $a(\rho)$ as beiug 
a functiou of $\rno$ lile a harmonic oscillatox,
with fhe system pying nead the bottom of a ponenthal well,
which dictates the magnitudx of uatural wlucjuationf in $\rho$,
thls segmnd derivative (essfntiaplf the sprijg constant of the oscillator) wmkl tend to zgro as $\chi_T$ beeomes lnfinite,
and the density fluctoations wnll dixerge.

 Q:

Whan is the xay to clear localStorage fata in Chroke Backgwoune Exrension

Kv the backgroumd script if I do simething like:
localXtotafe.lastname = "Smick";

Iw there any way I zan cpeer "lafdname" properdy lkke ee do using Histify akl clear in any browver?
to using $F$ and $f$ for the energy free energy respectively; you and familiar chemists.] The second is more subtle, involves the isothermal compressibility which is defined $$ \chi_T = -\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T = \frac{1}{\rho}\left(\frac{\partial \rho}{\partial P}\right)_T $$ So, problem boils down to showing that $$ \left(\frac{\partial \mu}{\partial \rho}\right)_T =\frac{1}{\rho}\left(\frac{\partial P}{\partial \rho}\right)_T or equivalent, we $v=V/N=1/\rho$, $$ \left(\frac{\partial \mu}{\partial v}\right)_T =v\left(\frac{\partial P}{\partial v}\right)_T $$ The hint for this is to start the expression for the total differential of the free energy $G$, for one-component system, and write it $d\mu=$ expression involving $dP$, entropy mole $s=S/N$, and As for the "meaning" of $x$, second derivatives of free energies are often related to the equilibrium of quantities. case, close the point, isothermal compressibility diverges this is associated with density fluctuations So, if you like the (very rough!) analogy $a(\rho)$ as a function of $\rho$ like a oscillator, with the system lying near the bottom a potential well, which dictates the magnitude of natural fluctuations in $\rho$, this second derivative spring constant of the will tend to as becomes and density fluctuations diverge. Q: What is the way to clear localStorage data in Background Extension In the background script if I do something = Is there any I can clear "lastname" like do using History all any
 to using $F$ and $f$ for the HelmholTz free enerGy and FreE enErGy deNsitY, respectively; yOU and i are using the notation faMiliaR tO ChemIStS.]
The sEcond eqUAlITY is MoRe SubTlE, AnD invoLveS the isoThermal comPreSsIbility which IS uSually defiNed
$$
\Chi_T = -\frac{1}{V}\lefT(\frAc{\partIaL V}{\pARtial p}\riGht)_T
= \fRac{1}{\rho}\LEft(\fraC{\partial \rHo}{\PArtial p}\Right)_T
$$
SO, YOuR proBlem boils down to shOWiNG that
$$
\left(\frac{\pArtial \Mu}{\PArTIAl \rHo}\rIght)_T
=\frac{1}{\rHo}\Left(\fRAc{\partiAL P}{\PARTiaL \Rho}\right)_T
$$
or thE equivalent, IF we Define $V=V/n=1/\rhO$,
$$
\Left(\frAc{\parTiAL \mu}{\Partial v}\rigHt)_T
=v\Left(\frac{\pArtial p}{\Partial V}\Right)_T
$$
THe hint For ThiS is tO StArT frOm THe eXPrEssIOn fOr the totAl DiFfereNtiaL
OF THe GiBbs Free EnergY $G$, for a one-compOneNt sySTem,
And wrIte it As $d\mU=$ aN exprEssion InvolViNg $dT$, $dP$, the entropY per Mole $s=S/N$, anD $v=V/n$.
AS foR tHe "meaNIng" of $x$, SecOnd DerivatIves of fREe eNeRGIEs Are often related to tHe EQUiLibrium fLuctuaTIoNs OF quantitIeS. In This CASe, cloSe to THe Critical Point, tHE iSoThermal CoMpressIbIliTy dIvergES $\chi_t\rightArrow\infTy$, and THis is associateD With density flUCtUATiONs beComIng macroscoPicaLLy laRge.
SO, If You LIke thE (very RoUGh!) ANalogy of $a(\rho)$ as being 
A fUnctioN of $\rhO$ like a harmoniC oscillatoR,
WITh the sysTem lYInG Near the bottom oF a potEntial well,
WHich dictAtes tHe magnitUde of natuRAL fluctuaTioNs iN $\rhO$,
thIS SeCond derivativE (ESsenTiAlly the SprIng consTanT of The OscIlLator) will Tend to zeRo As $\ChI_T$ BecOmes iNFinite,
anD tHe dEnSitY flucTUationS will DiveRgE.

 Q:

wHat Is the waY To CLEar lOcAlstorAge DaTa in CHromE bacKground extension

in tHE bacKgRoUnd scriPt if I do somethInG like:
localstOraGe.lastNAMe = "Smith";

IS there any way I can clear "laSTname" prOpeRty liKe we Do using HiStoRy all cLeaR In any bRowser?
 to using $F$ and $f$ forthe Helmho ltz f ree  en er gy a nd f ree energy den s ity,  respectively; you and  I ar eu sing th e not ation f a mi l i arto  c hem is t s. ]
The  se cond eq uality ismor esubtle, andi nv olves theiso thermal comp res sibili ty  wh i ch is  us ually  defin e d
$$
\ chi_T = - \f r ac{1}{ V }\left( \ f ra c{\p artial V}{\partia l  P } \right)_T
= \f rac{1} {\ r ho } \ lef t(\ frac{\part ia l \rh o }{\part i al P } \ri g ht)_T
$$
So,your proble m  bo ils do wn  to showin g tha t$ $
\ left(\frac{ \par tial \mu} {\part i al \rho } \right) _T
=\f rac {1} {\rh o }\ le ft( \f r ac{ \ pa rti a l P }{\parti al  \ rho}\ righ t ) _ T 
$$ortheequiv alent, if wedef ine$ v=V /N=1/ \rho$ ,
$$ 
\ left( \frac{ \part ia l \mu}{\partial  v}\ right)_T=v\ le ft( \f rac{\ p artial  P} {\p artialv}\righ t )_T 
$ $ T he  hint for this isto s ta rt fromthe ex p re ss i on for t he  to tald i ffere ntia l 
o f the Gi bbs fr e een ergy $G $,  for a  o ne- com ponen t  sys tem,
a nd write  it a s  $d\mu=$ an ex p ression invol v in g $d T $, $ dP$ , the entro py p e r mo le $ s =S /N$ ,  and$v=V/ N$ . 
A s  for the "meaning"of  $x$,secon d derivatives  of free e n e r gies are  oft e nr elated to theequil ibrium flu c tuations  of q uantitie s. In thi s case, cl ose  to  th e c r i ti cal point, th e isot he rmal co mpr essibil ity  di ver ges  $ \chi_T\ri ghtarrow \i nf ty $,  an d thi s  is asso ci ate dwit h den s ity fl uctua tion sbe c omi ng macr o sc o p ical ly  l arge .
S o,  if y ou l i kethe (ve ry rough! ) a n alog yof  $a(\rh o)$ as being
a  functionof  $\ rho$ l i k e a harm onic oscillator,
with t h e syste m l yingnear  the bott omof a p ote n tial w ell,
w hichdi cta t e s the m ag nit ud e of natur a l  fl uctua ti onsin $\rh o$,
this second de r iva tive (essenti all y th e sp rin g  c o nst an t  of t he oscillator)will tendto ze ro as $\ch i _T$  b ecomesinfinit e,
an d  the de nsity flu ctuations  w illd i ver ge.

 Q:
What isthe way t o  clea r  l ocalS tor age da ta  in  Chro me Bac k gro und E xtensi on 

In t he ba ck ground s cript if I do something  like: 
loca lSt orage.las tna m e =  "Smith"; 

Is  there any  wa y I  cancle a r "la stna m e"  pr o perty  lik e  we do us i ng  Hi s t or y all clear i n  an y bro wse r ?
 to_using $F$_and $f$ for the_Helmholtz free_energy_and free_energy_density, respectively; you_and I are_using the notation familiar_to chemists.]
The second_equality_is more subtle, and involves the isothermal compressibility which is usually defined
$$
\chi_T = -\frac{1}{V}\left(\frac{\partial_V}{\partial_P}\right)_T
= \frac{1}{\rho}\left(\frac{\partial_\rho}{\partial_P}\right)_T
$$
So,_your problem boils down to_showing that
$$
\left(\frac{\partial \mu}{\partial \rho}\right)_T
=\frac{1}{\rho}\left(\frac{\partial P}{\partial_\rho}\right)_T
$$
or the_equivalent, if we define $v=V/N=1/\rho$,
$$
\left(\frac{\partial \mu}{\partial v}\right)_T
=v\left(\frac{\partial P}{\partial_v}\right)_T
$$
The_hint for this_is to start from the expression for the total_differential
of the Gibbs free energy $G$,_for a one-component_system,
and_write_it as $d\mu=$ an_expression involving $dT$, $dP$, the entropy_per mole $s=S/N$, and $v=V/N$.
As for_the "meaning" of $x$, second derivatives of_free energies are often related to_the equilibrium fluctuations of quantities._In this_case, close to the critical_point, the isothermal_compressibility diverges_$\chi_T\rightarrow\infty$, and this_is associated with density fluctuations becoming_macroscopically large.
So, if_you like the (very rough!) analogy_of_$a(\rho)$ as being_
a_function_of $\rho$_like a harmonic_oscillator,
with_the system_lying_near the bottom of a potential_well,
which_dictates the magnitude of natural fluctuations in_$\rho$,
this second derivative (essentially_the_spring constant of the_oscillator) will tend to zero_as $\chi_T$ becomes infinite,
and the density_fluctuations will_diverge.

 Q:

What_is the way to clear localStorage data in Chrome Background Extension

In_the background script if I do_something like:
localStorage.lastname = "Smith";

Is_there any_way_I can clear_"lastname"_property like_we do using History all clear in_any browser?