The variation of Cp with pressure at constant temperature is given by :
\(-T\left(\frac{\partial^{2}V}{\partial T^{2}}\right)_{P}\)
The derivation is a neat illustration of how a Maxwell relation converts an awkward quantity into a measurable one.
Start from the definition of Cp: \(C_P = T\left(\frac{\partial S}{\partial T}\right)_P\).
Differentiate with respect to pressure at constant T:
\(\left(\frac{\partial C_P}{\partial P}\right)_T = T\left[\frac{\partial}{\partial P}\left(\frac{\partial S}{\partial T}\right)_P\right]_T\).
Swap the order of differentiation, which is permitted because S is a state function and its mixed second derivatives are equal:
\(\left(\frac{\partial C_P}{\partial P}\right)_T = T\left[\frac{\partial}{\partial T}\left(\frac{\partial S}{\partial P}\right)_T\right]_P\).
Apply the Maxwell relation derived from the Gibbs energy, \(\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P\). This is the key step, because it replaces an entropy derivative — which cannot be measured directly — with a volume derivative that can be obtained from the equation of state.
Substitute:
\(\left(\frac{\partial C_P}{\partial P}\right)_T = -T\left(\frac{\partial^{2}V}{\partial T^{2}}\right)_P\).
A useful corollary: for an ideal gas, \(V = \frac{nRT}{P}\) is linear in T, so the second derivative vanishes and \(C_P\) is independent of pressure — exactly as expected. Any pressure dependence of \(C_P\) is therefore a signature of non-ideality.
Hence the required expression is \(-T\left(\frac{\partial^{2}V}{\partial T^{2}}\right)_{P}\).
Similar Questions
The property measured in derivative thermogravimetric analysis (DTG) is :
Analysis of four different samples of an alloy yielded 12.12, 12.14, 12.10 and 12.22% of metal. The mean deviation of the result is :
For the first order consecutive reaction :
Which of the following statement is incorrect ?
A given reaction is fitted into the following Arrhenius form :
K2 = 6.0 × 1014 (S-1) . exp \(\left[-\frac{104.4\ (\mathrm{kJ\,mol^{-1}})}{RT}\right]\)
Value of the rate constant at very high temperature would be :
The following half cell represents, normal hydrogen electrode (NHE) :
If, we discharge the electrochemical cell rapidly, then :
E° value for various ions are as follows :
E°(MnO4-) = +1.51 V
E°(Ag/Ag+) = +0.7996 V
E°(Au/Au+) = +1.692 V
E°(Zn/Zn2+) = -0.761 V
Based on this data, permaganate can be used to oxidize :
Vapour pressure of 0.5 molal solution of non-volatile solute in organic solvent at 30°C would be :
(Given : vapour pressure of pure organic solvent is 100 torr; the molecular weight of organic solvent is 100 g mol-1).
Molar conductance of 0.01 M acetic acid was found to be 16 × 10-4 S m2 mol-1 at 30°C. Molar conductance of H+ and CH3COO- ions at infinite dilution are 350 × 10-4 S m2 mol-1 and 50 × 10-4 S m2 mol-1, respectively at same temperature. What percentage of acetic acid is dissociated at that concentration ?
Consider the reaction :
H2S(g) + \(\tfrac{3}{2}\) O2(g) → SO2(g) + H2O(g), ΔH° = 518.62 kJ
What is the effect of increase of temperature and pressure on the reaction ?
Important Questions from Thermodynamics
Two blocks of ice when pressed together join to form one block because
The ratio C p/C vof the specific heats at constant pressure and volume of a monoatomic ideal gas in two dimensions is
The total number of phonon modes in a solid of volume V is \(\int_{\rm{0}}^{{\rm{ω_ D}}} {{\rm{g}}\left( {\rm{ω }} \right)\,} {\rm{dω }}\) = 3N, where N is the number of primitive cells, ω Dis the Debye frequency and density of photon modes is g( ω ) = AV ω2 (with A > 0 a constant). If the density of the solid doubles in a phase transition, the Debye temperature θ D, will
The dispersion relation of a gas of non-interacting bosons in d dimensions is E(k) = ak s, where a and s are positive constants. Bose-Einstein condensation will occur for all values of
Example of thermoplastic among the following is
