Derive the following relation,
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Thermodynamics, Statistical Thermodynamics, & Kinetics
- Cp− ̄Cv=T ̄V β^2/κ reduces to ̄CP− ̄CV=R for an ideal gas.arrow_forwardSince we will be dealing with partial derivatives later in the semester, this is a good opportunity to review this topic (see appendix C). Then evaluate the following partial derivatives (a) PV = nRT; (∂ P/∂V)T (b) r = (x2 + y2 + z 2 )1/2; (∂ r/∂y)x,zarrow_forward(a) Write expressions for dV and dp given that V is a function of p and T and p is a function of V and T. (b) Deduce expressions for d ln V and d ln p in terms of the expansion coefficient and the isothermal compressibility.arrow_forward
- P2D.2 Starting from the expression Cp − CV = T(∂p/∂T)V(∂V/∂T)p, use theappropriate relations between partial derivatives (The chemist’s toolkit 9 inTopic 2A) to show thatC CT V TV p( / )( / ) p VpT2− = ∂ ∂∂ ∂ Use this expression to evaluate Cp − CV for a perfect gas.arrow_forward3. At T = 300K, 1bar of ¹60¹80 in a 1m³ box (lengths ax ay = az = 1m) can be considered as an ideal gas. In that case, the average translational energy in each dimension for a molecule is given by: Ex = Ex = Ex = 1kT, where k = 1.38 x 10-23 J/K is the Boltzmann constant. The average rotational energy about an axis perpendicular to the O=O bond is: Erot=kT, Evib = KT. and the average vibrational energy is: Given that the fundamental vibrational frequency for ¹60¹80 is w = 4.741 x 10¹³ Hz, find the values of the quantum numbers nx, J, and u for an average ¹60¹80 molecule in this system.arrow_forwardA 0.250 mol nitrogen initially at 50 °C with a volume of 8.00 L is allowed to expand reversibly and adiabatically until its volume has doubled. Calculate the value of ΔHwhen Cp = 7/2R.arrow_forward
- The equation of state of a certain gas is given by p = RT/Vm + (a + bT)/Vm2, where a and b are constants. Find (∂Vm/∂T)p.arrow_forwardEstimate the values of γ = Cp,m/CV,m for gaseous ammonia and methane. Do this calculation with and without the vibrational contribution to the energy. Which is closer to the experimental value at 25 °C? Hint: Note that Cp,m − CV,m = R for a perfect gas.arrow_forwardInvestigate the dependence of pV on V for real gases.arrow_forward
- Show that CP = VT α (∂P/∂T) Sarrow_forwardThe cohesive energy density, U, is defined as U/V, where U is the mean potential energy of attraction within the sample and V its volume. Show that U = 1/2N2∫V(R)dτ where N is the number density of the molecules and V(R) is their attractive potential energy and where the integration ranges from d to infinity and over all angles. Go on to show that the cohesive energy density of a uniform distribution of molecules that interact by a van der Waals attraction of the form −C6/R6 is equal to −(2π/3)(NA2/d3M2)ρ2C6, where ρ is the mass density of the solid sample and M is the molar mass of the molecules.arrow_forwardThe heat capacity ratio of a gas determines the speed of sound in it through the formula cs = (γRT/M)1/2, where γ = Cp,m/CV,m and M is the molar mass of the gas. Deduce an expression for the speed of sound in a perfect gas of (a) diatomic, (b) linear triatomic, (c) nonlinear triatomic molecules at high temperatures (with translation and rotation active). Estimate the speed of sound in air at 25 °C. Hint: Note that Cp,m − CV,m = R for a perfect gas.arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,