Interpretation:
The validation of the statement that the value of constant
Concept introduction:
The probability distribution function of the velocities of the gas molecules in each dimension is given by
Answer to Problem 19.22E
The given statement that the value of constant
Explanation of Solution
The distribution function
It is given that the value of constant
Substitute the value of
Keep all the constant terms out of the differentiation.
The differentiation of the exponential term with respect to
Substitute equation (3) into equation (2).
Cancel the common terms and thus, the value of
The value of
Similarly for
It is given that the value of constant
The term
Since
The given statement that the value of constant
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Chapter 19 Solutions
Physical Chemistry
- The 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_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_forward20.7 cm3 of a pure vapour, at 1.136 atm, and 98.3 °C has a mass of 0.0603 g.Calculate the molar mass of the substance.arrow_forward
- The volume of a certain liquid varies with temperature asV = V′{0.75 + 3.9 × 10−4(T/K) + 1.48 × 10−6(T/K)2}where V′ is its volume at 300 K. Calculate its expansion coefficient, α, at320 K.arrow_forwardCompute for Delta U, Delta H and W if 5 moles of an ideal diatomic gas undergoes an isochoric processes (V = k) whose Cv = (5 / 2) R and Cp = (7/ 2) R from T1 = 273.15 K to T2 = 298.15 K.arrow_forwardIdentify the systems for which it is essential to include a factor of 1/N! on going from Q to q : (i) a sample of carbon dioxide gas, (ii) a sample of graphite, (iii) a sample of diamond, (iv) ice.arrow_forward
- Rearrange the van der Waals equation of state, p = nRT/(V − nb) − n2a/V2(Topic 1C) to give an expression for T as a function of p and V (with n constant). Calculate (∂T/∂p)V and confirm that (∂T/∂p)V = 1/(∂p/∂T)V.arrow_forwardSuggest a physical interpretation of the dependence of the Gibbs energy on the pressure.arrow_forwardWhat is Isothermal Expansion of a van der Waals Gas?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_forwardA sample of nitrogen of mass 3.12 g at 23.0 °C is allowed to expand reversibly and adiabatically from 400 cm3 to 2.00 dm3. What is the work done by the gas?arrow_forwardThe differential for the Gibbs function, G, at constant composition is: ?G = −S?T + Vdp Using the criterion for exact differentials, write the Maxwell relation that is derived from this equation.arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,