(16.2) (a) Derive the following general relations (i) ()₁ = − [T(),−1], др U C'v (ii) () --() ᎥᎢ (iii) др Н = ¿[T(),-V]. Р In each case the quantity on the left-hand side is the appropriate thing to consider for a particular type of expansion. State what type of expansion each refers to. (b) Using these relations, verify that for an ideal gas (@T/V)₁ = 0 and (T/ P)H = 0, and that (T/ǝV)s leads to the familiar relation pv = constant along an isentrope (a curve of constant entropy).
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- Problem 1: In statistical mechanics, the internal energy of an ideal gas is given by: N. aNkB 2/3 (3NKB U = U(S,V) = е where a is a constant. 1- Show that the variation of the internal energy is given by: 2 dS - \3V 2 dU = dV \3NkB 2- Using the fundamental relation of thermodynamic dU = T.ds – p. dV, show that the equation of state PV = nRT follows from the first expression of U.For a one-dimensional problem of a gas medium confined between two parallel plates, determine the direct exchange factor gig; (i.e., per unit area), where medium zone i is an infinitely large slab of thickness Az (located between z, sz s zi + Az), and j is another medium zone (located at z, sz s 2; + Az).(3) Partition function of an ideal gas comprising N particles of mass m in a volume V, which is in equilibrium with a heat bath at temperature T (or Bß = 1/kT), was calculated in lecture as: %3D 3N/2 VN (2nmkT)³N/2 VN m Z = N!h3N N! (2nh?ß, using the integral e-cx2 Now consider an ideal gas comprising N particles of two-atom molecule of mass m. Kinetic energy is separated as E = Etransl + Erot; where 1 1 Etransl = 2m -(P를 +pg + p2); Erot (på + p),0 <θ< T,0<Φ< 2π. 21 sin? 0 (a) Calculate the partition function for ideal gas comprising two-atom molecules. (b) Calculate the heat capacity.
- Question 2 Quickly estimate the change in enthalpy /mol) to go from CO2 at 325K and 1 bar to 298K and 97 bar. Hint: of. Figure 2.6c.The vapour pressure of water can be given by: , being p the vapour pressure in mmHg and T in K. Assuming the normal boiling temperature of water is 100 ºC, calculate the value of A. Vacuum has been created in a 10 L vessel, and 10 g water have been introduced. We will assume that vapour is an ideal gas, and the density of liquid water is 1 g/cm3 at any temperature. Calculate the mass of water in liquid phase at 50 ºC: a) Assuming the volume of the liquid is very small and therefore we can ignore it; b) considering the volume occupied by the liquid water. Results:. A = 8.564; a) 9.105 g ; b) almost the same!the partition function of an ideal gas of diatomic molecules in an external electric field & is [g(V, T, 8)]" Q(N, V, T, 8) N! where (2mmkT 312 (87 IkT -hv/2kT e q(V,T, 8)= V{ h2 (kT' (µ8 sinh kT) h2 (1 – e-hv/kT) Here I is the moment of inertia of the molecule; v is its fundamental vibrational frequency; and u is its dipole moment. Using this partition function along with the thermodynamic relation, dA = -S dT –p dV – M de where M=Nū, where u is the average dipole moment of a molecule in the direction of the external field &, show that kT] coth kT, Sketch this result versus & from & =0 to & =∞ and interpret it.
- Help please answer this calculus Question: The pressure P and volume V of an expanding gas are related by the formula PV^b =c, where b and C are constants (this holds in adiabatic expansion, with or without loss). Find the dP/dt if b=1.6, P=10 kPa, V=60 cm^3, and dV/dt =80 cm^3/min. (Use symbolic notation and fractions where needed).Using the method of the preceding problem, estimate the fraction of nitric oxide (NO) molecules at a temperature of 250 K that have energies between 3.451021 J and 3.501021 J. `(a) What is the volume occupied by 1.00 mol of an ideal gas at standard conditions—that is, 1.00 atm (= 1.01* 10^5 Pa) and 273 K? (b) Show that the number of molecules per cubic centimeter (the Loschmidt number) at standard conditions is 2.69 *10^9.
- Ex. 50 The absolute temperature of a gas 1s increased three times. The root mean square velocity of the gas molecules will be ... ..... (a) 3 times (b) 9 times 1 (c) times (d) 3 times 3.40000cm, and at 1000C its length is 40060cm. (a) What is the linear expansion coefficient of the alloy? (b) what is the temperature when its length is 39975cm? (10) A hole with diameter of 2cm is drilled through a stainless steel pate at room temperature (20OC). By what fraction (Dr/r) will the hole be larger when the plate is heated to 1500C? (11) Calculate the fractional change in volume (Dv/v) of an aluminum bar that undergoes a change in temperature of 300C.Pressure and bulk modulus of an electron gas. (a) Derive a relation connecting the pressure and volume of an electron gas at 0 K. Hint: Use the result of Problem 1 and the relation between er and electron concentration. The result may be writ- ten as p =(U/V). (b) Show that the bulk modulus B = –V(dp/əV) of an electron gas at 0 K is B = 5p/3 = 10U/9V. (c) Estimate for potassium, using Table 1, the value of the electron gas contribution to B.