The diameter of a gas molecule is 2:4 x 1-10 m. Calculate the mean free path at NTP. Given Boltzmann constant k = 1-38 x 10-23 J molecule-l K-I.
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- Problem 3. The viral coefficients of a gas at 20 °C and 11.5 bar are B = -138 cm³ mol¹ and C=7222 cmº mol². Calculate the V (molar volume) Z (compressibility factor) of the gas. Use the equation below (R = 83.14 cm³ bar mol-¹ K-¹). PV 2 = ² = (1 + = + =) Z RTCalculate the internal energy, U internal -in Joules- of 3.2 moles of a monoatomic gas, if 3 Cv, monoatomic = R The temperature of the gas is T = 349K R = 8.31 J mol. K Hint: U₁ internal = n. Cy. TEstimate the mean free path of a nitrogen molecule in a cylinder nitrogen at 2 atm and a temperature of 17 oC. Take the radius to be roughly equal to 1 x10-10 m. Boltzmann constant 1.38 x10-23 J/K.
- The mean free path λ and the mean collision time T of molecules of a diatomic gas with molecular mass 6.00 x10^-25 kg and radius r=1.0x10^-10m are measured.From these microscopic data we can obtain macroscopic properties such as temperature T and pressure P? If yes, consider λ=4.32x10^-8m and T=3.00x10^-10s and calculate T and P.a)It's not possible.b)Yes,T=150K and P~2.04atm.c)Yes,T=150K and P~4.08atm.d)Yes,T=300K and P~4.08atm.e)Yes,T=300K and P~5.32atmf)Yes,T=400K and P~4.08atmg)Yes,T=400K and P~5.32atm.One cubic meter (1 m³) of mono-atomic ideal gas, is initially at room temperature and atmo- spheric pressure. The mass of a single molecule is 1.34 × 10-26 kg. 1. Find the root mean square speed, vrms, Oof the molecules by equating the kinetic energy of a single molecule to its average thermal energy. 2. Knowing that the gas obeys the Maxwell speed distribution 3/2 4nv²e-2T m mu2 D(v) = (0.2) 2πkT such that D(v)dv = 1. (0.3) Find the expression of the probability (do not do the integral) that a particular molecule is moving with a speed faster than 2000m/s. 3. The gas is heated at constant pressure until it triples in volume. Calculate the increase in its entropy.You are studying a gas known as "gopherine" and looking in the literature you find that someone has reported the partition function for one molecule of this gas, 5/2 AzT q(V, T) = ) %3D h?m Assume that the molecules are independent and indistinguishable. Derive the expressions for the energy, (E), for this gas. Give your answers in terms of N, kg, T. V and the constants A and B. O (E) = NkaT ㅇ (E) =D NkaT ㅇ (E) %3D NkaT- O (E) = ANKET - O (E) = - T ㅇ (E)=D 쑤-
- For one mole of Van der Waals gas, find Pc, Vc, and Tc from the condition T, dP / dV = 0 d'P / dV 2= 0 at constant temperature. Find the Pc Vc/ RTc ratio.Under the Equipartition Principle, in the high temperature limit, molecular heat capacity should be 1 -NR, where N is the number of atoms in 2 the molecule and R is the gas constant. O True O False8. The average energy of a mole of Einstein solid is given by: ӨЕ 3R0E Ē = 3R + 2 ӨЕ ет — 1 where E is the Einstein temperature and R the gas constant. Obtain expressions for the average energy at the limits of zero and infinite temperature and comment on your results.
- Consider 1 mole of a van der Waals gas. (i) Derive the expressions for the pressure, pc, temperature, Tc, and volume, Vc, in the critical point of a van der Waals gas in terms of parameters a, b and R. Derive the vdw equation in reduced coordinates p =,= 7, V = V/ (ii) (iii) Find how many times the gas temperature exceeds its critical temperature if the gas pressure is 4 times as high as critical pressure and the volume of gas is equal to twice the critical volume.For any gas, C C. - (+7), (7), Suppose you have one mole of a gas that obeys the equation of state, p(v- b) = represents the volume taken up by the molecules in the system, and 3.5 bars and T = 425 K for your gas sample. Hint: Use the equation of state to evaluate the partial derivatives. =RT, where b is the molar volume. Find the value of A if C-C₁=AR when P = 6What is the mean free path of molecules in an ideal gas in which the mean collision time is 3.00 x 1010 s, the temperature is 308K, and the mass of the molecules is 6.00 × 10-25 kg? Assume that the molecules are moving at their root- mean-square speeds. The Boltzmann constant is 1.38 x 10-23 J/K. Glve your answer in Angstroms ( 1 Angstrom = 10-10 m)