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. TThe 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.
- Estimate 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.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-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.
- 8. 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.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.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 = 6
- 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 FalseConsider 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.What 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)