Give the temperature T of 1 mole of ideal gas as a function of the pressure P, volume V, and the gas constant R and give the internal energy U of a rigid diatomic ideal gas as a function of its temperature T and the gas constant R.
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Give the temperature T of 1 mole of ideal gas as a function of the pressure P, volume V, and the gas constant R and give the internal energy U of a rigid diatomic ideal gas as a function of its temperature T and the gas constant R.
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- A diatomic ideal gas at pressure p and volume V is expanding to three times its initial volume under constant pressure. In terms of p and V, calculate the change in internal energy ΔU.In this problem you are to consider an adiabaticexpansion of an ideal diatomic gas, which means that the gas expands with no addition or subtraction of heat. Assume that the gas is initially at pressure p0, volume V0, and temperature T0. In addition, assume that the temperature of the gas is such that you can neglect vibrational degrees of freedom. Thus, the ratio of heat capacities is γ=Cp/CV=7/5. Note that, unless explicitly stated, the variable γshould not appear in your answers--if needed use the fact that γ=7/5 for an ideal diatomic gas. Find an analytic expression for p(V), the pressure as a function of volume, during the adiabatic expansion. Express the pressure in terms of V and any or all of the given initial values p0, T0, and V0. p(V) = __________In this problem you are to consider an adiabaticexpansion of an ideal diatomic gas, which means that the gas expands with no addition or subtraction of heat. Assume that the gas is initially at pressure p0, volume V0, and temperature T0. In addition, assume that the temperature of the gas is such that you can neglect vibrational degrees of freedom. Thus, the ratio of heat capacities is γ=Cp/CV=7/5. Note that, unless explicitly stated, the variable γshould not appear in your answers--if needed use the fact that γ=7/5 for an ideal diatomic gas. A) Find an analytic expression for p(V), the pressure as a function of volume, during the adiabatic expansion. Express the pressure in terms of V and any or all of the given initial values p0, T0, and V0. p(V) = __________ B) At the end of the adiabatic expansion, the gas fills a new volume V1, where V1>V0. Find W, the work done by the gas on the container during the expansion. Express the work in terms of p0, V0, and V1. Your…
- Consider an ideal monatomic gas. Here, take N as constant. We can take any two arguments like (p, V) or (E, V) or (p, T) and use them as variables representing the macro state. Using E = 3 / 2Nk (B) T for a monatomic ideal gas: A) Take (E, V) as macroscopic variables and express dW and dQ in terms of these variables (ie, dW = (...) dE + (...) dV and dQ = (...) dE + (. ..) Find the dV expressions). B) Check that dW and dQ are not full differentials. Prove that dQ / T is the exact differential. C) Repeat the above procedure, taking (p, T) as macroscopic variables.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.A sample of 2.37 moles of an ideal diatomic gas experiences a temperature increase of 65.2 K at constant volume. Find the increase in internal energy if translational, rotational, and vibrational motions are possible.
- Show that the isothermal compressibility KT and the adiabatic compressibility KS of a Bose ideal gas are given by in the picture. where n(=N/V) is the density of the particles in the gas.A sample of a monatomic ideal gas occupies 5.00 L at atmospheric pressure and 300 K (point A in the figure below). It is warmed at constant volume to 3.00 atm (point B). Then it is allowed to expand isothermally to 1.00 atm (point C) and at last compressed isobarically to its original state. Р (atm) 3 B 1 V (L) 5 10 15 (a) Find the number of moles in the sample. moles (b) Find the temperature at point B. K (c) Find the temperature at point C. (d) Find the volume at point C. L (e) Now consider the processes A - B, B → C, and C- A. Describe how to carry out each process experimentally.You measure the average free path λ and the average collision time τ of the molecules of a diatomic gas of molecular mass 6.00 × 10-²⁵ kg and radius r = 1.0 x 10-¹⁰ m. From these microscopic data can we obtain macroscopic properties such as temperature T and pressure P? If so, consider λ = 4.32 x 10-⁸ m and τ = 3.00 x 10-¹⁰ s and calculate T and P. indicate the correct answer: 1- Not possible2- Yes, T =150 K and P ~ 2.04 atm.3- Yes, T = 150 K and P ~ 4.08 atm.4- Yes, T = 300 K and P ~ 4.08 atm.5- Yes, T = 300 K and P ~ 5.32 atm6- Yes, T = 400 K and P ~ 4.08 atm.7- Yes, T = 400 K and P ~ 5.32 atm. obs.: If necessary, consider: R = 8.314 J/mol∙K1 cal = 4.19 Jkb =1,38 x 10⁻²³ m² kg s⁻² K⁻¹
- A sample of 2.37 moles of an ideal diatomic gas experiences a temperature increase of 65.2 K at constant volume. Find the increase in internal energy if only translational and rotational motions are possible.A monatomic ideal gas (γ = 5/3) is contained within a perfectly insulated cylinder that is fitted with a movable piston. The initial pressure of the gas is 1.5 bar. The piston is pushed so as to compress the gas, with the result that the Kelvin temperature doubles. What is the final pressure of the gas? (ANSWER IS 8.5 bar)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.