Given rms speed, Q = (3RT/M)^1/2 Determine the temperature and pressure of 3.011

Question

Given rms speed, Q = (3RT/M)^1/2 Determine the temperature and pressure of 3.011x10^23 gas molecules, each with a mass of 6.64x10^26 kg, in a 5x10^-3 m^3 container if the rms speed is 560 m/s. There are molecules are distributed in 2 state with energies of 0 and 5epsilon(epsilon=1.381x10^-21J). There is no degeneracy for the first level. The second level is doubly degenerated. Calculate number of molecules in each state at the temperature you determined. A boy had a dream of riding a helium balloon to fly around the world. Assuming the boy had a weight of 50 kg, what is the minimum diameter of the balloon to lift the boy at room temperature (25degree C) and atmospheric pressure. Ammonia oxidation is a key step in the production of nitric acid, an important commodity chemical. Industrially, the reaction occurs at 750 - 900degree C in the presence of a Pt catalyst. Determine the Delta_rH for 4NH_3 (g) + 3O_2(g) rightarrow 4NO(g) + 6H_2O (g) at 1 bar and 800 degree C, provided with the following data: Standard enthalpy of formation lord NH_3, NO and H_2O are -45.9, 90.25 and -241.82 kJ mol^-1; the heat capacities of NH_3, O_2, NO, NO and H_2O are 35.06, 29.36, 29.8 and 33.6 J K^-1 mol^-1, respectively, and arc constant. How much ice at 0degree C is needed to mix with 100 g steam at 100degree C, reaching a final temperature of 13degree C? (b) Calculate the total change in entropy. Given that C_p, m = 75.0 J K^-1 mol^-1 for liquid water and Delta_fusH = 6.0 kJ mol^-1 at the normal melting point and Delta_vapH = 40.8 kJ mol^-1 taking molecular weight of water M_water =18.0 g mol^-1. Consider 5.0 moles of a monatomic perfect gas initially at T_1 = 300 K and p_1 = 5.0 bar. The heat capacit.es at constant volume and constant pressure are C_v =3/2 R and C_p =5/2 R, respectively. The system first follows a reversible isothermal compression step to reduce its volume to half of its original, determine the of p_2, V_2, w, q, Delta U, Delta H, Delta S, Delta G for this step. Make sure to show your work and provide units. Step (a) is is following by a constant pressure step to its original volume, determine the value of T_2, w, q, Delta U, Delta H, Delta S for this step. Make sure to show your work and provide units. The loop is closed by following a reversible constant volume step. Determine the value of w, q, Delta U, Delta H, Delta S for this step. Make sure to show your work and provide units.

Explanation / Answer

Answer (1)

Here the rms speed is given by

v = { (3RT) / M } (1/2)

Where v = root mean square velocity in m/sec = 560 m/s
R = ideal gas constant = 8.3145 (kg·m2/sec2)/K·mol
T = absolute temperature in Kelvin = ?
M = mass of a mole of the gas in kilograms. ( Molecular weight = Weight of mass / mol )

Now putting all value in above equation,

v = { (3RT) / M } (1/2)

v = { (3RT) / (m xN) } (1/2)

v ={ 3RT / m x N}(1/2)

T = (v2 x m xN ) / (3 x R )

T = { (560 m/s) 2 x 6.64 x 10-26 kg x 3.01 x 1023 (mol } / { 3 x 8.3145 (kg·m2/sec2)/K·mol ) }

T = 251292 x 10-3 K

T = 251 K

Now we know PV = nRT

      P = nRT / V

     P = ( 3.01 x 1023 (mol ) x 0.08206 L·atm/(mol·K) x 251 K ) / ( 5 L)

    P = 1.23 x 1023 atm

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