Q.3 \"Mathematics is a language in which one cannot express unprecise or nebulou

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Q.3

"Mathematics is a language in which one cannot express unprecise or nebulous thoughts." - Henri Poincaire The pressure and temperature at the surface of the planet Mars are p = 600. Nt/m^2, T = 218. K. Based on this information, find the number density (N/V, in units of molecules/cm^3) of molecules in the Martian atmosphere at surface level. Consider 1.0000 mol of oxygen gas (O_2) at a temperature T = 300.0 K and confined in a volume V = 4.0000 L. Find the pressure of the gas using: a) the ideal gas law b) the van der Waals equation (for O_2, a = 1.364 L^2 middot atm/mol^2, b = 3.19 times 10^2 L/mol) Give your calculated values for pressure to 4 significant figures. Which of the above values would you expect to be closer to the true value for pressure for the conditions in the problem? There are two naturally occurring isotopes of chlorine, ^35Cl(34.9688 amu) and^37Cl(36.9651 amu). While modern measurements of the percentage of each isotope of chlorine in nature are based on mass spectrometry, an approximate value for this percentage can be found from measurements on gases containing chlorine. The density of hydrogen chloride gas (HCl), measured at T = 18.3 degree C and p = 774.5 torr, is p = 1.554 g/L. Based on this information, and assuming that HCl obeys the ideal gas law, find the percentage of the^35Cl isotope in the sample of HCl. Note that naturally occurring hydrogen can be assumed to be all the^1H isotope (1.0078 amu). Compare your result to the value determined using mass spectrometry, percent^35Cl = 75.5 %. The one dimensional Maxwell-Boltzmann distribution of speeds of gas molecules is given by the expression f(v_x)dV_x = (M/2 pi RT)^1/2 exp(-Mv^2_x/2RT)dv_x; - infinity

Explanation / Answer

Using ideal gas law we find molar mass of HCl for 1 mol . Then by subtracting mass of H ( 1.0078) from it we will get average mass of Cl.

Then by using average mass formula, percent of Cl-35 is determined .

Required conversions

Pressure into atm

P = 774.5 torr x 1 atm / 760 torr = 1.019079 atm

Temperature into K

T = 18.3 +273.15 = 291.45 K

Ideal gas law

PV = nRT

P is pressure , V is volume in L and R is gas constant = 0.08206 L atm / K mol

P = n/V RT

We know that n = m / M

Here m is the mass in g and M is he molar mass.

And m/V is the density

Lets plug it in above equation, we get

P M = d RT

Lets find M

M = dRT /p = 1.554 x 0.08206 x 291.45 K / 1.019079

=36.47025

Finding the average mass of Cl

Average mass of Cl = 36.47025 – mass of H = 36.47025-1.0078 = 35.46245

Lets use following formula to find the percent of Cl-35

We can assume x is the percent of Cl-35 and ( 1-x) would be the percent of Cl-37

35.46245 = mass of Cl-35 * x + mass of Cl-37 * ( 1-x)

35.46245 = 34.9688 * x + 36.9651 * ( 1-x)

x = 0.7527

Lets find it in 100

Percent of Cl-35 = 0.7527*100 = 75.3 %

The obtained value is close to what we get from spectroscopy

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