Question #75905

1 Answer
Nov 1, 2015

Answer:

Sodium.

Explanation:

The idea here is that you need to use the ideal gas law equation to find a relationship between the density of the gas and its molar mass.

As you know, the ideal gas law equation establishes a relationship between pressure and volume, on one side, and number of moles and temperature on the other.

#PV = nRT#

Here #R# represents the universal gas constant and is usually given as

#R = 0.082("atm" * "L")/("mol" * "K")#

Now, the number of moles of can be written as the ration between the mass of the sample and the gas' molar mass

#n = m/M_M#

Plug this into the idea lgas law equation to g et

#PV = m/M_M * RT#

Multiply both sides of the equation by #M_M# to get

#PV * M_M = m/color(red)(cancel(color(black)(M_M))) * color(red)(cancel(color(black)(M_M))) * RT#

#PV * M_M = m * RT#

Now look what happens when you divide both sides by the volume of the gas

#(Pcolor(red)(cancel(color(black)(V))) M_M)/color(red)(cancel(color(black)(V))) = m/V * RT#

#P * M_M = m/V * RT#

But since density is defined as mass per unit of volume, you will have

#P * M_M = rho * RT#

Now plug in your values and solve this equation for #M_M#, the molar mass of the gas - do not forget to convert the temperature from degrees Celsius to Kelvin and the pressure from torr to atm!

#M_M = (rho * RT)/P#

#M_M = (2.9 * 10^(-3)"g"/color(red)(cancel(color(black)("L"))) * 0.082(color(red)(cancel(color(black)("atm"))) * color(red)(cancel(color(black)("L"))))/("mol" * color(red)(cancel(color(black)("K")))) * (1000 + 273.15)color(red)(cancel(color(black)("K"))))/((10.)/760color(red)(cancel(color(black)("atm"))))#

#M_M = "23.009 g/mol" ~~ "23 g/mol"#

Now, the important thing to realize here is that you're dealing with a gaseous element, not a compound.

The high temperature at which the element is kept is a clue - in this case, sodium has a molar mass of approximately #"23 g/mol"#, and a boiling point of about #883^@"C"#.

This means that at #1000^@"C"#, sodium atoms will exist in the gaseous state. Therefore, the element you're looking for is sodium.