# When the temperature of a rigid hollow sphere containing 685 L of helium gas is held at 621 K, the pressure of the gas is 1.89*10^3 kPa. How many moles of helium does the sphere contain?

Mar 8, 2016

$\text{251 moles}$

#### Explanation:

In order to find the number of moles of gas present in that sample of helium under those conditions for pressure and temperature, you must use the ideal gas law equation

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} P V = n R T \textcolor{w h i t e}{\frac{a}{a}} |}}} \text{ }$, where

$P$ - the pressure of the gas
$V$ - the volume it occupies
$n$ - the number of moles of gas
$R$ - the universal gas constant, usually given as $0.0821 \left(\text{atm" * "L")/("mol" * "K}\right)$
$T$ - the absolute temperature of the gas

Now, before plugging in your values into the ideal gas law equation, make sure that the units given to you for pressure, temperature, and volume match the units used in the expression of the universal gas constant.

As you can see, $R$ uses atm as the unit for pressure. The problem gives you the pressure of the gas expressed in kPa. This means that you're going to have to convert the pressure from kPa to atm by using the conversion factor

$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\text{1 atm" = 1.01325 * 10^3"kPa}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

The units for volume and temperature match those used by $R$, so rearrange the ideal gas law equation to solve for $n$

$P V = n R T \implies n = \frac{P V}{R T}$

Plug in your values to get

n = ((1.89 * 10^3color(red)(cancel(color(black)("atm"))))/(1.01325 * 10^2color(red)(cancel(color(black)("atm")))) * 685color(red)(cancel(color(black)("L"))))/(0.0821(color(red)(cancel(color(black)("atm"))) * color(red)(cancel(color(black)("L"))))/("mol" * color(red)(cancel(color(black)("K")))) * 621color(red)(cancel(color(black)("K")))) = "250.61 moles"

Rounded to three sig figs, the answer will be

$n = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{251 moles} \textcolor{w h i t e}{\frac{a}{a}} |}}}$