# A container has a volume of 4 L and holds 3 mol of gas. If the container is expanded such that its new volume is 9 L, how many moles of gas must be injected into the container to maintain a constant temperature and pressure?

Jun 14, 2017

$3.75$ $\text{mol}$

#### Explanation:

To solve this problem, we can use the volume-quantity relationship of gases, illustrated by Avogadro's law:

$\frac{{n}_{1}}{{V}_{1}} = \frac{{n}_{2}}{{V}_{2}}$

where

• ${n}_{1}$ and ${n}_{2}$ are the initial and final quantities, in $\text{mol}$, of the gas, and

• ${V}_{1}$ and ${V}_{2}$ are the initial and final volumes of the gas, usually in $\text{L}$

Since we're trying to find how many moles of gas are needed to inject, we'll rearrange the equation to solve for ${n}_{2}$:

${n}_{2} = \frac{{n}_{1} {V}_{2}}{{V}_{1}}$

Our known quantities are

• ${n}_{1} = 3$ $\text{mol}$

• ${V}_{1} = 4$ $\text{L}$

• ${V}_{2} = 9$ $\text{L}$

Plugging these values into the equation, we have

${n}_{2} = \frac{{n}_{1} {V}_{2}}{{V}_{1}} = \left(\left(3 \text{mol")(9cancel("L")))/(4cancel("L}\right)\right) = 6.75$ $\text{mol}$

This is the total final volume; we were asked to find how many moles needed to be injected. To find this, we simply subtract the initial value from the final value:

${n}_{\text{need}} = 6.75$ $\text{mol} - 3$ "mol" = color(red)(3.75 color(red)("mol"

Thus, to maintain a constant temperature and pressure, you must inject color(red)(3.75 moles of gas into the container.

If you follow the rules for significant figures, the answer is technically $4$ moles, but questions like this with one significant figure are universally disliked... one significant figure is too uncertain for most people :)