Learning Goal:

To practice Problem$-$Solving Strategy $18\mathrm{.}1$ Ideal Gases.

Lethal concentrations of gases are often expressed in terms of volume fraction, that is$,$ the fraction of available space occupied by the hazardous gas. A hazardous$-$materials worker wants to calculate the volume fraction of chlorine gas in a storage room after all of the contents of a $0\mathrm{.}650-$LL cylinder of chlorine gas at $115$ atmatm $$is accidentally released in the room. The environment in the storage room is kept at a constant pressure of $1\mathrm{.}00$ atmatm and a constant temperature of $20\mathrm{.}0$C$$C$.$ The room's dimensions are $5\mathrm{.}50$m$5\mathrm{.}00$m$2\mathrm{.}50$m$5\mathrm{.}50$m$5\mathrm{.}00$m$2\mathrm{.}50$m$.$ What is the volume fraction of chlorine gas present in the storage room?

Problem Solving Strategy $18\mathrm{.}1$: Ideal Gases

IDENTIFY the relevant concepts:

Unless the problem explicitly states otherwise, you can use the ideal$-$gas equation for any situation in which you need to find the state $($pressure$,$ volume, temperature, and$/$or number of moles$)$ of a gas.

SET UP the problem using the following steps:

Identify the variables.

In some problems you will be concerned with only one state of the system, in which case the following equation is the relationship to use.

pV$=$nRTpV$=$nRT

Some of the quantities in this equation will be known; others will be unknown. Make a list of what you know and what you have to find.

In other problems you will compare two different states of the same amount of gas. Decide which is state $1$ and which is state $2,$ and make a list of the quantities for each: p$1$p$1,$ p$2$p$2,$ V$1$V$1,$ V$2$V$2,$ T$1$T$1,$ T$2$T$2.$ If all but one of these quantities are known, you can use the following equation.

p$1$V$1$T$1=$p$2$V$2$T$2=$constantp$1$V$1$T$1=$p$2$V$2$T$2=$constant $($ideal gas, constant mass$)$

Otherwise, use this equation.

pV$=$nRTpV$=$nRT

Some problems involve the density $($mass per volume$)$ rather than the number of moles nn and the volume VV$.$ In this case it$$s most convenient to use the following equation.

$=$pMRT$=$pMRT

EXECUTE the solution as follows:

Use a consistent set of units. Sometimes the problem statement will make one system of units clearly more convenient than others. Decide on your system and stick to it$.$

Don't forget that TT must always be an absolute temperature. If you are given temperatures in degrees Celsius be sure to convert to Kelvin temperatures by adding $273\mathrm{.}15.$ Likewise, pp is always the absolute pressure, never the gauge pressure.

You may sometimes have to convert between mass and number of moles nn$.$ The relationship is

mtotal$=$Mnmtotal$=$Mn

where MM is the molar mass. If you use the following equation, you must use the same mass units for mtotalmtotal and MM$.$

pV$=$mtotalMRTpV$=$mtotalMRT

Once you have taken care of steps $13,$ solve for the target variables.

EVALUATE your answer:

Look carefully at your results and see whether they make physical sense.

IDENTIFY the relevant concepts

In this problem you are asked to find the volume fraction of chlorine gas in a room. To find the volume fraction you will need to find the state of the chlorine gas under the conditions of the room. The ideal$-$gas equation can be used to obtain this information.

SET UP the problem using the following steps

Part A

Which of the following statements about what happens to the chlorine gas when it leaves the cylinder is true?

The pressure of chlorine decreases.The pressure of chlorine increases.The pressure of chlorine stays the same.